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  • Articles  (158)
  • Global optimization  (158)
  • Mathematics  (158)
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  • Articles  (158)
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  • 1
    Electronic Resource
    Electronic Resource
    Concurrent stochastic methods for global optimization (1990)
    Springer
    Mathematical programming 46 (1990), S. 1-29 
    Details
    ISSN: 1436-4646
    Keywords: Global optimization ; concurrent ; parallel ; stochastic ; network of computers
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract The global optimization problem, finding the lowest minimizer of a nonlinear function of several variables that has multiple local minimizers, appears well suited to concurrent computation. This paper presents a new parallel algorithm for the global optimization problem. The algorithm is a stochastic method related to the multi-level single-linkage methods of Rinnooy Kan and Timmer for sequential computers. Concurrency is achieved by partitioning the work of each of the three main parts of the algorithm, sampling, local minimization start point selection, and multiple local minimizations, among the processors. This parallelism is of a coarse grain type and is especially well suited to a local memory multiprocessing environment. The paper presents test results of a distributed implementation of this algorithm on a local area network of computer workstations. It also summarizes the theoretical properties of the algorithm.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01585724
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  • 2
    Electronic Resource
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    Convergence rates of a global optimization algorithm (1992)
    Springer
    Mathematical programming 54 (1992), S. 223-232 
    Details
    ISSN: 1436-4646
    Keywords: Global optimization ; Lipschitz functions
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract This paper presents a best and worst case analysis of convergence rates for a deterministic global optimization algorithm. Superlinear convergence is proved for Lipschitz functions which are convex in the direction of the global maximum (concave in the direction of the global minimum). Computer results are given, which confirm the theoretical convergence rates.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01586051
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  • 3
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    Global optimization and simulated annealing (1991)
    Springer
    Mathematical programming 50 (1991), S. 367-393 
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    ISSN: 1436-4646
    Keywords: Global optimization ; continuous variables ; simulated annealing
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract In this paper we are concerned with global optimization, which can be defined as the problem of finding points on a bounded subset of ℝ n in which some real valued functionf assumes its optimal (maximal or minimal) value. We present a stochastic approach which is based on the simulated annealing algorithm. The approach closely follows the formulation of the simulated annealing algorithm as originally given for discrete optimization problems. The mathematical formulation is extended to continuous optimization problems, and we prove asymptotic convergence to the set of global optima. Furthermore, we discuss an implementation of the algorithm and compare its performance with other well-known algorithms. The performance evaluation is carried out for a standard set of test functions from the literature.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01594945
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  • 4
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    Global minimization by reducing the duality gap (1994)
    Springer
    Mathematical programming 63 (1994), S. 193-212 
    Details
    ISSN: 1436-4646
    Keywords: Global optimization ; nonconvex programming ; duality gap ; branch and bound method ; decomposition ; nonsmooth optimization ; pooling problem
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract We derive a general principle demonstrating that by partitioning the feasible set, the duality gap, existing between a nonconvex program and its lagrangian dual, can be reduced, and in important special cases, even eliminated. The principle can be implemented in a Branch and Bound algorithm which computes an approximate global solution and a corresponding lower bound on the global optimal value. The algorithm involves decomposition and a nonsmooth local search. Numerical results for applying the algorithm to the pooling problem in oil refineries are given.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01582066
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  • 5
    Electronic Resource
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    An analytical approach to global optimization (1991)
    Springer
    Mathematical programming 52 (1991), S. 227-254 
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    ISSN: 1436-4646
    Keywords: Global optimization ; analytical methods ; computer algebra
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract Global optimization problems with a few variables and constraints arise in numerous applications but are seldom solved exactly. Most often only a local optimum is found, or if a global optimum is detected no proof is provided that it is one. We study here the extent to which such global optimization problems can be solved exactly using analytical methods. To this effect, we propose a series of tests, similar to those of combinatorial optimization, organized in a branch-and-bound framework. The first complete solution of two difficult test problems illustrates the efficiency of the resulting algorithm. Computational experience with the programbagop, which uses the computer algebra systemmacsyma, is reported on. Many test problems from the compendiums of Hock and Schittkowski and others sources have been solved.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01582889
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  • 6
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    Optimal and sub-optimal stopping rules for the Multistart algorithm in global optimization (1992)
    Springer
    Mathematical programming 57 (1992), S. 445-458 
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    ISSN: 1436-4646
    Keywords: Global optimization
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract In this paper the problem of stopping the Multistart algorithm for global optimization is considered. The algorithm consists of repeatedly performing local searches from randomly generated starting points. The crucial point in this algorithmic scheme is the development of a stopping criterion; the approach analyzed in this paper consists in stopping the sequential sampling as soon as a measure of the trade-off between the cost of further local searches is greater than the expected benefit, i.e. the possibility of discovering a better optimum. Stopping rules are thoroughly investigated both from a theoretical point of view and from a computational one via extensive simulation. This latter clearly shows that the simple1-step look ahead rule may achieve surprisingly good results in terms of computational cost vs. final accuracy.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01581094
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  • 7
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    An adaptive stochastic global optimization algorithm for one-dimensional functions (1995)
    Springer
    Annals of operations research 58 (1995), S. 261-278 
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    ISSN: 1572-9338
    Keywords: Global optimization ; Wiener process ; sequential stopping rules ; one-step look-ahead
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Economics
    Notes: Abstract In this paper a new algorithm is proposed, based upon the idea of modeling the objective function of a global optimization problem as a sample path from a Wiener process. Unlike previous work in this field, in the proposed model the parameter of the Wiener process is considered as a random variable whose conditional (posterior) distribution function is updated on-line. Stopping criteria for Bayesian algorithms are discussed and detailed proofs on finite-time stopping are provided.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02096402
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  • 8
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    Parallel computing in nonconvex programming (1993)
    Springer
    Annals of operations research 43 (1993), S. 87-107 
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    ISSN: 1572-9338
    Keywords: Global optimization ; parallel computing ; minimum-concave cost network flow problems
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Economics
    Notes: Abstract In this paper, we are concerned with the development of parallel algorithms for solving some classes of nonconvex optimization problems. We present an introductory survey of parallel algorithms that have been used to solve structured problems (partially separable, and large-scale block structured problems), and algorithms based on parallel local searches for solving general nonconvex problems. Indefinite quadratic programming posynomial optimization, and the general global concave minimization problem can be solved using these approaches. In addition, for the minimum concave cost network flow problem, we are going to present new parallel search algorithms for large-scale problems. Computational results of an efficient implementation on a multi-transputer system will be presented.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02024487
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  • 9
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    A Bayesian algorithm for global optimization (1984)
    Springer
    Annals of operations research 1 (1984), S. 111-128 
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    ISSN: 1572-9338
    Keywords: Global optimization ; Bayesian nonparametric inference ; random distributions ; cluster analysis
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Economics
    Notes: Abstract A crucial step in global optimization algorithms based on random sampling in the search domain is decision about the achievement of a prescribed accuracy. In order to overcome the difficulties related to such a decision, the Bayesian Nonparametric Approach has been introduced. The aim of this paper is to show the effectiveness of the approach when an ad hoc clustering technique is used for obtaining promising starting points for a local search algorithm. Several test problems are considered.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01876142
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  • 10
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    A survey on the global optimization problem: General theory and computational approaches (1984)
    Springer
    Annals of operations research 1 (1984), S. 87-110 
    Details
    ISSN: 1572-9338
    Keywords: Global optimization ; statistical optimization ; Bayesian statistics ; random search
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Economics
    Notes: Abstract Several different approaches have been suggested for the numerical solution of the global optimization problem: space covering methods, trajectory methods, random sampling, random search and methods based on a stochastic model of the objective function are considered in this paper and their relative computational effectiveness is discussed. A closer analysis is performed of random sampling methods along with cluster analysis of sampled data and of Bayesian nonparametric stopping rules.
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    URL: http://dx.doi.org/10.1007/BF01876141
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  • 11
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    Convex-Concave Extensions (2000)
    Springer
    BIT 40 (2000), S. 291-313 
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    ISSN: 1572-9125
    Keywords: Global optimization ; interval arithmetic ; range of functions
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In this paper we introduce a new notion which we call convex-concave extensions. Convex-concave extensions provide for given nonlinear functions convex lower bound functions and concave upper bound functions, and can be viewed as a generalization of interval extensions. Convex-concave extensions can approximate the shape of a given function in a better way than interval extensions which deliver only constant lower and upper bounds for the range. Therefore, convex-concave extensions can be applied in a more flexible manner. For example, they can be used to construct convex relaxations. Moreover, it is demonstrated that in many cases the overestimation which is due to interval extensions can be drastically reduced. Applications and some numerical examples, including constrained global optimization problems of large scale, are presented.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1022343007844
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  • 12
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    An algorithm for a class of nonlinear fractional problems using ranking of the vertices (1986)
    Springer
    BIT 26 (1986), S. 392-395 
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    ISSN: 1572-9125
    Keywords: 65K05 ; 90C30 ; Fractional programming ; Vertex ranking ; Global optimization
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In this note we consider an algorithm for quasiconcave nonlinear fractional programming problems, based on ranking the vertices of a linear fractional programming problem and techniques from global optimization.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01933719
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  • 13
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    Quadratic problems defined on a convex hull of points (1988)
    Springer
    BIT 28 (1988), S. 323-328 
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    ISSN: 1572-9125
    Keywords: 90C30 ; 65K05 ; Global optimization ; Quadratic programming ; Convex hull
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In this paper we consider an algorithm for a class of quadratic problems defined on a polytope which is described as the convex hull of a set of points. The algorithm is based on simplex partitions using convex underestimating functions.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01934095
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  • 14
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    An algorithm for a singly constrained class of quadratic programs subject to upper and lower bounds (1990)
    Springer
    Mathematical programming 46 (1990), S. 321-328 
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    ISSN: 1436-4646
    Keywords: Global optimization ; separable programming ; quadratic programming
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract This paper gives an O(n) algorithm for a singly constrained convex quadratic program using binary search to solve the Kuhn-Tucker system. Computational results indicate that a randomized version of this algorithm runs in expected linear time and is suitable for practical applications. For the nonconvex case anε-approximate algorithm is proposed which is based on convex and piecewise linear approximations of the objective function.
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    URL: http://dx.doi.org/10.1007/BF01585748
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  • 15
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    Convergence and restart in branch-and-bound algorithms for global optimization. Application to concave minimization and D.C. Optimization problems (1988)
    Springer
    Mathematical programming 41 (1988), S. 161-183 
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    ISSN: 1436-4646
    Keywords: Global optimization ; nonconvex programming ; branch-and-bound ; restart procedure ; decomposition ; outer approximation ; concave minimization ; d.c. optimization
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract A general branch-and-bound conceptual scheme for global optimization is presented that includes along with previous branch-and-bound approaches also grid-search techniques. The corresponding convergence theory, as well as the question of restart capability for branch-and-bound algorithms used in decomposition or outer approximation schemes are discussed. As an illustration of this conceptual scheme, a finite branch-and-bound algorithm for concave minimization is described and a convergent branch-and-bound algorithm, based on the previous one, is developed for the minimization of a difference of two convex functions.
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    URL: http://dx.doi.org/10.1007/BF01580762
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  • 16
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    An algorithm for solving convex programs with an additional convex—concave constraint (1993)
    Springer
    Mathematical programming 61 (1993), S. 75-87 
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    ISSN: 1436-4646
    Keywords: Global optimization ; branch-and-bound ; convex—concave constraint ; d.c. programming ; the product of two affine functions ; decomposition method ; algorithm
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract An implementable decomposition method based on branch-and-bound techniques is proposed for finding a global optimal solution of certain convex programs with an additional convex—concave constrainth(x, y) ⩽ 0. A nonadaptive simplicial and an adaptive bisection are used for the branching operation, which is performed iny-space only. The bounding operation is based on a relaxation of the convex—concave constrainth(x, y) ⩽ 0. The algorithm can be applied efficiently for linear programs with an additional affine multiplicative constraint.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01582140
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  • 17
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    Pure adaptive search for finite global optimization (1995)
    Springer
    Mathematical programming 69 (1995), S. 443-448 
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    ISSN: 1436-4646
    Keywords: Global optimization ; Discrete optimization ; Algorithm complexity ; Random search ; Markov chains
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract Pure Adaptive Search is a stochastic algorithm which has been analyzed for continuous global optimization. When a uniform distribution is used in PAS, it has been shown to have complexity which is linear in dimension. We define strong and weak variations of PAS in the setting of finite global optimization and prove analogous results. In particular, for then-dimensional lattice {1,⋯,k} n , the expected number of iterations to find the global optimum is linear inn. Many discrete combinatorial optimization problems, although having intractably large domains, have quite small ranges. The strong version of PAS for all problems, and the weak version of PAS for a limited class of problems, has complexity the order of the size of the range.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01585570
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  • 18
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    Global optimization of univariate Lipschitz functions: I. Survey and properties (1992)
    Springer
    Mathematical programming 55 (1992), S. 251-272 
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    ISSN: 1436-4646
    Keywords: Global optimization ; algorithm ; univariate function ; Lipschitz function ; convergence
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract We consider the following global optimization problems for a univariate Lipschitz functionf defined on an interval [a, b]: Problem P: find a globally optimal value off and a corresponding point; Problem P′: find a globallyε-optimal value off and a corresponding point; Problem Q: localize all globally optimal points; Problem Q′: find a set of disjoint subintervals of small length whose union contains all globally optimal points; Problem Q″: find a set of disjoint subintervals containing only points with a globallyε-optimal value and whose union contains all globally optimal points. We present necessary conditions onf for finite convergence in Problem P and Problem Q, recall the concepts necessary for a worst-case and an empirical study of algorithms (i.e., those ofpassive and ofbest possible algorithms), summarize and discuss algorithms of Evtushenko, Piyavskii-Shubert, Timonov, Schoen, Galperin, Shen and Zhu, presenting them in a simplified and uniform way, in a high-level computer language. We address in particular the problems of using an approximation for the Lipschitz constant, reducing as much as possible the expected length of the region of indeterminacy which contains all globally optimal points and avoiding remaining subintervals without points with a globallyε-optimal value. New algorithms for Problems P′ and Q″ and an extensive computational comparison of algorithms are presented in a companion paper.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01581202
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  • 19
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    Global optimization of univariate Lipschitz functions: II. New algorithms and computational comparison (1992)
    Springer
    Mathematical programming 55 (1992), S. 273-292 
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    ISSN: 1436-4646
    Keywords: Global optimization ; algorithm ; univariate function ; Lipschitz function ; computation
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract We consider the following global optimization problems for a Lipschitz functionf implicitly defined on an interval [a, b]. Problem P′: find a globallyε-optimal value off and a corresponding point; Problem Q″: find a set of disjoint subintervals of [a, b] containing only points with a globallyε-optimal value and the union of which contains all globally optimal points. A two-phase algorithm is proposed for Problem P′. In phase I, this algorithm obtains rapidly a solution which is often globallyε-optimal. Moreover, a sufficient condition onf for this to be the case is given. In phase II, the algorithm proves theε-optimality of the solution obtained in phase I or finds a sequence of points of increasing value containing one with a globallyε-optimal value. The new algorithm is empirically compared (on twenty problems from the literature) with a best possible algorithm (for which the optimal value is assumed to be known), with a passive algorithm and with the algorithms of Evtushenko, Galperin, Shen and Zhu, Piyavskii, Timonov and Schoen. For smallε, the new algorithm requires only a few percent more function evaluations than the best possible one. An extended version of Piyavskii's algorithm is proposed for problem Q″. A sufficient condition onf is given for the globally optimal points to be in one-to-one correspondance with the obtained intervals. This result is achieved for all twenty test problems.
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    URL: http://dx.doi.org/10.1007/BF01581203
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  • 20
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    Global one-dimensional optimization using smooth auxiliary functions (1998)
    Springer
    Mathematical programming 81 (1998), S. 127-146 
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    ISSN: 1436-4646
    Keywords: Global optimization ; Lipschitzean first derivatives ; Numerical algorithms ; Convergence
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract In this paper new global optimization algorithms are proposed for solving problems where the objective function is univariate and has Lipschitzean first derivatives. To solve this problem, smooth auxiliary functions, which are adaptively improved during the course of the search, are constructed. Three new algorithms are introduced: the first used the exact a priori known Lipschitz constant for derivatives; the second, when this constant is unknown, estimates it during the course of the search and finally, the last method uses neither the exact global Lipschitz constant nor its estimate but instead adaptively estimates the local Lipschitz constants in different sectors of the search region during the course of optimization. Convergence conditions of the methods are investigated from a general viewpoint and some numerical results are also given. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.
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    URL: http://dx.doi.org/10.1007/BF01584848
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  • 21
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    Analytical solutions to the optimization of a quadratic cost function subject to linear and quadratic equality constraints (1996)
    Springer
    Applied mathematics & optimization 34 (1996), S. 161-182 
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    ISSN: 1432-0606
    Keywords: Global optimization ; Quadratic cost ; Quadratic equality constraints ; Multivariate polynomials ; Resultants ; 49N99 ; 90C26 ; 90C30 ; 65K05
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In the area of broad-band antenna array signal processing, the global minimum of a quadratic equality constrained quadratic cost minimization problem is often required. The problem posed is usually characterized by a large optimization space (around 50–90 tuples), a large number of linear equality constraints, and a few quadratic equality constraints each having very low rank quadratic constraint matrices. Two main difficulties arise in this class of problem. Firstly, the feasibility region is nonconvex and multiple local minima abound. This makes conventional numerical search techniques unattractive as they are unable to locate the global optimum consistently (unless a finite search area is specified). Secondly, the large optimization space makes the use of decision-method algorithms for the theory of the reals unattractive. This is because these algorithms involve the solution of the roots of univariate polynomials of order to the square of the optimization space. In this paper we present a new algorithm which exploits the structure of the constraints to reduce the optimization space to a more manageable size. The new algorithm relies on linear-algebra concepts, basic optimization theory, and a multivariate polynomial root-solving tool often used by decision-method algorithms.
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    URL: http://dx.doi.org/10.1007/BF01182622
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  • 22
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    On global optimization in two-dimensional scaling (1993)
    Springer
    Acta applicandae mathematicae 33 (1993), S. 109-118 
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    ISSN: 1572-9036
    Keywords: 60F05 ; 60F17 ; 62L20 ; Global optimization ; multidimensional scaling ; evolutionary strategies ; genetic algorithms
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We consider multidimensional scaling for embedding dimension two, which allows the detection of structures in dissimilarity data by simply drawing two-dimensional figures. The corresponding objective function, called STRESS, is generally nondifferentiable and has many local minima. In this paper we investigate several features of this function, and discuss the application of different global optimization procedures. A method based on combining a local search algorithm with an evolutionary strategy of generating new initial points is proposed, and its efficiency is investigated by numerical examples.
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    URL: http://dx.doi.org/10.1007/BF00995497
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  • 23
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    Metric unidimensional scaling and global optimization (1996)
    Springer
    Journal of classification 13 (1996), S. 3-18 
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    ISSN: 1432-1343
    Keywords: Unidimensional scaling ; Seriation ; Local minima ; Global optimization ; Smoothing technique ; Multidimensional scaling
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract For the problem of metric unidimensional scaling, the number of local minima is estimated. For locating the globally optimal solution we develop an approach, called the “smoothing technique.” Although not guaranteed inevitably to locate the global optimum, the smoothing technique did so in all computational experiments where the global optimum was known.
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    URL: http://dx.doi.org/10.1007/BF01202579
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  • 24
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    Semiparametric statistical inference in global random search (1993)
    Springer
    Acta applicandae mathematicae 33 (1993), S. 69-88 
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    ISSN: 1572-9036
    Keywords: 60F05 ; 60F17 ; 62L20 ; Global optimization ; global random search ; Statistical inference ; branch and bound ; endpoint of a distribution ; stratified sampling
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The multivariate multiextremal optimization problem is considered. Various statistical procedures based on the use of the asymptotic theory of extreme order statistics are thoroughly described. These procedures are used to infer about the maximal value of a function by its values at random points. A class of global random search methods underlying the procedures is considered. These methods generalize the well-known branch and bound methods. The article is mainly of a survey nature. It also contains new results.
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    URL: http://dx.doi.org/10.1007/BF00995495
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  • 25
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    Global optimization using interval analysis: The one-dimensional case (1979)
    Springer
    Journal of optimization theory and applications 29 (1979), S. 331-344 
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    ISSN: 1573-2878
    Keywords: Global optimization ; interval analysis ; global minimization ; one-dimensional optimization
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We show how interval analysis can be used to compute the minimum value of a twice continuously differentiable function of one variable over a closed interval. When both the first and second derivatives of the function have a finite number of isolated zeros, our method never fails to find the global minimum.
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    URL: http://dx.doi.org/10.1007/BF00933139
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  • 26
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    Outer approximation algorithm for nondifferentiable optimization problems (1984)
    Springer
    Journal of optimization theory and applications 42 (1984), S. 19-30 
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    ISSN: 1573-2878
    Keywords: Global optimization ; nondifferentiable optimization ; Lipschitz continuous functions ; outer-approximation algorithms
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract It is known that the problem of minimizing a convex functionf(x) over a compact subsetX of ℝ n can be expressed as minimizing max{g(x, y)|y ∈X}, whereg is a support function forf[f(x) ≥g(x, y), for ally ∈X andf(x)=g(x, x)]. Standard outer-approximation theory can then be employed to obtain outer-approximation algorithms with procedures for dropping previous cuts. It is shown here how this methodology can be extended to nonconvex nondifferentiable functions.
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    URL: http://dx.doi.org/10.1007/BF00934131
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    Outer approximation algorithm for nondifferentiable optimization problems (1984)
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    Journal of optimization theory and applications 44 (1984), S. 365-365 
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    ISSN: 1573-2878
    Keywords: Global optimization ; nondifferentiable optimization ; Lipschitz continuous functions ; outer-approximation algorithms
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Omissions from the list of references of Ref. 1 are corrected.
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    Properties of the random search in global optimization (1975)
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    Journal of optimization theory and applications 16 (1975), S. 383-398 
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    Keywords: Global optimization ; uniformly random search ; uniform grid search ; gaps in random points
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    Notes: Abstract From theorems which we prove about the behavior of gaps in a set ofN uniformly random points on the interval [0, 1], we determine properties of the random search procedure in one-dimensional global optimization. In particular, we show that the uniform grid search is better than the random search when the optimum is chosen using the deterministic strategy, that a significant proportion of large gaps are contained in the uniformly random search, and that the error in the determination of the point at which the optimum occurs, assuming that it is unique, will on the average be twice as large using the uniformly random search compared with the uniform grid. In addition, some of the properties of the largest gap are verified numerically, and some extensions to higher dimensions are discussed. The latter show that not all of the conclusions derived concerning the inadequacies of the one-dimensional random search extend to higher dimensions, and thaton average the random search is better than the uniform grid for dimensions greater than 6.
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    Deterministic global optimization with partition sets whose feasibility is not known: Application to concave minimization, reverse convex constraints, DC-programming, and Lipschitzian optimization (1988)
    Springer
    Journal of optimization theory and applications 58 (1988), S. 11-37 
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    ISSN: 1573-2878
    Keywords: Global optimization ; nonconvex programming ; mathematical programming ; concave minimization ; DC-programming ; Lipschitzian optimization ; branch-and-bound methods
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    Notes: Abstract A crucial problem for many global optimization methods is how to handle partition sets whose feasibility is not known. This problem is solved for broad classes of feasible sets including convex sets, sets defined by finitely many convex and reverse convex constraints, and sets defined by Lipschitzian inequalities. Moreover, a fairly general theory of bounding is presented and applied to concave objective functions, to functions representable as differences of two convex functions, and to Lipschitzian functions. The resulting algorithms allow one to solve any global optimization problem whose objective function is of one of these forms and whose feasible set belongs to one of the above classes. In this way, several new fields of optimization are opened to the application of global methods.
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    Terminal repeller unconstrained subenergy tunneling (trust) for fast global optimization (1993)
    Springer
    Journal of optimization theory and applications 77 (1993), S. 97-126 
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    ISSN: 1573-2878
    Keywords: Global optimization ; dynamical systems ; terminal repellers ; subenergy tunneling function ; artificial neural networks
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    Notes: Abstract A new method for unconstrained global function optimization, acronymedtrust, is introduced. This method formulates optimization as the solution of a deterministic dynamical system incorporating terminal repellers and a novel subenergy tunneling function. Benchmark tests comparing this method to other global optimization procedures are presented, and thetrust algorithm is shown to be substantially faster. Thetrust formulation leads to a simple stopping criterion. In addition, the structure of the equations enables an implementation of the algorithm in analog VLSI hardware, in the vein of artificial neural networks, for further substantial speed enhancement.
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    Cord-slope form of Taylor's expansion in univariate global optimization (1994)
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    Journal of optimization theory and applications 80 (1994), S. 441-464 
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    ISSN: 1573-2878
    Keywords: Global optimization ; univariate functions ; interval arithmetic ; Taylor's formula
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    Notes: Abstract Interval arithmetic and Taylor's formula can be used to bound the slope of the cord of a univariate function at a given point. This leads in turn to bounding the values of the function itself. Computing such bounds for the function, its first and second derviatives, allows the determination of intervals in which this function cannot have a global minimum. Exploiting this information together with a simple branching rule yields an efficient algorithm for global minimization of univariate functions. Computational experience is reported.
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    Deletion-by-infeasibility rule for DC-constrained global optimization (1995)
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    Journal of optimization theory and applications 84 (1995), S. 443-455 
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    Keywords: Global optimization ; nonconvex programming ; mathematical programming ; DC-programming ; branch-and-bound methods
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    Notes: Abstract In Ref. 1, a general class of branch-and-bound methods was proposed by Horst for solving global optimization problems. One of the main contributions of Ref. 1 was the opportunity of handling partition elements whose feasibility is not known. Deletion-by-infeasibility rules were presented for problems where the feasible set is convex, is defined by finitely many convex and reverse convex constraints, or is defined by Lipschitzian inequalities. In this note, we propose a new deletion-by-infeasibility rule for problems whose feasible set is defined by functions representable as differences of convex functions.
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    Decomposition approach for the global minimization of biconcave functions over polytopes (1996)
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    Journal of optimization theory and applications 88 (1996), S. 561-583 
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    Keywords: Global optimization ; biconcave programming ; concave minimization ; bilinear and quadratic programming ; branch-and-bound algorithms ; outer approximations.
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    Notes: Abstract A decomposition approach is proposed for minimizing biconcave functions over polytopes. Important special cases include concave minimization, bilinear and indefinite quadratic programming for which new algorithms result. The approach introduces a new polyhedral partition and combines branch-and-bound techniques, outer approximation, and projection of polytopes in a suitable way.
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    Generalized primal-relaxed dual approach for global optimization (1996)
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    Journal of optimization theory and applications 90 (1996), S. 417-434 
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    Keywords: Global optimization ; primal-relaxed dual approach ; penalty methods ; nonsmooth optimization
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    Notes: Abstract A generalized primal-relaxed dual algorithm for global optimization is proposed and its convergence is proved. The (GOP) algorithm of Floudas and Visweswaran (Refs. 1–2) is shown to be a special case of this general algorithm. Within the proposed framework, the algorithm of Floudas and Visweswaran (Refs. 1–2) is further extended to the nonsmooth case. A penalty implementation of the extended (GOP) algorithm is studied to improve its efficiency.
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    Stochastic Methods for Practical Global Optimization (1998)
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    Journal of global optimization 13 (1998), S. 433-444 
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    ISSN: 1573-2916
    Keywords: Adaptive search ; Global optimization ; Multi-disciplinary optimization ; Simulated annealing
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    Notes: Abstract Engineering design problems often involve global optimization of functions that are supplied as ‘black box’ functions. These functions may be nonconvex, nondifferentiable and even discontinuous. In addition, the decision variables may be a combination of discrete and continuous variables. The functions are usually computationally expensive, and may involve finite element methods. An engineering example of this type of problem is to minimize the weight of a structure, while limiting strain to be below a certain threshold. This type of global optimization problem is very difficult to solve, yet design engineers must find some solution to their problem – even if it is a suboptimal one. Sometimes the most difficult part of the problem is finding any feasible solution. Stochastic methods, including sequential random search and simulated annealing, are finding many applications to this type of practical global optimization problem. Improving Hit-and-Run (IHR) is a sequential random search method that has been successfully used in several engineering design applications, such as the optimal design of composite structures. A motivation to IHR is discussed as well as several enhancements. The enhancements include allowing both continuous and discrete variables in the problem formulation. This has many practical advantages, because design variables often involve a mixture of continuous and discrete values. IHR and several variations have been applied to the composites design problem. Some of this practical experience is discussed.
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    Two Strategies of Adaptive Cluster Covering with Descent and Their Comparison to Other Algorithms (1999)
    Springer
    Journal of global optimization 14 (1999), S. 55-78 
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    ISSN: 1573-2916
    Keywords: Adaptation ; Clustering ; Covering ; Descent ; Global optimization ; Parameter identification
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    Topics: Mathematics
    Notes: Abstract Two strategies of randomized search, namely adaptive cluster covering (ACCO), and adaptive cluster covering with descent (ACD), are introduced and positioned in the group of the global optimization techniques. Several algorithms based on these new strategies are compared with other techniques of global randomized search in terms of effectiveness, efficiency and reliability. The other techniques include two versions of multistart, two versions of the controlled random search (CRS2 and CRS4) and the canonical genetic algorithm. Thirteen minimization problems including two parameter identification problems (for a flexible membrane mirror model and a hydrologic model) are solved. The algorithm ACCO, and a version of CRS4 algorithm (Ali and Storey 1994) show the highest efficiency, effectiveness and reliability. The second new algorithm, ACD, is in some runs very efficient and effective, but its reliability needs further improvement.
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    Global Optimization on Funneling Landscapes (2000)
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    Journal of global optimization 18 (2000), S. 367-383 
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    Keywords: Global optimization ; Lennard–Jones clusters ; Basin-hopping ; Energy landscape ; Folding funnel ; Molecular conformation
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    Notes: Abstract Molecular conformation problems arising in computational chemistry require the global minimization of a non-convex potential energy function representing the interactions of, for example, the component atoms in a molecular system. Typically the number of local minima on the potential energy surface grows exponentially with system size, and often becomes enormous even for relatively modestly sized systems. Thus the simple multistart strategy of randomly sampling local minima becomes impractical. However, for many molecular conformation potential energy surfaces the local minima can be organized by a simple adjacency relation into a single or at most a small number of funnels. A distinguished local minimum lies at the bottom of each funnel and a monotonically descending sequence of adjacent local minima connects every local minimum in the funnel with the funnel bottom. Thus the global minimum can be found among the comparatively small number of funnel bottoms, and a multistart strategy based on sampling funnel bottoms becomes viable. In this paper we present such an algorithm of the basin-hopping type and apply it to the Lennard–Jones cluster problem, an intensely studied molecular conformation problem which has become a benchmark for global optimization algorithms. Results of numerical experiments are presented which confirm both the multifunneling character of the Lennard–Jones potential surface as well as the efficiency of the algorithm. The algorithm has found all of the current putative global minima in the literature up to 110 atoms, as well as discovered a new global minimum for the 98-atom cluster of a novel geometrical class.
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    A Note on a Penalty Function Approach for Solving Bilevel Linear Programs (2000)
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    Journal of global optimization 16 (2000), S. 245-255 
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    ISSN: 1573-2916
    Keywords: Bilevel linear programming ; Global optimization ; Penalty function
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    Topics: Mathematics
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    Unconstrained and constrained global optimization of polynomial functions in one variable (1992)
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    Journal of global optimization 2 (1992), S. 73-99 
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    ISSN: 1573-2916
    Keywords: Global optimization ; polynomial functions ; unconstrained and constrained optimization ; the GOP algorithm
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    Topics: Mathematics
    Notes: Abstract In Floudas and Visweswaran (1990), a new global optimization algorithm (GOP) was proposed for solving constrained nonconvex problems involving quadratic and polynomial functions in the objective function and/or constraints. In this paper, the application of this algorithm to the special case of polynomial functions of one variable is discussed. The special nature of polynomial functions enables considerable simplification of the GOP algorithm. The primal problem is shown to reduce to a simple function evaluation, while the relaxed dual problem is equivalent to the simultaneous solution of two linear equations in two variables. In addition, the one-to-one correspondence between the x and y variables in the problem enables the iterative improvement of the bounds used in the relaxed dual problem. The simplified approach is illustrated through a simple example that shows the significant improvement in the underestimating function obtained from the application of the modified algorithm. The application of the algorithm to several unconstrained and constrained polynomial function problems is demonstrated.
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    Nonlinear coordinate transformations for unconstrained optimization II. Theoretical background (1993)
    Springer
    Journal of global optimization 3 (1993), S. 359-375 
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    Keywords: Global optimization ; smooth unconstrained optimization ; geodesic convexity ; unimodal function ; nonlinear coordinate transformation ; 65K05 ; 90C30 ; 54H00
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In this two-part article, nonlinear coordinate transformations are discussed in order to simplify global unconstrained optimization problems and to test their unimodality on the basis of the analytical structure of the objective functions. If the transformed problems can be quadratic in some or all the variables, then the optimum can be calculated directly, without an iterative procedure, or the number of variables to be optimized can be reduced. Otherwise, the analysis of the structure can serve as a first phase for solving global unconstrained optimization problems. The first part treats real-life problems where the presented technique is applied and the transformation steps are constructed. The second part of the article deals with the differential geometrical background and the conditions of the existence of such transformations.
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    An adaptive simulated annealing algorithm for global optimization over continuous variables (1995)
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    Journal of global optimization 6 (1995), S. 1-37 
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    Keywords: Global optimization ; simulated annealing ; Monte Carlo optimization
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    Notes: Abstract A method is presented for attempting global minimization for a function of continuous variables subject to constraints. The method, calledAdaptive Simulated Annealing (ASA), is distinguished by the fact that the fixed temperature schedules and step generation routines that characterize other implementations are here replaced by heuristic-based methods that effectively eliminate the dependence of the algorithm's overall performance on user-specified control parameters. A parallelprocessing version of ASA that gives increased efficiency is presented and applied to two standard problems for illustration and comparison.
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    Parallel methods for verified global optimization practice and theory (1996)
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    Journal of global optimization 9 (1996), S. 1-22 
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    Keywords: Global optimization ; parallel computing ; interval arithmetic ; branch and bound ; dynamic load balancing
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    Notes: Abstract We present a new parallel method for verified global optimization, using a centralized mediator for the dynamic load balancing. The new approach combines the advantages of two previous models, the master slave model and the processor farm. Numerical results show the efficiency of this new method. For a large number of problems at least linear speedup is reached. The efficiency of this new method is also confirmed by a comparison with other parallel methods for verified global optimization. A theoretical study proves that using the best-first strategy to choose the next box for subdivision, no real superlinear speedup may be expected concerning the number of iterations. Moreover, the potential of parallelization of methods of verified global optimization is discussed in general.
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    Second-order sufficient optimality conditions for local and global nonlinear programming (1996)
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    Journal of global optimization 9 (1996), S. 141-151 
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    Keywords: Global optimization ; optimality condition ; second-order sufficient condition ; verification of convexity ; interval analysis ; primary 90C30 ; secondary 65G10
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This paper presents a new approach to the sufficient conditions of nonlinear programming. Main result is a sufficient condition for the global optimality of a Kuhn-Tucker point. This condition can be verified constructively, using a novel convexity test based on interval analysis, and is guaranteed to prove global optimality of strong local minimizers for sufficiently narrow bounds. Hence it is expected to be a useful tool within branch and bound algorithms for global optimization.
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    Quadratic programming with one negative eigenvalue is NP-hard (1991)
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    Journal of global optimization 1 (1991), S. 15-22 
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    Keywords: Global optimization ; quadratic programming ; NP-hard
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    Notes: Abstract We show that the problem of minimizing a concave quadratic function with one concave direction is NP-hard. This result can be interpreted as an attempt to understand exactly what makes nonconvex quadratic programming problems hard. Sahni in 1974 [8] showed that quadratic programming with a negative definite quadratic term (n negative eigenvalues) is NP-hard, whereas Kozlov, Tarasov and Hačijan [2] showed in 1979 that the ellipsoid algorithm solves the convex quadratic problem (no negative eigenvalues) in polynomial time. This report shows that even one negative eigenvalue makes the problem NP-hard.
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    On Timonov's algorithm for global optimization of univariate Lipschitz functions (1991)
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    Journal of global optimization 1 (1991), S. 37-46 
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    Keywords: Global optimization ; univariate function ; Lipschitz function
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    Notes: Abstract Timonov proposes an algorithm for global maximization of univariate Lipschitz functions in which successive evaluation points are chosen in order to ensure at each iteration a maximal expected reduction of the “region of indeterminacy”, which contains all globally optimal points. It is shown that such an algorithm does not necessarily converge to a global optimum.
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    An all-linear programming relaxation algorithm for optimizing over the efficient set (1991)
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    Journal of global optimization 1 (1991), S. 83-104 
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    ISSN: 1573-2916
    Keywords: Global optimization ; multiple criteria decision making ; relaxation algorithms ; efficient set ; multiple objective linear programming
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The problem (P) of optimizing a linear function over the efficient set of a multiple objective linear program has many important applications in multiple criteria decision making. Since the efficient set is in general a nonconvex set, problem (P) can be classified as a global optimization problem. Perhaps due to its inherent difficulty, it appears that no precisely-delineated implementable algorithm exists for solving problem (P) globally. In this paper a relaxation algorithm is presented for finding a globally optimal solution for problem (P). The algorithm finds an exact optimal solution to the problem after a finite number of iterations. A detailed discussion is included of how to implement the algorithm using only linear programming methods. Convergence of the algorithm is proven, and a sample problem is solved.
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    What can interval analysis do for global optimization? (1991)
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    Journal of global optimization 1 (1991), S. 111-130 
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    Keywords: 65K05 ; 65G10 ; 90C30 ; Global optimization ; interval analysis
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    Topics: Mathematics
    Notes: Abstract An overview of interval arithmetical tools and basic techniques is presented that can be used to construct deterministic global optimization algorithms. These tools are applicable to unconstrained and constrained optimization as well as to nonsmooth optimization and to problems over unbounded domains. Since almost all interval based global optimization algorithms use branch-and-bound methods with iterated bisection of the problem domain we also embed our overview in such a setting.
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    Global optimization of fractional programs (1991)
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    Journal of global optimization 1 (1991), S. 173-182 
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    Keywords: Global optimization ; fractional programming
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    Topics: Mathematics
    Notes: Abstract Dinkelbach's global optimization approach for finding the global maximum of the fractional programming problem is discussed. Based on this idea, a modified algorithm is presented which provides both upper and lower bounds at each iteration. The convergence of the lower and upper bounds to the global maximum function value is shown to be superlinear. In addition, the special case of fractional programming when the ratio involves only linear or quadratic terms is considered. In this case, the algorithm is guaranteed to find the global maximum to within any specified tolerance, regardless of the definiteness of the quadratic form.
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    Bounding the set of solutions of a perturbed global optimization problem (1991)
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    Journal of global optimization 1 (1991), S. 359-374 
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    Keywords: Global optimization ; sensitivity analysis ; error analysis ; interval analysis ; perturbations
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Consider a global optimization problem in which the objective function and/or the constraints are expressed in terms of parameters. Suppose we wish to know the set of global solutions as the parameters vary over given intervals. In this paper we discuss procedures using interval analysis for computing guaranteed bounds on the solution set. This provides a means for doing a sensitivity analysis or simply bounding the effect of errors in data.
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    Global and Local Quadratic Minimization (1997)
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    Journal of global optimization 10 (1997), S. 77-90 
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    ISSN: 1573-2916
    Keywords: Global optimization ; parametric quadratic programming ; non-convex quadratic program.
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    Notes: Abstract We present a method which when applied to certain non-convex QP will locatethe globalminimum, all isolated local minima and some of the non-isolated localminima. The method proceeds by formulating a (multi) parametric convex QP interms ofthe data of the given non-convex QP. Based on the solution of the parametricQP,an unconstrained minimization problem is formulated. This problem ispiece-wisequadratic. A key result is that the isolated local minimizers (including theglobalminimizer) of the original non-convex problem are in one-to-one correspondencewiththose of the derived unconstrained problem. The theory is illustrated with several numerical examples. A numericalprocedure isdeveloped for a special class of non-convex QP's. It is applied to a problemfrom theliterature and verifies a known global optimum and in addition, locates apreviously unknown local minimum.
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    Global Optimization of Nonconvex Polynomial Programming Problems Having Rational Exponents (1998)
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    Journal of global optimization 12 (1998), S. 267-283 
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    ISSN: 1573-2916
    Keywords: Polynomial programs ; Reformulation-Linearization Technique (RLT) ; Nonconvex programming ; Global optimization
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This paper considers the solution of nonconvex polynomial programming problems that arise in various engineering design, network distribution, and location-allocation contexts. These problems generally have nonconvex polynomial objective functions and constraints, involving terms of mixed-sign coefficients (as in signomial geometric programs) that have rational exponents on variables. For such problems, we develop an extension of the Reformulation-Linearization Technique (RLT) to generate linear programming relaxations that are embedded within a branch-and-bound algorithm. Suitable branching or partitioning strategies are designed for which convergence to a global optimal solution is established. The procedure is illustrated using a numerical example, and several possible extensions and algorithmic enhancements are discussed.
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    Global Optimization in Practice: An Application to Interactive Multiple Objective Linear Programming (1998)
    Springer
    Journal of global optimization 12 (1998), S. 353-372 
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    ISSN: 1573-2916
    Keywords: Multiple objective linear programming ; Optimization over the efficient set ; Interactive methods ; Global optimization ; Citrus rootstockselection
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A multiple objective linear programming problem (P) involves the simultaneous maximization of two or more conflicting linear objective functions over a nonempty polyhedron X. Many of the most popular methods for solving this type of problem, including many well-known interactive methods, involve searching the efficient set X E of the problem. Generally, however, X E is a complicated, nonconvex set. As a result, concepts and methods from global optimization may be useful in searching X E. In this paper, we will explain in theory, and show via an actual application to citrus rootstock selection in Florida, how the potential usefulness of the well-known interactive method STEM for solving problem (P) in this way, can depend crucially upon how accurately certain global optimization problems involving minimizations over X E are solved. In particular, we will show both in theory and in practice that the choice of whether to use the popular but unreliable ‘payoff table’ approach or to use one of the lesser known, more accurate global optimization methods to solve these problems can determine whether STEM succeeds or fails as a decision aid. Several lessons and conclusions of transferable value derived from this research are also given.
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    A Generic Global Optimization Algorithm for the Chemical and Phase Equilibrium Problem (1998)
    Springer
    Journal of global optimization 12 (1998), S. 325-351 
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    ISSN: 1573-2916
    Keywords: Chemical and phase equilibrium ; convexity ; Gibbs free energy ; Global optimization ; Non-convex optimization ; Tangent-plane criterion
    Source: Springer Online Journal Archives 1860-2000
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    Notes: Abstract This paper addresses the problem of finding the number, K, of phases present at equilibrium and their composition, in a chemical mixture of n s substances. This corresponds to the global minimum of the Gibbs free energy of the system, subject to constraints representing m b independent conserved quantities, where m b=n s when no reaction is possible and m b ≤ n e +1 when reaction is possible and n e is the number of elements present. After surveying previous work in the field and pointing out the main issues, we extend the necessary and sufficient condition for global optimality based on the ‘reaction tangent-plane criterion’, to the case involving different thermodynamical models (multiple phase classes). We then present an algorithmic approach that reduces this global optimization problem (involving a search space of m b(n s-1) dimensions) to a finite sequence of local optimization steps inK(n s-1) -space, K ≤ m b, and global optimization steps in (n s-1)-space. The global step uses the tangent-plane criterion to determine whether the current solution is optimal, and, if it is not, it finds an improved feasible solution either with the same number of phases or with one added phase. The global step also determines what class of phase (e.g. liquid or vapour) is to be added, if any phase is to be added. Given a local minimization procedure returning a Kuhn–Tucker point and a global optimization procedure (for a lower-dimensional search space) returning a global minimum, the algorithm is proved to converge to a global minimum in a finite number of the above local and global steps. The theory is supported by encouraging computational results.
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    On the convergence of global methods in multiextremal optimization (1987)
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    Journal of optimization theory and applications 54 (1987), S. 253-271 
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    ISSN: 1573-2878
    Keywords: Global optimization ; multiextremal optimization ; optimization algorithms ; nonlinear programming ; branch-and-bound methods ; Lipschitzian optimization
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A general class of derivative-free optimization procedures is presented including the corresponding convergence theory. This theory turns out to be very constructive, in the sense that the convergence conditions not only can be verified easily for many existing algorithms, but also allow one to construct new procedures. It is shown that popular methods such as branch-and-bound concepts, Pintér's general class of procedures, the algorithms of Pijavskii, Shubert, and Mladineo, and the approach of Zheng and Galperin can not only be subsumed under this class of methods, but also partly be improved by regarding them within the framework presented.
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    An algorithm for global optimization of Lipschitz continuous functions (1988)
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    Journal of optimization theory and applications 57 (1988), S. 307-322 
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    ISSN: 1573-2878
    Keywords: Global optimization ; nondifferentiable optimization ; Lipschitz continuous functions
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    Topics: Mathematics
    Notes: Abstract An algorithm is presented which locates the global minimum or maximum of a function satisfying a Lipschitz condition. The algorithm uses lower bound functions defined on a partitioned domain to generate a sequence of lower bounds for the global minimum. Convergence is proved, and some numerical results are presented.
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    Conical algorithm for the global minimization of linearly constrained decomposable concave minimization problems (1992)
    Springer
    Journal of optimization theory and applications 74 (1992), S. 469-486 
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    ISSN: 1573-2878
    Keywords: Global optimization ; concave minimization ; conical algorithms ; branch-and-bound methods ; decomposition
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    Notes: Abstract In this paper, we are concerned with the linearly constrained global minimization of the sum of a concave function defined on ap-dimensional space and a linear function defined on aq-dimensional space, whereq may be much larger thanp. It is shown that a conical algorithm can be applied in a space of dimensionp + 1 that involves only linear programming subproblems in a space of dimensionp +q + 1. Some computational results are given.
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    Primal-relaxed dual global optimization approach (1993)
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    Journal of optimization theory and applications 78 (1993), S. 187-225 
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    ISSN: 1573-2878
    Keywords: Global optimization ; quadratic programming ; polynomial functions ; ∈-optimal solutions
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    Notes: Abstract A deterministic global optimization approach is proposed for nonconvex constrained nonlinear programming problems. Partitioning of the variables, along with the introduction of transformation variables, if necessary, converts the original problem into primal and relaxed dual subproblems that provide valid upper and lower bounds respectively on the global optimum. Theoretical properties are presented which allow for a rigorous solution of the relaxed dual problem. Proofs of ∈-finite convergence and ∈-global optimality are provided. The approach is shown to be particularly suited to (a) quadratic programming problems, (b) quadratically constrained problems, and (c) unconstrained and constrained optimization of polynomial and rational polynomial functions. The theoretical approach is illustrated through a few example problems. Finally, some further developments in the approach are briefly discussed.
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    A parallel method for finding the global minimum of univariate functions (1994)
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    Journal of optimization theory and applications 80 (1994), S. 513-536 
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    ISSN: 1573-2878
    Keywords: Global optimization ; parallel numerical methods ; convergence ; speedup
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    Notes: Abstract We describe a new parallel method for solving global optimization problems. The formulation of the decision rules of this method is presented. We examine convergence conditions of the proposed algorithm and establish conditions which guarantee a considerable speedup with respect to the sequential version of the algorithm. We also present some numerical experiments executed on Alliant FX/80 for one class of multiextremal functions.
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    New LP bound in multivariate Lipschitz optimization: Theory and applications (1995)
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    Journal of optimization theory and applications 86 (1995), S. 369-388 
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    ISSN: 1573-2878
    Keywords: Global optimization ; Lipschitz optimization ; systems of inequalities ; branch-and-bounds methods
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    Notes: Abstract Most numerically promising methods for solving multivariate unconstrained Lipschitz optimization problems of dimension greater than two use rectangular or simplicial branch-and-bound techniques with computationally cheap but rather crude lower bounds. Generalizations to constrained problems, however, require additional devices to detect sufficiently many infeasible partition sets. In this article, a new lower bounding procedure is proposed for simplicial methods yielding considerably better bounds at the expense of two linear programs in each iteration. Moreover, the resulting approach can solve easily linearly constrained problems, since in this case infeasible partition sets are automatically detected by the lower bounding procedure. Finally, it is shown that the lower bounds can be further improved when the method is applied to solve systems of inequalities. Implementation issues, numerical experiments, and comparisons are discussed in some detail.
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    Linearly constrained global minimization of functions with concave minorants (1996)
    Springer
    Journal of optimization theory and applications 88 (1996), S. 751-763 
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    ISSN: 1573-2878
    Keywords: Global optimization ; functions with concave minorants ; d.c. optimization ; branch-and-bounds methods ; systems of inequalities
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    Notes: Abstract In this note, we show how a recent approach for solving linearly constrained multivariate Lipschitz optimization problems and corresponding systems of inequalities can be generalized to solve optimization problems where the objective function is only assumed to possess a concave minorant at each point. This class of functions includes not only Lipschitz functions and some generalizations, such as certain ρ-convex functions and Hölder functions with exponent greater than one, but also all functions which can be expressed as differences of two convex functions (d.c. functions). Thus, in particular, a new approach is obtained for the important problem of minimizing a d.c. function over a polytope.
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    Lagrange Duality and Partitioning Techniques in Nonconvex Global Optimization (1997)
    Springer
    Journal of optimization theory and applications 95 (1997), S. 347-369 
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    ISSN: 1573-2878
    Keywords: Global optimization ; duality gap ; branch-and-bound techniques ; partly convex programs ; bilinear constraints ; sum of ratios ; reverse convex constraints
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    Notes: Abstract It is shown that, for very general classes of nonconvex global optimization problems, the duality gap obtained by solving a corresponding Lagrangian dual in reduced to zero in the limit when combined with suitably refined partitioning of the feasible set. A similar result holds for partly convex problems where exhaustive partitioning is applied only in the space of nonconvex variables. Applications include branch-and-bound approaches for linearly constrained problems where convex envelopes can be computed, certain generalized bilinear problems, linearly constrained optimization of the sum of ratios of affine functions, and concave minimization under reverse convex constraints.
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    Global Optimization Requires Global Information (1998)
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    Journal of optimization theory and applications 96 (1998), S. 575-588 
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    ISSN: 1573-2878
    Keywords: Global optimization ; convergence ; stochastic algorithms ; deterministic algorithms
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    Notes: Abstract There are many global optimization algorithms which do not use global information. We broaden previous results, showing limitations on such algorithms, even if allowed to run forever. We show that deterministic algorithms must sample a dense set to find the global optimum value and can never be guaranteed to converge only to global optimizers. Further, analogous results show that introducing a stochastic element does not overcome these limitations. An example is simulated annealing in practice. Our results show that there are functions for which the probability of success is arbitrarily small.
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    Global Interval Methods for Local Nonsmooth Optimization (1999)
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    Journal of global optimization 14 (1999), S. 157-179 
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    ISSN: 1573-2916
    Keywords: Global optimization ; Interval methods ; Local nonsmooth optimization ; Scientific computing
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    Notes: Abstract An interval method for determining local solutions of nonsmooth unconstrained optimization problems is discussed. The objective function is assumed to be locally Lipschitz and to have appropriate interval inclusions. The method consists of two parts, a local search and a global continuation and termination. The local search consists of a globally convergent descent algorithm showing similarities to ε-bundle methods. While ε-bundle methods use polytopes as inner approximations of the ε-subdifferentials, which are the main tools of almost all bundle concepts, our method uses axes parallel boxes as outer approximations of the ε-subdifferentials. The boxes are determined almost automatically with inclusion techniques of interval arithmetic. The dimension of the boxes is equal to the dimension of the problem and remains constant during the whole computation. The application of boxes does not suffer from the necessity to invest methodical and computational efforts to adapt the polytopes to the latest state of the computation as well as to simplify them when the number of vertices becomes too large, as is the case with the polytopes. The second part of the method applies interval techniques of global optimization to the approximative local solution obtained from the search of the first part in order to determine guaranteed error bounds or to improve the solution if necessary. We present prototype algorithms for both parts of the method as well as a complete convergence theory for them and demonstrate how outer approximations can be obtained.
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    Extension of Piyavskii's Algorithm to Continuous Global Optimization (1999)
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    Journal of global optimization 14 (1999), S. 205-216 
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    ISSN: 1573-2916
    Keywords: Global optimization ; Lipschitz continuity ; Piyavskii's algorithm ; Uniform Continuity
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    Notes: Abstract We use the simple, but little-known, result that a uniformly continuous function on a convex set is ε-Lipschitz (as defined below) to extend Piyavskii's algorithm for Lipschitz global optimization to the larger domain of continuous (not-necessarily-Lipschitz) global optimization.
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    A Nonsmooth Global Optimization Technique Using Slopes: The One-Dimensional Case (1999)
    Springer
    Journal of global optimization 14 (1999), S. 365-393 
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    ISSN: 1573-2916
    Keywords: Branch-and-bound ; Global optimization ; Interval arithmetic ; Interval slopes
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    Notes: Abstract In this paper we introduce a pruning technique based on slopes in the context of interval branch-and-bound methods for nonsmooth global optimization. We develop the theory for a slope pruning step which can be utilized as an accelerating device similar to the monotonicity test frequently used in interval methods for smooth problems. This pruning step offers the possibility to cut away a large part of the box currently investigated by the optimization algorithm. We underline the new technique's efficiency by comparing two variants of a global optimization model algorithm: one equipped with the monotonicity test and one equipped with the pruning step. For this reason, we compared the required CPU time, the number of function and derivative or slope evaluations, and the necessary storage space when solving several smooth global optimization problems with the two variants. The paper concludes on the test results for several nonsmooth examples.
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    A New Semidefinite Programming Bound for Indefinite Quadratic Forms Over a Simplex (1999)
    Springer
    Journal of global optimization 14 (1999), S. 357-364 
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    ISSN: 1573-2916
    Keywords: Global optimization ; Nonconvex quadratic programming ; Semidefinite programming
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    Notes: Abstract The paper describes a method for computing a lower bound of the global minimum of an indefinite quadratic form over a simplex. The bound is derived by computing an underestimator of the convex envelope by solving a semidefinite program (SDP). This results in a convex quadratic program (QP). It is shown that the optimal value of the QP is a lower bound of the optimal value of the original problem. Since there exist fast (polynomial time) algorithms for solving SDP's and QP's the bound can be computed in reasonable time. Numerical experiments indicate that the relative error of the bound is about 10 percent for problems up to 20 variables, which is much better than a known SDP bound.
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    A Parallel Grasp for the Steiner Tree Problem in Graphs Using a Hybrid Local Search Strategy (2000)
    Springer
    Journal of global optimization 17 (2000), S. 267-283 
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    ISSN: 1573-2916
    Keywords: Combinatorial optimization ; Global optimization ; GRASP ; Heuristics ; Local search ; Network design ; Steiner problem in graphs
    Source: Springer Online Journal Archives 1860-2000
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    Notes: Abstract In this paper, we present a parallel greedy randomized adaptive search procedure (GRASP) for the Steiner problem in graphs. GRASP is a two-phase metaheuristic. In the first phase, solutions are constructed using a greedy randomized procedure. Local search is applied in the second phase, leading to a local minimum with respect to a specified neighborhood. In the Steiner problem in graphs, feasible solutions can be characterized by their non-terminal nodes (Steiner nodes) or by their key-paths. According to this characterization, two GRASP procedures are described using different local search strategies. Both use an identical construction procedure. The first uses a node-based neighborhood for local search, while the second uses a path-based neighborhood. Computational results comparing the two procedures show that while the node-based variant produces better quality solutions, the path-based variant is about twice as fast. A hybrid GRASP procedure combining the two neighborhood search strategies is then proposed. Computational experiments with a parallel implementation of the hybrid procedure are reported, showing that the algorithm found optimal solutions for 45 out of 60 benchmark instances and was never off by more than 4% of the optimal solution value. The average speedup results observed for the test problems show that increasing the number of processors reduces elapsed times with increasing speedups. Moreover, the main contribution of the parallel algorithm concerns the fact that larger speedups of the same order of the number of processors are obtained exactly for the most difficult problems.
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    Stochastic Global Optimization: Problem Classes and Solution Techniques (1999)
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    Journal of global optimization 14 (1999), S. 437-447 
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    ISSN: 1573-2916
    Keywords: Global optimization ; Problem features ; Problem classes ; Test problems ; Solution techniques
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    Topics: Mathematics
    Notes: Abstract There is a lack of a representative set of test problems for comparing global optimization methods. To remedy this a classification of essentially unconstrained global optimization problems into unimodal, easy, moderately difficult, and difficult problems is proposed. The problem features giving this classification are the chance to miss the region of attraction of the global minimum, embeddedness of the global minimum, and the number of minimizers. The classification of some often used test problems are given and it is recognized that most of them are easy and some even unimodal. Global optimization solution techniques treated are global, local, and adaptive search and their use for tackling different classes of problems is discussed. The problem of fair comparison of methods is then adressed. Further possible components of a general global optimization tool based on the problem classes and solution techniques is presented.
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    Multisection in Interval Branch-and-Bound Methods for Global Optimization – I. Theoretical Results (2000)
    Springer
    Journal of global optimization 16 (2000), S. 371-392 
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    ISSN: 1573-2916
    Keywords: Branch-and-bound method ; Global optimization ; Interval arithmetic ; Multisection ; Accelerating devices
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    Topics: Mathematics
    Notes: Abstract We have investigated variants of interval branch-and-bound algorithms for global optimization where the bisection step was substituted by the subdivision of the current, actual interval into many subintervals in a single iteration step. The convergence properties of the multisplitting methods, an important class of multisection procedures are investigated in detail. We also studied theoretically the convergence improvements caused by multisection on algorithms which involve the accelerating tests (like e.g. the monotonicity test). The results are published in two papers, the second one contains the numerical test result.
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    Pivoting in an Outcome Polyhedron (2000)
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    Journal of global optimization 16 (2000), S. 301-323 
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    ISSN: 1573-2916
    Keywords: Outcome polyhedron ; Linear programming ; Pivoting ; Nonlinear programming ; Global optimization ; Extreme point mathematical programming ; Neighborhood problem
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    Notes: Abstract In many types of linear, convex and nonconvex optimization problems over polyhedra, a global optimal solution can be found by searching the extreme points of the outcome polyhedron Y instead of the extreme points of the decision set polyhedron Z. Since the dimension of Y is often significantly smaller than the dimension of Z, and since the structure of Y is often much simpler than the structure of Z, such an approach has the potential to often yield significant computational savings. This article seeks to motivate these potential savings through both general theory and concrete examples. The article then develops two new procedures. The first procedure is linear-programming based and finds an initial extreme point of an outcome polyhedron Y. The second procedure provides a mechanism for moving from a given extreme point y of Y along any chosen edge of Y emanating from y until a neighboring extreme point to y is reached. As a by-product of the second procedure, as in the pivoting process of the simplex method, a complete algebraic description of the chosen edge can also be easily obtained.
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    Convexity analysis in detecting a steel plant hidden global optimum (1993)
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    Journal of global optimization 3 (1993), S. 117-131 
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    ISSN: 1573-2916
    Keywords: Global optimization ; convex analysis ; convex-concave ; blast furnace ; steel plant
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    Topics: Mathematics
    Notes: Abstract This case study demonstrates the value of classical analysis and to a lesser degree, system decomposition for finding a global optimum missed by a sequential linear programming scheme which converges to a non-global local minimum. The example is a 20 variable steelmaking problem in which the variable annual cost to be minimized is linear, as are all constraints except a non-convex one in each blast furnace. The sequential linear programming method gives a provenlocal minimum, although the non-convex nonlinearity prevents any proof of global optimality. The provenglobal minimum found here has a 4% lower cost. The local minimum costs only 0.2% per annum less than the rather flat global maximum, so the original local minimization only achieved about 5% of the economy possible. In the overall plant, the cost saving is over three million US$ (1972) annually.
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    On a new stochastic global optimization algorithm based on censored observations (1994)
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    Journal of global optimization 4 (1994), S. 17-35 
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    ISSN: 1573-2916
    Keywords: Global optimization ; Multistart algorithm ; sequential stopping rules ; nonparametric models ; clustering techniques ; censored observations
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    Topics: Mathematics
    Notes: Abstract In this paper a new algorithm is proposed for global optimization problems. The main idea is that of modifying a standard clustering approach by sequentially sampling the objective function while adaptively deciding an appropriate sample size. Theoretical as well as computational results are presented.
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    A wide class of test functions for global optimization (1993)
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    Journal of global optimization 3 (1993), S. 133-137 
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    Keywords: Global optimization ; test functions ; metric interpolation
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    Multistart method with estimation scheme for global satisfycing problems (1993)
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    Journal of global optimization 3 (1993), S. 139-156 
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    ISSN: 1573-2916
    Keywords: Global optimization ; satisfycing problem ; linear convergence
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    Notes: Abstract We present a multistart method for solving global satisfycing problems. The method uses data generated by linearly converging local algorithms to estimate the cost value at the local minimum to which the local search is converging. When the estimate indicates that the local search is converging to a value higher than the satisfycing value, the local search is interrupted and a new local search is initiated from a randomly generated point. When the satisfycing problem is difficult and the estimation scheme is fairly accurate, the new method is superior over a straightforward adaptation of classical multistart methods.
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    Global optimization of arborescent multilevel inventory systems (1995)
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    Journal of global optimization 6 (1995), S. 269-292 
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    Keywords: Global optimization ; inventory problems ; discrete Hamilton-Jacobi-Bellman equations ; quasi-variational inequalities ; subsolutions and supersolutions
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    Notes: Abstract We consider the numerical resolution of hierarchical inventory problems under global optimization. First we describe the model as well as the dynamical stochastic system and the impulse controls involved. Next we characterize the optimal cost function and we formulate the Hamilton-Jacobi-Bellman equations. We present a numerical scheme and a fast algorithm of resolution, with a result on the speed of convergence. Finally, we apply the discretization method to some examples where we show the usefulness of the proposed numerical method as well as the advantages of operating under global optimization.
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    A branch and bound algorithm for bound constrained optimization problems without derivatives (1995)
    Springer
    Journal of global optimization 7 (1995), S. 297-331 
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    ISSN: 1573-2916
    Keywords: 49D37 ; 65G10 ; Global optimization ; interval arithmetic
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    Notes: Abstract In this paper, we give a new branch and bound algorithm for the global optimization problem with bound constraints. The algorithm is based on the use of inclusion functions. The bounds calculated for the global minimum value are proved to be correct, all rounding errors are rigorously estimated. Our scheme attempts to exclude most “uninteresting” parts of the search domain and concentrates on its “promising” subsets. This is done as fast as possible (by involving local descent methods), and uses little information as possible (no derivatives are required). Numerical results for many well-known problems as well as some comparisons with other methods are given.
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    Topographical Multilevel Single Linkage (1994)
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    Journal of global optimization 5 (1994), S. 349-358 
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    ISSN: 1573-2916
    Keywords: Global optimization ; multilevel single linkage ; topographs ; graph minima
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract An iterative topographical Multilevel Single Linkage (TMSL) method has been introduced. The approach uses topographical information on the objective function, in particular theg-nearest-neighbour graph. The algorithm uses evenly distributed points from a Halten sequence of uniform limiting density. We discuss the implementation of the algorithm and compare its performance with other well-known algorithms. The new algorithm performs much better (in some cases several times) than the Multilevel Single Linkage method in terms of number of function evaluations but is not quite so competitive with respect to CPU time.
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    URL: http://dx.doi.org/10.1007/BF01096684
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    Global optimization conditions for certain nonconvex minimization problems (1994)
    Springer
    Journal of global optimization 5 (1994), S. 359-370 
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    ISSN: 1573-2916
    Keywords: Global optimization ; necessary and sufficient optimality conditions ; Mathematics Subject Classifications (1991) ; 49K27 ; 90C26 ; 90C48
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract By means of suitable dual problems to the following global optimization problems: extremum{f(x): x εM ⊂X}, wheref is a proper convex and lower-semicontinuous function andM a nonempty, arbitrary subset of a reflexive Banach spaceX, we derive necessary and sufficient optimality conditions for a global minimizer. The method is also applicable to other nonconvex problems and leads to at least necessary global optimality conditions.
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    URL: http://dx.doi.org/10.1007/BF01096685
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    Finding maximum likelihood estimators for the three-parameter Weibull distribution (1994)
    Springer
    Journal of global optimization 5 (1994), S. 373-397 
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    ISSN: 1573-2916
    Keywords: Global optimization ; decomposition ; maximum likelihood estimation ; Weibull distribution
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Much work has been devoted to the problem of finding maximum likelihood estimators for the three-parameter Weibull distribution. This problem has not been clearly recognized as a global optimization one and most methods from the literature occasionally fail to find a global optimum. We develop a global optimization algorithm which uses first order conditions and projection to reduce the problem to a univariate optimization one. Bounds on the resulting function and its first order derivative are obtained and used in a branch-and-bound scheme. Computational experience is reported. It is also shown that the solution method we propose can be extended to the case of right censored samples.
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    URL: http://dx.doi.org/10.1007/BF01096687
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    Simple global minimization algorithm for one-variable rational functions (1995)
    Springer
    Journal of global optimization 6 (1995), S. 293-311 
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    ISSN: 1573-2916
    Keywords: Global optimization ; univariate optimization ; polynomials ; rational functions ; Storm's chain
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Sturm's chain technique for evaluation of a number of real roots of polynomials is applied to construct a simple algorithm for global optimization of polynomials or generally for rational functions of finite global minimal value. The method can be applied both to find the global minimum in an interval or without any constraints. It is shown how to use the method to minimize globally a truncated Fourier series. The results of numerical tests are presented and discussed. The cost of the method scales as the square of the degree of the polynomial.
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    URL: http://dx.doi.org/10.1007/BF01099466
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    A class of test functions for global optimization (1994)
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    Journal of global optimization 5 (1994), S. 195-199 
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    ISSN: 1573-2916
    Keywords: Global optimization ; test functions ; multidimensional scaling
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We suggest weighted least squares scaling, a basic method in multidimensional scaling, as a class of test functions for global optimization. The functions are easy to code, cheap to calculate, and have important applications in data analysis. For certain data these functions have many local minima. Some characteristic features of the test functions are investigated.
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    URL: http://dx.doi.org/10.1007/BF01100693
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    On the selection of subdivision directions in interval branch-and-bound methods for global optimization (1995)
    Springer
    Journal of global optimization 7 (1995), S. 183-207 
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    ISSN: 1573-2916
    Keywords: Global optimization ; interval arithmetic ; branch-and-bound ; interval subdivision
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This paper investigates the influence of the interval subdivision selection rule on the convergence of interval branch-and-bound algorithms for global optimization. For the class of rules that allows convergence, we study the effects of the rules on a model algorithm with special list ordering. Four different rules are investigated in theory and in practice. A wide spectrum of test problems is used for numerical tests indicating that there are substantial differences between the rules with respect to the required CPU time, the number of function and derivative evaluations, and the necessary storage space. Two rules can provide considerable improvements in efficiency for our model algorithm.
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    URL: http://dx.doi.org/10.1007/BF01097060
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    Global optimization of Hölder functions (1996)
    Springer
    Journal of global optimization 8 (1996), S. 323-348 
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    ISSN: 1573-2916
    Keywords: Global optimization ; Hölder functions ; Lipschitz optimization
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We propose a branch-and-bound framework for the global optimization of unconstrained Hölder functions. The general framework is used to derive two algorithms. The first one is a generalization of Piyavskii's algorithm for univariate Lipschitz functions. The second algorithm, using a piecewise constant upper-bounding function, is designed for multivariate Hölder functions. A proof of convergence is provided for both algorithms. Computational experience is reported on several test functions from the literature.
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    URL: http://dx.doi.org/10.1007/BF00554012
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    An algorithm for maximizing a convex function over a simple set (1996)
    Springer
    Journal of global optimization 8 (1996), S. 379-391 
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    ISSN: 1573-2916
    Keywords: Global optimization ; optimality condition ; convex function ; numerical algorithm ; simple set
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The problem of maximizing a convex function on a so-called simple set is considered. Based on the optimality conditions [19], an algorithm for solving the problem is proposed. This numerical algorithm is shown to be convergent. The proposed algorithm has been implemented and tested on a variety of test problems.
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    URL: http://dx.doi.org/10.1007/BF00554014
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    A branch-and-reduce approach to global optimization (1996)
    Springer
    Journal of global optimization 8 (1996), S. 107-138 
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    ISSN: 1573-2916
    Keywords: Global optimization ; range reduction ; branch-and-bound ; polynomial programming ; multiplicative programming ; mixed-integer nonlinear programming ; quadratic programming
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This paper presents valid inequalities and range contraction techniques that can be used to reduce the size of the search space of global optimization problems. To demonstrate the algorithmic usefulness of these techniques, we incorporate them within the branch-and-bound framework. This results in a branch-and-reduce global optimization algorithm. A detailed discussion of the algorithm components and theoretical properties are provided. Specialized algorithms for polynomial and multiplicative programs are developed. Extensive computational results are presented for engineering design problems, standard global optimization test problems, univariate polynomial programs, linear multiplicative programs, mixed-integer nonlinear programs and concave quadratic programs. For the problems solved, the computer implementation of the proposed algorithm provides very accurate solutions in modest computational time.
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    URL: http://dx.doi.org/10.1007/BF00138689
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    A Global Optimization Algorithm for Multivariate Functions with Lipschitzian First Derivatives (1997)
    Springer
    Journal of global optimization 10 (1997), S. 257-281 
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    ISSN: 1573-2916
    Keywords: Global optimization ; multiextremal algorithms ; Lipschitzian first derivatives ; convergence ; numerical experiments.
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In this paper we propose a new multi-dimensional methodto solve unconstrained global optimization problems with Lipschitzianfirst derivatives. The method is based on apartition scheme that subdivides the search domain into a set of hypercubesin the course of optimization. This partitioning is regulated by thedecision rule that provides evaluation of the "importance"of each generated hypercube and selection of some partition element to performthe next iteration. Sufficient conditions of global convergence for the newmethod are investigated. Results of numerical experiments are alsopresented.
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    URL: http://dx.doi.org/10.1023/A:1008290629896
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    On nonconvex optimization problems with separated nonconvex variables (1992)
    Springer
    Journal of global optimization 2 (1992), S. 133-144 
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    ISSN: 1573-2916
    Keywords: Global optimization ; separated nonconvex variables ; reverse convex programming ; Lipschitz optimization ; decomposition ; lower linearization ; outer approximation ; branch and bound ; relief indicator method
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A mathematical programming problem is said to have separated nonconvex variables when the variables can be divided into two groups: x=(x 1,...,x n ) and y=( y 1,...,y n ), such that the objective function and any constraint function is a sum of a convex function of (x, y) jointly and a nonconvex function of x alone. A method is proposed for solving a class of such problems which includes Lipschitz optimization, reverse convex programming problems and also more general nonconvex optimization problems.
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    URL: http://dx.doi.org/10.1007/BF00122050
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    On the Convergence of the P-Algorithm for One-Dimensional Global Optimization of Smooth Functions (1999)
    Springer
    Journal of optimization theory and applications 102 (1999), S. 479-495 
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    ISSN: 1573-2878
    Keywords: Global optimization ; Gaussian processes
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The Wiener process is a widely used statistical model for stochastic global optimization. One of the first optimization algorithms based on a statistical model, the so-called P-algorithm, was based on the Wiener process. Despite many advantages, this process does not give a realistic model for many optimization problems, particularly from the point of view of local behavior. In the present paper, a version of the P-algorithm is constructed based on a stochastic process with smooth sampling functions. It is shown that, in such a case, the algorithm has a better convergence rate than in the case of the Wiener process. A similar convergence rate is proved for a combination of the Wiener model-based P-algorithm with quadratic fit-based local search.
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    URL: http://dx.doi.org/10.1023/A:1022677121193
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    Generalized descent for global optimization (1981)
    Springer
    Journal of optimization theory and applications 34 (1981), S. 11-39 
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    ISSN: 1573-2878
    Keywords: Global optimization ; generalized descent ; search trajectories ; target level
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This paper introduces a new method for the global unconstrained minimization of a differentiable objective function. The method is based on search trajectories, which are defined by a differential equation and exhibit certain similarities to the trajectories of steepest descent. The trajectories depend explicitly on the value of the objective function and aim at attaining a given target level, while rejecting all larger local minima. Convergence to the gloal minimum can be proven for a certain class of functions and appropriate setting of two parameters.
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    Global optimization and stochastic differential equations (1985)
    Springer
    Journal of optimization theory and applications 47 (1985), S. 1-16 
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    ISSN: 1573-2878
    Keywords: Global optimization ; stochastic differential equations
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Let ℝ n be then-dimensional real Euclidean space,x=(x 1,x 2, ...,x n)T ∈ ℝ n , and letf:ℝ n → R be a real-valued function. We consider the problem of finding the global minimizers off. A new method to compute numerically the global minimizers by following the paths of a system of stochastic differential equations is proposed. This method is motivated by quantum mechanics. Some numerical experience on a set of test problems is presented. The method compares favorably with other existing methods for global optimization.
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    URL: http://dx.doi.org/10.1007/BF00941312
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    Construction of large-scale global minimum concave quadratic test problems (1986)
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    Journal of optimization theory and applications 48 (1986), S. 303-313 
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    ISSN: 1573-2878
    Keywords: Global optimization ; test problems ; concave minimization
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Construction of problems with known global solutions is important for the computational testing of constrained global minimization algorithms. In this paper, it is shown how to construct a concave quadratic function which attains its global minimum at a specified vertex of a polytope inR n+k. The constructed function is strictly concave in the variablesx ∈R n and is linear in the variablesy ∈R k. The number of linear variablesk may be much larger thann, so that large-scale global minimization test problems can be constructed by the methods described here.
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    URL: http://dx.doi.org/10.1007/BF00940675
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    A graphical method for a class of Branin trajectories (1986)
    Springer
    Journal of optimization theory and applications 49 (1986), S. 367-374 
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    ISSN: 1573-2878
    Keywords: Global optimization ; graphical methods
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The construction of Branin trajectories for locating the stationary points of a scalar function of many variables involves, in the general case, the numerical solution of a set of simultaneous ordinary differential equations, or some equivalent numerical procedure. For a function of only two variables which is separable in either the multiplicative or additive sense, it is shown that Branin trajectories may be obtained by a graphical method due to Volterra.
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    URL: http://dx.doi.org/10.1007/BF00941067
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    A general class of branch-and-bound methods in global optimization with some new approaches for concave minimization (1986)
    Springer
    Journal of optimization theory and applications 51 (1986), S. 271-291 
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    ISSN: 1573-2878
    Keywords: Global optimization ; nonconvex programming ; mathematical programming ; nonlinear optimization ; branch-and-bound methods ; concave minimization
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Based on a review of existing algorithms, a general branch-and-bound concept in global optimization is presented. A sufficient and necessary convergence condition is established, and a broad class of realizations is derived that include existing and several new approaches for concave minimization problems.
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    Internal modelling of objective functions for global optimization (1986)
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    Journal of optimization theory and applications 51 (1986), S. 345-353 
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    ISSN: 1573-2878
    Keywords: Global optimization ; stochastic process models ; multidimensional objective functions
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Global optimization requires an adequate internal representation of the objective function for success in a reasonable number of function evaluations. A method for determining the location of a new function evaluation, based on a representation using a stationary stochastic process model, is investigated and some results are given.
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    A search technique for global optimization in a chaotic environment (1989)
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    Journal of optimization theory and applications 61 (1989), S. 111-121 
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    ISSN: 1573-2878
    Keywords: Global optimization ; dynamical systems ; search trajectories ; chaos
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We describe a new algorithm which uses the trajectories of a discrete dynamical system to sample the domain of an unconstrained objective function in search of global minima. The algorithm is unusually adept at avoiding nonoptimal local minima and successfully converging to a global minimum. Trajectories generated by the algorithm for objective functions with many local minima exhibit chaotic behavior, in the sense that they are extremely sensitive to changes in initial conditions and system parameters. In this context, chaos seems to have a beneficial effect: failure to converge to a global minimum from a given initial point can often be rectified by making arbitrarily small changes in the system parameters.
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    On consistency of bounding operations in deterministic global optimization (1989)
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    Journal of optimization theory and applications 61 (1989), S. 143-146 
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    ISSN: 1573-2878
    Keywords: Global optimization ; nonconvex programming ; mathematical programming ; branch-and-bound methods
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This technical comment refers to the discussion of strong consistency of several bounding procedures in Lemma 2.1 and Proposition 2.1 of Ref. 1. A necessary clarification is given of the notion of convergence φq → φ in Lemma 2.1, and a derivation of Proposition 2.1 is presented that includes a new and simple consistency proof of the classical bounding by convex envelopes used in many branch-and-bound procedures.
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    Efficient domain partitioning algorithms for global optimization of rational and Lipschitz continuous functions (1989)
    Springer
    Journal of optimization theory and applications 61 (1989), S. 247-270 
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    ISSN: 1573-2878
    Keywords: Global optimization ; nondifferentiable optimization ; rational functions ; Lipschitz continuous functions
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A domain partitioning algorithm for minimizing or maximizing a Lipschitz continuous function is enhanced to yield two new, more efficient algorithms. The use of interval arithmetic in the case of rational functions and the estimates of Lipschitz constants valid in subsets of the domain in the case of others and the addition of local optimization have resulted in an algorithm which, in tests on standard functions, performs well.
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    Random tunneling by means of acceptance-rejection sampling for global optimization (1989)
    Springer
    Journal of optimization theory and applications 62 (1989), S. 255-277 
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    ISSN: 1573-2878
    Keywords: Global optimization ; acceptance-rejection sampling ; Boltzmann distribution ; random tunneling
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Any global minimization algorithm is made by several local searches performed sequentially. In the classical multistart algorithm, the starting point for each new local search is selected at random uniformly in the region of interest. In the tunneling algorithm, such a starting point is required to have the same function value obtained by the last local minimization. We introduce the class of acceptance-rejection based algorithms in order to investigate intermediate procedures. A particular instance is to choose at random the new point approximately according to a Boltzmann distribution, whose temperatureT is updated during the algorithm. AsT → 0, such distribution peaks around the global minima of the cost function, producing a kind of random tunneling effect. The motivation for such an approach comes from recent works on the simulated annealing approach in global optimization. The resulting algorithm has been tested on several examples proposed in the literature.
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    On using estimates of Lipschitz constants in global optimization (1992)
    Springer
    Journal of optimization theory and applications 75 (1992), S. 195-200 
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    ISSN: 1573-2878
    Keywords: Global optimization ; univariate functions ; Lipschitz constants
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Several authors have proposed estimating Lipschitz constants in global optimization by a multiple of the largest slope (in absolute value) between successive evaluation points. A class of univariate functions is exhibited for which the global optimum will be missed when using such a procedure, even if the multiple is arbitrarily large.
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    Convex minimization under Lipschitz constraints (1990)
    Springer
    Journal of optimization theory and applications 64 (1990), S. 595-614 
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    ISSN: 1573-2878
    Keywords: Global optimization ; convex minimization ; Lipschitz constraints ; reverse convex constraints
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We consider the problem of minimizing a convex functionf(x) under Lipschitz constraintsf i (x)≤0,i=1,...,m. By transforming a system of Lipschitz constraintsf i (x)≤0,i=l,...,m, into a single constraints of the formh(x)-∥x∥2≤0, withh(·) being a closed convex function, we convert the problem into a convex program with an additional reverse convex constraint. Under a regularity assumption, we apply Tuy's method for convex programs with an additional reverse convex constraint to solve the converted problem. By this way, we construct an algorithm which reduces the problem to a sequence of subproblems of minimizing a concave, quadratic, separable function over a polytope. Finally, we show how the algorithm can be used for the decomposition of Lipschitz optimization problems involving relatively few nonconvex variables.
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    URL: http://dx.doi.org/10.1007/BF00939426
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