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  • Articles  (57)
  • Global optimization  (57)
  • 1990-1994  (57)
  • Mathematics  (57)
  • Energy, Environment Protection, Nuclear Power Engineering
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  • Articles  (57)
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  • 1
    Electronic Resource
    Electronic Resource
    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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  • 2
    Electronic Resource
    Electronic Resource
    Cord-slope form of Taylor's expansion in univariate global optimization (1994)
    Springer
    Journal of optimization theory and applications 80 (1994), S. 441-464 
    Details
    ISSN: 1573-2878
    Keywords: Global optimization ; univariate functions ; interval arithmetic ; Taylor's formula
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02207774
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  • 3
    Electronic Resource
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    A parallel method for finding the global minimum of univariate functions (1994)
    Springer
    Journal of optimization theory and applications 80 (1994), S. 513-536 
    Details
    ISSN: 1573-2878
    Keywords: Global optimization ; parallel numerical methods ; convergence ; speedup
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02207778
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  • 4
    Electronic Resource
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    On a new stochastic global optimization algorithm based on censored observations (1994)
    Springer
    Journal of global optimization 4 (1994), S. 17-35 
    Details
    ISSN: 1573-2916
    Keywords: Global optimization ; Multistart algorithm ; sequential stopping rules ; nonparametric models ; clustering techniques ; censored observations
    Source: Springer Online Journal Archives 1860-2000
    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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01096532
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  • 5
    Electronic Resource
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    Topographical Multilevel Single Linkage (1994)
    Springer
    Journal of global optimization 5 (1994), S. 349-358 
    Details
    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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01096684
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  • 6
    Electronic Resource
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    Global optimization conditions for certain nonconvex minimization problems (1994)
    Springer
    Journal of global optimization 5 (1994), S. 359-370 
    Details
    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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01096685
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  • 7
    Electronic Resource
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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 
    Details
    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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01096687
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  • 8
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    A class of test functions for global optimization (1994)
    Springer
    Journal of global optimization 5 (1994), S. 195-199 
    Details
    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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01100693
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  • 9
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    Branch-and-bound decomposition approach for solving quasiconvex-concave programs (1994)
    Springer
    Journal of optimization theory and applications 82 (1994), S. 267-293 
    Details
    ISSN: 1573-2878
    Keywords: Global optimization ; convex-concave programming ; branch-and-bound methods ; optimization of differences of convex functions ; fractional multiplicative programming
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A class of branch-and-bound methods is proposed for minimizing a quasiconvex-concave function subject to convex and quasiconvex-concave inequality constraints. Several important special cases where the subproblems involved by the bounding-and-branching operations can be solved quite effectively include certain d.c. programming problems, indefinite quadratic programming with one negative eigenvalue, affine multiplicative problems, and fractional multiplicative optimization.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02191854
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  • 10
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    Computational comparison of two methods for constrained global optimization (1994)
    Springer
    Journal of global optimization 5 (1994), S. 325-332 
    Details
    ISSN: 1573-2916
    Keywords: Global optimization ; stochastic methods ; deterministic methods
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract For constrained concave global minimization problems, two very different solution techniques have been investigated. The first such method is a stochastic mulitstart approach which typically finds, with high probability, all local minima for the problem. The second method is deterministic and guarantees a global minimum solution to within any user specified tolerance. It is the purpose of this paper to make a careful comparison of these two methods on a range of test problems using separable concave objectives over compact polyhedral sets, and to investigate in this way the advantages and disadvantages of each method. A direct computational comparison, on the same set of over 140 problems, is presented.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01096682
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  • 11
    Electronic Resource
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    A counterexample to a global optimization algorithm (1994)
    Springer
    Journal of global optimization 5 (1994), S. 371-372 
    Details
    ISSN: 1573-2916
    Keywords: Global optimization ; quasiconcavity ; reverse-convex contraints
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A counterexample to an algorithm of Dien (1988) for solving a minimization problem with a quasiconcave objective function and both linear and a reverse-convex constraint shows that this algorithm needn't lead to a solution of the given problem.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01096686
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  • 12
    Electronic Resource
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    Accelerations for global optimization covering methods using second derivatives (1994)
    Springer
    Journal of global optimization 4 (1994), S. 329-341 
    Details
    ISSN: 1573-2916
