Tibor Csendes

Subdivision Direction Selection in Interval Methods for Global Optimization

This paper gives a short overview of the latest results on the role of the interval subdivision selection rule in branch-and-bound algorithms for global optimization. The class of rules that allow convergence for two slightly different model algorithms is characterized, and it is shown that the four rules investigated satisfy the conditions of convergence. An extensive numerical study with a wide spectrum of test problems indicates that there are substantial differences between the rules in terms of the required CPU time, the number of function and derivative evaluations and space complexity. Two of the rules can provide substantial improvements in efficiency.

ON THE SELECTION OF SUBDIVISION DIRECTIONS IN INTERVAL BRANCH-AND-BOUND METHODS FOR GLOBAL OPTIMIZATION

This paper investigates the influence of the interval subdivision selection rule on the convergence of interval branch-and-bound algorithms for global optimization. For the class of rules that allows convergence, we study the effects of the rules on a model algorithm with special list ordering. Four different rules are investigated in theory and in practice. A wide spectrum of test problems is used for numerical tests indicating that there are substantial differences between the rules with respect to the required CPU time, the number of function and derivative evaluations, and the necessary storage space. Two rules can provide considerable improvements in efficiency for our model algorithm.

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.

A New Verified Optimization Technique for the Packing Circles in a Unit Square Problems

This paper presents a new verified optimization method for the problem of finding the densest packings of nonoverlapping equal circles in a square. In order to provide reliable numerical results, the developed algorithm is based on interval analysis. As one of the most efficient parts of the algorithm, an interval-based version of a previous elimination procedure is introduced. This method represents the remaining areas still of interest as polygons fully calculated in a reliable way. Currently the most promising strategy of finding optimal circle packing configurations is to partition the original problem into subproblems. Still, as a result of the highly increasing number of subproblems, earlier computer-aided methods were not able to solve problem instances where the number of circles was greater than 27. The present paper provides a carefully developed technique resolving this difficulty by eliminating large groups of subproblems together. As a demonstration of the capabilities of the new algorithm the problems of packing 28, 29, and 30 circles were solved within very tight tolerance values. Our verified procedure decreased the uncertainty in the location of the optimal packings by more than 700 orders of magnitude in all cases.

New interval methods for constrained global optimization

Interval analysis is a powerful tool which allows to design branch-and-bound algorithms able to solve many global optimization problems. In this paper we present new adaptive multisection rules which enable the algorithm to choose the proper multisection type depending on simple heuristic decision rules. Moreover, for the selection of the next box to be subdivided, we investigate new criteria. Both the adaptive multisection and the subinterval selection rules seem to be specially suitable for being used in inequality constrained global optimization problems. The usefulness of these new techniques is shown by computational studies.

Global Optimization in Geometry — Circle Packing into the Square

The present review paper summarizes the research work done mostly by the authors on packing equal circles in the unit square in the last years.

Multisection in Interval Branch-and-Bound Methods for Global Optimization – I. Theoretical Results

We have investigated variants of interval branch-and-bound algorithms for global optimization where the bisection step was substituted by the subdivision of the current, actual interval into many subintervals in a single iteration step. The convergence properties of the multisplitting methods, an important class of multisection procedures are investigated in detail. We also studied theoretically the convergence improvements caused by multisection on algorithms which involve the accelerating tests (like e.g. the monotonicity test). The results are published in two papers, the second one contains the numerical test result.

A new multisection technique in interval methods for global optimization

Abstract A new multisection technique in interval methods for global optimization is investigated, and numerical tests demonstrate that the efficiency of the underlying global optimization method can be improved substantially. The heuristic rule is based on experiences that suggest the subdivision of the current subinterval into a larger number of pieces only if it is located in the neighbourhood of a minimizer point. An estimator of the proximity of a subinterval to the region of attraction to a minimizer point is utilized. According to the numerical study made, the new multisection strategies seem to be indispensable, and can improve both the computational and the memory complexity substantially.

New Subinterval Selection Criteria for Interval Global Optimization

AbstractThe theoretical convergence properties of interval global optimization algorithms that select the next subinterval to be subdivided according to a new class of interval selection criteria are investigated. The latter are based on variants of the RejectIndex: