Open Access
Global optimization using special ordered sets
E. Martín Beale,J.J.H. Forrest +1 more
- 01 Jan 1977
- pp 2
209
TL;DR: In this article, an improved branching strategy for general special-ordered-set problems is presented, which is similar to an interpolation scheme as used in separable programming, but its incorporation in a branch and bound method for global optimization is not entirely straightforward.
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Abstract: The task of finding global optima to general classes of nonconvex optimization problem is attracting increasing attention. McCormick points out that many such problems can conveniently be expressed in separable form, when they can be tackled by the special methods of Falk and Soland or Soland, or by Special Ordered Sets. Special Ordered Sets, introduced by Beale and Tomlin, have lived up to their early promise of being useful for a wide range of practical problems. Forrest, Hirst and Tomlin show how they have benefitted from the last few years, as a result of being incorporated in a general mathematical programming system.
Nevertheless, Special Ordered Sets in their original form require that any continuous functions arising in the problem be approximated by piecewise linear functions at the start of the analysis. The motivation for the new work described in this paper is the relaxation of this requirement by allowing automatic interpolation of additional relevant points in the course of the analysis.
This is similar to an interpolation scheme as used in separable programming, but its incorporation in a branch and bound method for global optimization is not entirely straightforward. Two bt-products of the work are of interest. One is an improved branching strategy for general special-ordered-set problems. The other is a method for finding a global minimum of a function of a scalar variable in a finite interval, assuming that one can calculate function values and first derivatives, and also bounds on the second derivatives within any subinterval.
The paper describes these methods, their implementation in the UMPIRE system, and preliminary computational experience.
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Citations
Mixed-integer nonlinear optimization
TL;DR: An emerging area of mixed-integer optimal control that adds systems of ordinary differential equations to MINLP is described and a range of approaches for tackling this challenging class of problems are discussed, including piecewise linear approximations, generic strategies for obtaining convex relaxations for non-convex functions, spatial branch-and-bound methods, and a small sample of techniques that exploit particular types of non- Convex structures to obtain improved convex Relaxations.
Branching and bounds tighteningtechniques for non-convex MINLP
TL;DR: An sBB software package named couenne (Convex Over- and Under-ENvelopes for Non-linear Estimation) is developed and used for extensive tests on several combinations of BT and branching techniques on a set of publicly available and real-world MINLP instances and is compared with a state-of-the-art MINLP solver.
Non-convex mixed-integer nonlinear programming: A survey
Samuel Burer,Adam N. Letchford +1 more
TL;DR: In this paper, the authors survey the literature on non-convex mixed-integer nonlinear programs, discussing applications, algorithms, and software, and special attention is paid to the case in which the objective and constraint functions are quadratic.
631
Branch-and-bound algorithms
TL;DR: A description of recent research advances in the design of B&B algorithms is presented, particularly with regards to the search strategy, the branching strategy, and the pruning rules.
527
Complete search in continuous global optimization and constraint satisfaction
TL;DR: This survey covers the state of the art of techniques for solving general-purpose constrained global optimization problems and continuous constraint satisfaction problems, with emphasis on complete techniques that provably find all solutions (if there are finitely many).
References
Practical Solution of Large Mixed Integer Programming Problems with Umpire
TL;DR: This paper discusses some branch and bound methods implemented in the UMPIRE mathematical programming system for solving practical integer programming problems and gives details of computational experience with these methods.
153
An Algorithm for Separable Nonconvex Programming Problems II: Nonconvex Constraints
TL;DR: In this article, a branch and bound algorithm was proposed to solve mathematical programming problems of the form: find x = (x1, …, xn) to minimize ∆φi0(xi) subject to x ∈ G, l ≦ x ≦ L and ∆ij(xi), j = 1, m. Each φij is assumed to be lower semicontinuous, possibly nonconvex, and G is considered to be closed.
104
Attempts to Calculate Global Solutions of Problems that May Have Local Minima
Garth P. McCormick
- 01 Dec 1971
TL;DR: In this article, the authors examined several global solutions for not necessarily convex programming problems with emphasis on the associated pitfalls, including penalty function methods, Lagrangian methods, grid methods, heuristic methods, random methods, and a branch and bound technique for separable programming problems.
28
An Algorithm for Separable Nonconvex Programming Problems
James E. Falk,Richard M. Soland +1 more
TL;DR: An algorithm for solving mathematical programming problems of the form Find x = x1,..., xn to minimize Σφixi subject to x ∈ G and l ≤ x ≤ L, which solves a sequence of problems in each of which the objective function is convex.
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