    Keywords: Global optimization ; covering methods ; deterministic ; mathematical programming ; 90C30 ; 65K05
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Two improvements for the algorithm of Breiman and Cutler are presented. Better envelopes can be built up using positive quadratic forms. Better utilization of first and second derivative information is attained by combining both global aspects of curvature and local aspects near the global optimum. The basis of the results is the geometric viewpoint developed by the first author and can be applied to a number of covering type methods. Improvements in convergence rates are demonstrated empirically on standard test functions.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01098365
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  • 13
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    Global optimization by random perturbation of the gradient method with a fixed parameter (1994)
    Springer
    Journal of global optimization 5 (1994), S. 159-180 
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    ISSN: 1573-2916
    Keywords: Global optimization ; random perturbations ; Monte Carlo methods
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The paper deals with the global minimization of a differentiable cost function mapping a ball of a finite dimensional Euclidean space into an interval of real numbers. It is established that a suitable random perturbation of the gradient method with a fixed parameter generates a bounded minimizing sequence and leads to a global minimum: the perturbation avoids convergence to local minima. The stated results suggest an algorithm for the numerical approximation of global minima: experiments are performed for the problem of fitting a sum of exponentials to discrete data and to a nonlinear system involving about 5000 variables. The effect of the random perturbation is examined by comparison with the purely deterministic gradient method.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01100691
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  • 14
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    Topographical global optimization using pre-sampled points (1994)
    Springer
    Journal of global optimization 5 (1994), S. 267-276 
    Details
    ISSN: 1573-2916
    Keywords: Global optimization ; topography graph ; parallel algorithms
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A method for global minimization of a functionf(x), x εA ⊂R n by using presampled global points inA is presented. The global points are obtained by uniform sampling, discarding points too near an already accepted point to obtain a very uniform covering. The accepted points and their nearest-neighbours matrix are stored on a file. When optimzing a given function these pre-sampled points and the matrix are read from file. Then the function value of each point is computed and itsk nearest neighbours that have larger function values are marked. The points for which all its neighbours are marked are extracted as promising starting points for local minimizations. Results from a parallel implementation are presented. The working of a sequential version in Fortran is illustrated.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01096456
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  • 15
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    Constraint decomposition algorithms in global optimization (1994)
    Springer
    Journal of global optimization 5 (1994), S. 333-348 
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    ISSN: 1573-2916
    Keywords: Global optimization ; decomposition ; canonical d.c. program ; conical branch and bound algorithms ; outer approximation ; cutting plane algorithms
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Many global optimization problems can be formulated in the form min{c(x, y): x εX, y εY, (x, y) εZ, y εG} where X, Y are polytopes in ℝ p , ℝ n , respectively, Z is a closed convex set in ℝp+n, while G is the complement of an open convex set in ℝ n . The function c:ℝ p+n → ℝ is assumed to be linear. Using the fact that the nonconvex constraints depend only upon they-variables, we modify and combine basic global optimization techniques such that some new decomposition methods result which involve global optimization procedures only in ℝ n . Computational experiments show that the resulting algorithms work well for problems with smalln.
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    URL: http://dx.doi.org/10.1007/BF01096683
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  • 16
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    Decomposition based and branch and bound global optimization approaches for the phase equilibrium problem (1994)
    Springer
    Journal of global optimization 5 (1994), S. 205-251 
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    ISSN: 1573-2916
    Keywords: Global optimization ; phase equilibrium ; biconvex and DC programming problems
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract An increasingly popular approach when solving the phase and chemical equilibrium problem is to pose it as an optimization problem. However, difficulties are encountered due to the highly nonlinear nature of the models used to represent the behavior of the fluids, and because of the existence of multiple local solutions. This work shows how it is possible to guarantee ε-global solutions for a certain important class of the phase and chemical equilibrium problem, namely when the liquid phase can be modeled using neither the Non-Random Two-Liquid (NRTL) equation, or the UNIversal QUAsi Chemical (UNIQUAC) equation. Ideal vapor phases are easily incorporated into the global optimization framework. A numberof interesting properties are described which drastically alter the structure of the respective problems. For the NRTL equation, it is shown that the formulation can be converted into a biconvex optimization problem. The GOP algorithm of Floudas and Visweswaran [8, 9] can then be used to obtain ε-global solutions in this case. For the UNIQUAC equation, the new properties show how the objective function can be transformed into the difference of two convex functions (i.e. a D.C. programming problem is obtained), where the concave portion is separable. A branch and bound algorithm based on that of Falk and Soland [6] is used to guarantee convergence to an ε-global solution. Examples are presented which demonstrate the performance of both algorithms.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01096454
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  • 17
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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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  • 18
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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.
