Reiner Horst

Handbook of Global Optimization

Preface. 1. Tight relaxations for nonconvex optimization problems using the Reformulation-Linearization/Convexification Technique (RLT) H.D. Sherali. 2. Exact algorithms for global optimization of mixed-integer nonlinear programs M. Tawarmalani, N.V. Sahinidis. 3. Algorithms for global optimization and discrete problems based on methods for local optimization W. Murray, Kien-Ming Ng. 4. An introduction to dynamical search L. Pronzato, et al. 5. Two-phase methods for global optimization F. Schoen. 6. Simulated annealing algorithms for continuous global optimization M. Locatelli. 7. Stochastic Adaptive Search G.R. Wood, Z.B. Zabinsky. 8. Implementation of Stochastic Adaptive Search with Hit-and-Run as a generator Z.B. Zabinsky, G.R. Wood. 9. Genetic algorithms J.E. Smith. 10. Dataflow learning in coupled lattices: an application to artificial neural networks J.C. Principe, et al. 11. Taboo Search: an approach to the multiple-minima problem for continuous functions D. Cvijovic, J. Klinowski. 12. Recent advances in the direct methods of X-ray crystallography H.A. Hauptman. 13. Deformation methods of global optimization in chemistry and physics L. Piela. 14. Experimental analysis of algorithms C.C. McGeoch. 15. Global optimization: software, test problems, and applications J.D. Pinter.

DC programming: overview

Mathematical programming problems dealing with functions, each of which can be represented as a difference of two convex functions, are called DC programming problems. The purpose of this overview is to discuss main theoretical results, some applications, and solution methods for this interesting and important class of programming problems. Some modifications and new results on the optimality conditions and development of algorithms are also presented.

An algorithm for nonconvex programming problems

Branch and bound approaches for nonconvex programming problems had been given in [1] and [4]. Crucial for both are the use of rectangular partitions, convex envelopes and separable nonconvex portions of the objective function and constraints. We want to propose a similar algorithm which solves a sequence of problems in each of which the objective function is convex or even linear. The main difference between this approach and previous approaches is the use of general compact partitions instead of rectangular ones and a different refining rule such that the algorithm does not rely on the concept of convex envelopes and handles non-separable functions.First we describe a general algorithm and prove a convergence theorem under suitable regularity assumptions. Then we give as example an algorithm for concave minimization problems.

Developments in Global Optimization

Preface. 1. NOP - A Compact Input Format for Nonlinear Optimization Problems A. Neumaier. 2. GLOPT - A Program for Constrained Global Optimization S. Dallwig, et al. 3. Global Optimization for Imprecise Problems M.N. Vrahatis, et al. 4. New Results on Gap-Treating Techniques in Extended Interval Newton Gauss-Seidel Steps for Global Optimization D. Ratz. 5. Quadratic Programming with Box Constraints P.L. De Angelis. 6. Evolutionary Approach to The Maximum Clique Problem: Empirical Evidence on a Larger Scale I. Bomze, et al. 7. Interval and Bounding Hessians C. Stephens. 8. On Global Search for Non-Convex Optimal Control Problems A. Strekalovsky, I. Vasiliev. 9. A Multistart Linkage Algorithm Using First Derivatives C.J. Price. 10. Convergence Speed of an Integral Method for Computing the Essential Supremum J. Hichert, et al. 11. Complexity Analysis Integrating Pure Adaptive Search (PAS) and Pure Random Search (PRS) Z.B. Zabinsky, B.P. Kristinsdottir. 12. LGO - A Program System for Continuous and Lipschitz Global Optimization J.D. Pinter. 13. A Method Using Local Tuning For Minimizing Functions with Lipschitz Derivatives Ya.D. Sergeyev. 14. Molecular Structure Prediction by Global Optimization K.A. Dill, et al. 15. Optimal Renewal Policy for Slowly Degrading Systems A. Pfening, M. Telek. 16. Numerical Prediction of Crystal Structures by Simulated Annealing W. Bollweg, et al. 17. Multidimensional Optimization in Image Reconstruction from Projections I.Garcia, et al. 18. Greedy Randomized Adaptive Search for a Location Problem with Economies of Scale K. Holmqvist, et al. 19. An Algorithm for Improving the Bounding Procedure in Solving Process Network Synthesis by a B&B Method B. Imreh, et al.

On finding new vertices and redundant constraints in cutting plane algorithms for global optimization

Cutting plane algorithms belong to the basic tools for solving certain classes of global multiextremal optimization problems such as the global minimization of a concave function on a compact, convex set. The computationally most expensive part of these algorithms consists in the calculation of all new vertices of the polytope P created from a given polytope P (with known vertex set) by a cut. We first present a new procedure for solving this subproblem that allows to handle degeneracy and give a theoretical and numerical comparison with existing approaches. Then we show how redundant constraints can be eliminated by a criterion based on the known vertex set of a polytope.

Convergence and restart in branch-and-bound algorithms for global optimization. application to concave minimization and d.c. optimization problems

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.

Global optimization of concave functions subject to quadratic constraints: an application in nonlinear bilevel programming

When the followers optimality conditions are both necessary and sufficient, the nonlinear bilevel program can be solved as a global optimization problem. The complementary slackness condition is usually the complicating constraint in such problems. We show how this constraint can be replaced by an equivalent system of convex and separable quadratic constraints. In this paper, we propose different methods for finding the global minimum of a concave function subject to quadratic separable constraints. The first method is of the branch and bound type, and is based on rectangular partitions to obtain upper and lower bounds. Convergence of the proposed algorithm is also proved. For computational purposes, different procedures that accelerate the convergence of the proposed algorithm are analysed. The second method is based on piecewise linear approximations of the constraint functions. When the constraints are convex, the problem is reduced to global concave minimization subject to linear constraints. In the case of non-convex constraints, we use zero-one integer variables to linearize the constraints. The number of integer variables depends only on the concave parts of the constraint functions.

A new simplicial cover technique in constrained global optimization

A simplicial branch and bound-outer approximation technique for solving nonseparable, nonlinearly constrained concave minimization problems is proposed which uses a new simplicial cover rather than classical simplicial partitions. Some geometric properties and convergence results are demonstrated. A report on numerical aspects and experiments is given which shows that the most promising variant of the cover technique can be expected to be more efficient than comparable previous simplicial procedures.

Concave minimization via conical partitions and polyhedral outer approximation

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.

On solving a D.C. programming problem by a sequence of linear programs

We are dealing with a numerical method for solving the problem of minimizing a difference of two convex functions (a d.c. function) over a closed convex set in ℝn. This algorithm combines a new prismatic branch and bound technique with polyhedral outer approximation in such a way that only linear programming problems have to be solved.