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  • Articles  (110)
  • Articles: DFG German National Licenses  (110)
  • 16W30  (55)
  • global optimization  (55)
  • 1990-1994  (110)
  • 1965-1969
  • Mathematics  (110)
  • Architecture, Civil Engineering, Surveying
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  • Articles  (110)
Source
  • Articles: DFG German National Licenses  (110)
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  • 1
    Electronic Resource
    Electronic Resource
    Concave minimization via conical partitions and polyhedral outer approximation (1991)
    Springer
    Mathematical programming 50 (1991), S. 259-274 
    Details
    ISSN: 1436-4646
    Keywords: Concave minimization ; global optimization ; nonlinear programming ; nonconvex programming ; branch-and-bound ; outer approximation ; cutting planes
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract An algorithm is proposed for globally minimizing a concave function over a compact convex set. This algorithm combines typical branch-and-bound elements like partitioning, bounding and deletion with suitably introduced cuts in such a way that the computationally most expensive subroutines of previous methods are avoided. In each step, essentially only few linear programming problems have to be solved. Some preliminary computational results are reported.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01594938
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  • 2
    Electronic Resource
    Electronic Resource
    Using copositivity for global optimality criteria in concave quadratic programming problems (1993)
    Springer
    Mathematical programming 62 (1993), S. 575-580 
    Details
    ISSN: 1436-4646
    Keywords: Non-convex non-linear programming ; global optimization ; second-order optimality conditions ; copositive matrices ; quadratic programming ; convex maximization problem
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract In this note we specify a necessary and sufficient condition for global optimality in concave quadratic minimization problems. Using this condition, it follows that, from the perspective of worst-case complexity of concave quadratic problems, the difference between local and global optimality conditions is not as large as in general. As an essential ingredient, we here use theε-subdifferential calculus via an approach of Hiriart-Urruty and Lemarechal (1990).
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01585185
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  • 3
    Electronic Resource
    Electronic Resource
    Pure adaptive search in global optimization (1992)
    Springer
    Mathematical programming 53 (1992), S. 323-338 
    Details
    ISSN: 1436-4646
    Keywords: Random search ; Monte Carlo optimization ; global optimization ; complexity
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract Pure adaptive seach iteratively constructs a sequence of interior points uniformly distributed within the corresponding sequence of nested improving regions of the feasible space. That is, at any iteration, the next point in the sequence is uniformly distributed over the region of feasible space containing all points that are strictly superior in value to the previous points in the sequence. The complexity of this algorithm is measured by the expected number of iterations required to achieve a given accuracy of solution. We show that for global mathematical programs satisfying the Lipschitz condition, its complexity increases at mostlinearly in the dimension of the problem.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01585710
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  • 4
    Electronic Resource
    Electronic Resource
    Linear multiplicative programming (1992)
    Springer
    Mathematical programming 56 (1992), S. 51-64 
    Details
    ISSN: 1436-4646
    Keywords: Non-convex quadratic programming problem ; global optimization ; parametric simplex algorithm
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract An algorithm for solving a linear multiplicative programming problem (referred to as LMP) is proposed. LMP minimizes the product of two linear functions subject to general linear constraints. The product of two linear functions is a typical non-convex function, so that it can have multiple local minima. It is shown, however, that LMP can be solved efficiently by the combination of the parametric simplex method and any standard convex minimization procedure. The computational results indicate that the amount of computation is not much different from that of solving linear programs of the same size. In addition, the method proposed for LMP can be extended to a convex multiplicative programming problem (CMP), which minimizes the product of two convex functions under convex constraints.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01580893
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  • 5
    Electronic Resource
    Electronic Resource
    Deciding uniqueness in norm maximization (1992)
    Springer
    Mathematical programming 57 (1992), S. 203-214 
    Details
    ISSN: 1436-4646
    Keywords: 90C30 ; 68Q15 ; 90C20 ; 90C25 ; 52A25 ; global optimization ; quadratic programming ; unique optimum ; polytope ; parallelotope ; norm ; NP-hardness ; diameter ; width ; inradius ; circumradius
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract NP-hardness is established for the problem whose instance is a system of linear inequalities defining a polytopeP, and whose question is whether, onP, the global maximum of the Euclidean norm is attained at more than one vertex ofP. The NP-hardness persists even for the restricted problem in whichP is a full-dimensional parallelotope with one vertex at the origin. This makes it possible to establish NP-hardness for other uniqueness problems, including some from pseudoboolean programming and computational convexity.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01581081
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  • 6
    Electronic Resource
    Electronic Resource
    Minimum concave-cost network flow problems: Applications, complexity, and algorithms (1990)
    Springer
    Annals of operations research 25 (1990), S. 75-99 
    Details
    ISSN: 1572-9338
    Keywords: Concave-cost network flow ; global optimization ; complexity theory ; NP-hard transportation problems
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Economics