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    URL: http://dx.doi.org/10.1007/BF01582140
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  • 19
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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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  • 20
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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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  • 21
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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
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    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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    URL: http://dx.doi.org/10.1007/BF00940781
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  • 22
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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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    ISSN: 1573-2916
    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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    URL: http://dx.doi.org/10.1007/BF01096776
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  • 23
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    Primal-relaxed dual global optimization approach (1993)
    Springer
    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
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    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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    URL: http://dx.doi.org/10.1007/BF00939667
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  • 24
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    Convexity analysis in detecting a steel plant hidden global optimum (1993)
    Springer
    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
    Source: Springer Online Journal Archives 1860-2000
    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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    URL: http://dx.doi.org/10.1007/BF01096733
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  • 25
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    A wide class of test functions for global optimization (1993)
    Springer
    Journal of global optimization 3 (1993), S. 133-137 
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    ISSN: 1573-2916
    Keywords: Global optimization ; test functions ; metric interpolation
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
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    URL: http://dx.doi.org/10.1007/BF01096734
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  • 26
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    Multistart method with estimation scheme for global satisfycing problems (1993)
    Springer
    Journal of global optimization 3 (1993), S. 139-156 
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    ISSN: 1573-2916
    Keywords: Global optimization ; satisfycing problem ; linear convergence
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01096735
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  • 27
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    Lipschitzian optimization without the Lipschitz constant (1993)
    Springer
    Journal of optimization theory and applications 79 (1993), S. 157-181 
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    ISSN: 1573-2878
    Keywords: Global optimization ; Lipschitzian optimization ; space covering ; space partitioning
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We present a new algorithm for finding the global minimum of a multivariate function subject to simple bounds. The algorithm is a modification of the standard Lipschitzian approach that eliminates the need to specify a Lipschitz constant. This is done by carrying out simultaneous searches using all possible constants from zero to infinity. On nine standard test functions, the new algorithm converges in fewer function evaluations than most competing methods. The motivation for the new algorithm stems from a different way of looking at the Lipschitz constant. In particular, the Lipschitz constant is viewed as a weighting parameter that indicates how much emphasis to place on global versus local search. In standard Lipschitzian methods, this constant is usually large because it must equal or exceed the maximum rate of change of the objective function. As a result, these methods place a high emphasis on global search and exhibit slow convergence. In contrast, the new algorithm carries out simultaneous searches using all possible constants, and therefore operates at both the global and local level. Once the global part of the algorithm finds the basin of convergence of the optimum, the local part of the algorithm quickly and automatically exploits it. This accounts for the fast convergence of the new algorithm on the test functions.
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    Decomposition and interval arithmetic applied to global minimization of polynomial and rational functions (1993)
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    Journal of global optimization 3 (1993), S. 421-437 
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    Keywords: Global optimization ; decomposition ; interval arithmetic ; Optimisation globale ; décomposition ; arithmétique d'intervalles
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Description / Table of Contents: Résumé On montre que l'algorithme récent d'optimisation globale basé sur la décomposition du à Floudas et Visweswaran, lorsqu'on le spécialise au cas de fonctions polynômiales, est équivalent à une méthode d'optimisation globale basée sur l'arithmétique d'intervalles, qui applique l'extension naturelle à la forme de la pente de la corde du développement de Taylor. Plusieurs variantes plus efficaces utilisant d'autres formes de l'arithmétique d'intervalles sont explorées. On propose des extensions au cas des fonctions fractionnaires. On présente des résultats de calcul comparatifs.
    Notes: Abstract A recent global optimization algorithm using decomposition (GOP), due to Floudas and Visweswaran, when specialized to the case of polynomial functions is shown to be equivalent to an interval arithmetic global optimization algorithm which applies natural extension to the cord-slope form of Taylor's expansion. Several more efficient variants using other forms of interval arithmetic are explored. Extensions to rational functions are presented. Comparative computational experiences are reported.
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    New properties and computational improvement of the GOP algorithm for problems with quadratic objective functions and constraints (1993)
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    Journal of global optimization 3 (1993), S. 439-462 
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    ISSN: 1573-2916
    Keywords: Global optimization ; indefinite quadratic programming quadratic constraints ; multiperiod tankage quality problems
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In Floudas and Visweswaran (1990, 1993), a deterministic global optimization approach was proposed for solving certain classes of nonconvex optimization problems. An algorithm, GOP, was presented for the solution of the problem through a series ofprimal andrelaxed dual problems that provide valid upper and lower bounds respectively on the global solution. The algorithm was proved to have finite convergence to an ∈-global optimum. In this paper, new theoretical properties are presented that help to enhance the computational performance of the GOP algorithm applied to problems of special structure. The effect of the new properties is illustrated through application of the GOP algorithm to a difficult indefinite quadratic problem, a multiperiod tankage quality problem that occurs frequently in the modeling of refinery processes, and a set of pooling/blending problems from the literature. In addition, extensive computational experience is reported for randomly generated concave and indefinite quadratic programming problems of different sizes. The results show that the properties help to make the algorithm computationally efficient for fairly large problems.
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    Experiments using interval analysis for solving a circuit design problem (1993)
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    Journal of global optimization 3 (1993), S. 501-518 
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    Keywords: Global optimization ; least squares problems ; global zero search ; electrical circuits ; transistor modeling ; interval computations ; branch-and-bound
    Source: Springer Online Journal Archives 1860-2000
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    Notes: Abstract An already classical attempt at solving a circuit design problem leads to a system of 9 nonlinear equations in 9 variables. The sensitivity of the problem to small perturbations is extraordinarily high. Since 1974 several investigations have been made into this problem and they hint at one solution in the restricted domain of the nonnegative reals. The investigations did not give error estimates nor did they present conclusive evidence that the solution found is the only one in the domain of the nonnegative reals. Our paper reports on experimental computations which used various kinds of interval analytic methods while also sometimes reflecting on Wright-Cutteridge's philosophy and theses. The computations resulted in a guarantee that in the domain of consideration, that is, the interval [0,10] for each of the 9 variables, exactly one solution did exist, which was near the solution known up to now. Finally, our solution could be localized within a parallelpiped with edge lengths between 10−6 and 3.2-10−4.
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    Parallel algorithms for global optimization (1993)
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    Journal of optimization theory and applications 79 (1993), S. 385-395 
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    ISSN: 1573-2878
    Keywords: Global optimization ; parallel computing ; transputer performance ; interval methods
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    Notes: Abstract In this paper, we report results of implementations of algorithms designed (i) to solve the global optimization problem (GOP) and (ii) to run on a parallel network of transputers. There have always been two alternative approaches to the solution of the GOP, probabilistic and deterministic. Interval methods can be implemented on our network of transputers using Concurrent ADA, and a secondary objective of the tests reported was to investigate the relative computer times required by parallel interval algorithms compared to probabilistic methods.
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    Nonlinear coordinate transformations for unconstrained optimization I. Basic transformations (1993)
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    Journal of global optimization 3 (1993), S. 213-221 
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    Keywords: Global optimization ; nonlinear parameter transformation ; unconstrained optimization ; unimodal function
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    Notes: Abstract In this two-part article, nonlinear coordinate transformations are discussed to simplify unconstrained global optimization problems and to test their unimodality on the basis of the analytical structure of the objective functions. If the transformed problems are 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 unconstrained global 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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    Convergence rates of a global optimization algorithm (1992)
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    Mathematical programming 54 (1992), S. 223-232 
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    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.
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    Optimal and sub-optimal stopping rules for the Multistart algorithm in global optimization (1992)
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    Mathematical programming 57 (1992), S. 445-458 
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    Keywords: Global optimization
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    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.
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    Global optimization of univariate Lipschitz functions: I. Survey and properties (1992)
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    Mathematical programming 55 (1992), S. 251-272 
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    Keywords: Global optimization ; algorithm ; univariate function ; Lipschitz function ; convergence
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    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.
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    Global optimization of univariate Lipschitz functions: II. New algorithms and computational comparison (1992)
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    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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    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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    Keywords: Global optimization ; polynomial functions ; unconstrained and constrained optimization ; the GOP algorithm
    Source: Springer Online Journal Archives 1860-2000
    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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    Conical algorithm for the global minimization of linearly constrained decomposable concave minimization problems (1992)
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    Journal of optimization theory and applications 74 (1992), S. 469-486 
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    Keywords: Global optimization ; concave minimization ; conical algorithms ; branch-and-bound methods ; decomposition
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    Topics: Mathematics
    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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    On nonconvex optimization problems with separated nonconvex variables (1992)
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    Journal of global optimization 2 (1992), S. 133-144 
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    Keywords: Global optimization ; separated nonconvex variables ; reverse convex programming ; Lipschitz optimization ; decomposition ; lower linearization ; outer approximation ; branch and bound ; relief indicator method
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    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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    On using estimates of Lipschitz constants in global optimization (1992)
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    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
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    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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    A review of statistical models for global optimization (1992)
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    Journal of global optimization 2 (1992), S. 145-153 
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    Keywords: Global optimization ; stochastic models ; optimal design ; rational choice
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    Notes: Abstract A review of statistical models for global optimization is presented. Rationality of the search for a global minimum is formulated axiomatically and the features of the corresponding algorithm are derived from the axioms. Furthermore the results of some applications of the proposed algorithm are presented and the perspectives of the approach are discussed.
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    An optimality criterion for global quadratic optimization (1992)
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    Journal of global optimization 2 (1992), S. 201-208 
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    Keywords: Global optimization ; quadratic program ; lower bound ; branch and bound
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    Notes: Abstract In this paper we prove a sufficient condition that a strong local minimizer of a bounded quadratic program is the unique global minimizer. This sufficient condition can be verified computationally by solving a linear and a convex quadratic program and can be used as a quality test for local minimizers found by standard indefinite quadratic programming routines.
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    Global optimization and simulated annealing (1991)
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    Mathematical programming 50 (1991), S. 367-393 
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    Keywords: Global optimization ; continuous variables ; simulated annealing
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    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.
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    An analytical approach to global optimization (1991)
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    Mathematical programming 52 (1991), S. 227-254 
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    Keywords: Global optimization ; analytical methods ; computer algebra
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    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.
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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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    Topics: Mathematics
    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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    Keywords: Global optimization ; multiple criteria decision making ; relaxation algorithms ; efficient set ; multiple objective linear programming
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    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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    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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    Diffusion equation method of global minimization: Performance for standard test functions (1991)
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    Journal of optimization theory and applications 69 (1991), S. 269-284 
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    Keywords: Global optimization ; global minimum ; diffusion ; multiple minima
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    Notes: Abstract Recently, we have proposed a method of global minimization that is based on solving the diffusion equation with the objective function as a boundary condition. In the present paper, the performance of the method is examined when applied to the standard test functions of the Goldstein-Price, Hartman, Shekel, and Griewank families. It turns out that the method succeeds in all these cases. A comparison of the effectiveness of our method and other methods is also reported.
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    Bayesian approach to global optimization and application to multiobjective and constrained problems (1991)
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    Journal of optimization theory and applications 70 (1991), S. 157-172 
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    ISSN: 1573-2878
    Keywords: Global optimization ; Bayesian approach ; multiobjective optimization ; linear and nonlinear constraints ; density of observations
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    Notes: Abstract In this paper, the Bayesian methods of global optimization are considered. They provide the minimal expected deviation from the global minimum. It is shown that, using the Bayesian methods, the asymptotic density of calculations of the objective function is much greater around the point of global minimum. The relation of this density to the parameters of the method and to the function is defined. Algorithms are described which apply the Bayesian methods to problems with linear and nonlinear constraints. The Bayesian approach to global multiobjective optimization is defined. Interactive procedures and reduction of multidimensional data in the case of global optimization are discussed.
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    Electronic Resource
    Method for minimizing a convex-concave function over a convex set (1991)
    Springer
    Journal of optimization theory and applications 70 (1991), S. 377-384 
    Details
    ISSN: 1573-2878
    Keywords: Global optimization ; branch-and-bound methods ; convex-concave functions ; DC-problems
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A branch-and-bound method is proposed for minimizing a convex-concave function over a convex set. The minimization of a DC-function is a special case, where the subproblems connected with the bounding operation can be solved effectively.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00940633
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  • 54
    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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  • 55
    Electronic Resource
    Electronic Resource
    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 
    Details
    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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01585748
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  • 56
    Electronic Resource
    Electronic Resource
    Convex minimization under Lipschitz constraints (1990)
    Springer
    Journal of optimization theory and applications 64 (1990), S. 595-614 
    Details
    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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00939426
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  • 57
    Electronic Resource
    Electronic Resource
    An extended continuous Newton method (1990)
    Springer
    Journal of optimization theory and applications 67 (1990), S. 57-77 
    Details
    ISSN: 1573-2878
    Keywords: Global optimization ; continuous Newton method ; trajectory method ; multiple solutions ; critical points
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This paper describes a numerical realization of an extended continuous Newton method defined by Diener. It traces a connected set of locally one-dimensional trajectories which contains all critical points of a smooth functionf:ℝ n →ℝ. The results show that the method is effectively applicable.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00939735
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