    Notes: Abstract We discuss a wide range of results for minimum concave-cost network flow problems, including related applications, complexity issues, and solution techniques. Applications from production and inventory planning, and transportation and communication network design are discussed. New complexity results are proved which show that this problem is NP-hard for cases with cost functions other than fixed charge. An overview of solution techniques for this problem is presented, with some new results given regarding the implementation of a particular branch-and-bound approach.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02283688
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  • 7
    Electronic Resource
    Electronic Resource
    An explicit characterization of the convex envelope of a bivariate bilinear function over special polytopes (1990)
    Springer
    Annals of operations research 25 (1990), S. 197-209 
    Details
    ISSN: 1572-9338
    Keywords: Convex envelopes ; bilinear functions ; bilinear programming problems ; global optimization
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Economics
    Notes: Abstract In this paper, we constructively derive an explicit characterization of the convex envelope of a bilinear function over a special type of polytope in ℝ2. Our motivation stems from the use of such functions for deriving strengthened lower bounds within the context of a branch-and-bound algorithm for solving bilinear programming problems. For the case of polytopes with no edges having finite positive slopes, that is polytopes with “downward” sloping edges (which we call D-polytopes), we obtain a direct, explicit characterization of the convex envelope. This case subsumes the analysis of Al-Khayyal and Falk (1983) for constructing the convex envelope of a bilinear function over a rectangle in ℝ2. For non-D-polytopes, the analysis is more complex. We propose three strategies for this case based on (i) encasing the region in a D-polytope, (ii) employing a discretization technique, and (iii) providing an explicit characterization over a triangle along with a triangular decomposition approach. The analysis is illustrated using numerical examples.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02283695
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  • 8
    Electronic Resource
    Electronic Resource
    Interval Newton/generalized bisection when there are singularities near roots (1990)
    Springer
    Annals of operations research 25 (1990), S. 181-196 
    Details
    ISSN: 1572-9338
    Keywords: Nonlinear algebraic systems ; Newton's method ; interval arithmetic ; Gauss-Seidel method ; global optimization ; singularities
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Economics
    Notes: Abstract Interval Newton methods in conjunction with generalized bisection are important elemetns of algorithms which find theglobal optimum within a specified box X ⊂ ℝn of an objective function ϕ whose critical points are solutions to the system of nonlinear equationsF(X)=0with mathematical certainty, even in finite presision arithmetic. The overall efficiency of such a scheme depends on the power of the interval Newton method to reduce the widths of the coordinate intervals of the box. Thus, though the generalized bisection method will still converge in a box which contains a critical point at which the Jacobian matrix is singular, the process is much more costly in that case. Here, we propose modifications which make the generalized bisection method isolate singular solutions more efficiently. These modifications are based on an observation about the verification property of interval Newton methods and on techniques for detecting the singularity and removing the region containing it. The modifications assume no special structure forF. Additionally, one of the observations should also make the algorithm more efficient when finding nonsingular solutions. We present results of computational experiments.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02283694
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  • 9
    Electronic Resource
    Electronic Resource
    Quantum homogeneous spaces, duality and quantum 2-spheres (1994)
    Springer
    Geometriae dedicata 52 (1994), S. 291-315 
    Details
    ISSN: 1572-9168
    Keywords: 16W30 ; 17B37 ; 22E99 ; quantum groups ; quantum homogeneous spaces ; coaction ; dual Hopf algebras ; two-sided coideals ; infinitesimal invariance ; quantum SU(2) ; sl(2) quantized universal enveloping algebra ; quantum 2-spheres
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract For a quantum groupG the notion of quantum homogeneousG-space is defined. Two methods to construct such spaces are discussed. The first one makes use of quantum subgroups, the second more general one is based upon the notion of infinitesimal invariance with respect to certain two-sided coideals in the Hopf algebra dual to the Hopf algebra ofG. These methods are applied to the quantum group SU(2). As two-sided coideals we take the subspaces spanned by twisted primitive elements in the sl(2) quantized universal enveloping algebra. A one-parameter series of mutually non-isomorphic quantum 2-spheres is obtained, together with the spectral decomposition of the corresponding right regular representation of quantum SU(2). The link with the quantum spheres defined by Podleś is established.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01278478
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  • 10
    Electronic Resource
    Electronic Resource
    Interval method for bounding level sets: Revisited and tested with global optimization problems (1990)
    Springer
    BIT 30 (1990), S. 650-657 
    Details
    ISSN: 1572-9125
    Keywords: 90C30 ; 65K05 ; Inclusion function ; interval arithmetic ; level set ; global optimization
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract An interval method for bounding level sets, modified to increase its efficiency and to get sharper bounding boxes, is presented. The new algorithm was tested with standard global optimization test problems. The test results show that, while the modified method gives a more valuable, guaranteed reliability result set, it is competitive with non-interval methods in terms of CPU time and number of function evaluations.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01933213
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