Fabio Schoen

A Population-based Approach for Hard Global Optimization Problems based on Dissimilarity Measures

When dealing with extremely hard global optimization problems, i.e. problems with a large number of variables and a huge number of local optima, heuristic procedures are the only possible choice. In this situation, lacking any possibility of guaranteeing global optimality for most problem instances, it is quite difficult to establish rules for discriminating among different algorithms. We think that in order to judge the quality of new global optimization methods, different criteria might be adopted like, e.g.: 1.efficiency – measured in terms of the computational effort necessary to obtain the putative global optimum2.robustness – measured in terms of “percentage of successes”, i.e. of the number of times the algorithm, re-started with different seeds or starting points, is able to end up at the putative global optimum3.discovery capability – measured in terms of the possibility that an algorithm discovers, for the first time, a putative optimum for a given problem which is better than the best known up to now. Of course the third criterion cannot be considered as a compulsory one, as it might be the case that, for a given problem, the best known putative global optimum is indeed the global one, so that no algorithm will ever be able to discover a better one. In this paper we present a computational framework based on a population-based stochastic method in which different candidate solutions for a single problem are maintained in a population which evolves in such a way as to guarantee a sufficient diversity among solutions. This diversity enforcement is obtained through the definition of a dissimilarity measure whose definition is dependent on the specific problem class. We show in the paper that, for some well known and particularly hard test classes, the proposed method satisfies the above criteria, in that it is both much more efficient and robust when compared with other published approaches. Moreover, for the very hard problem of determining the minimum energy conformation of a cluster of particles which interact through short-range Morse potential, our approach was able to discover four new putative optima.

Global Optimization: Theory, Algorithms, and Applications

This volume contains a thorough overview of the rapidly growing field of global optimization, with chapters on key topics such as complexity, heuristic methods, derivation of lower bounds for minimization problems, and branch-and-bound methods and convergence. The final chapter offers both benchmark test problems and applications of global optimization, such as finding the conformation of a molecule or planning an optimal trajectory for interplanetary space travel. An appendix provides fundamental information on convex and concave functions. Audience: Global Optimization is intended for Ph.D. students, researchers, and practitioners looking for advanced solution methods to difficult optimization problems. It can be used as a supplementary text in an advanced graduate-level seminar. Contents: Chapter 1: Introduction; Chapter 2: Complexity; Chapter 3: Heuristics; Chapter 4: Lower Bounds; Chapter 5: Branch and Bound; Chapter 6: Problems; Appendix A: Basic Definitions and Results on Convexity; Appendix B: Notation.

Solving the problem of packing equal and unequal circles in a circular container

In this paper we propose a Monotonic Basin Hopping approach and its population-based variant Population Basin Hopping to solve the problem of packing equal and unequal circles within a circular container with minimum radius. Extensive computational experiments have been performed both to analyze the problem at hand, and to choose in an appropriate way the parameter values for the proposed methods. Different improvements with respect to the best results reported in the literature have been detected.

Fast Global Optimization of Difficult Lennard-Jones Clusters

The minimization of the potential energy function of Lennard-Jones atomic clusters has attracted much theoretical as well as computational research in recent years. One reason for this is the practical importance of discovering low energy configurations of clusters of atoms, in view of applications and extensions to molecular conformation research; another reason of the success of Lennard Jones minimization in the global optimization literature is the fact that this is an extremely easy-to-state problem, yet it poses enormous difficulties for any unbiased global optimization algorithm.In this paper we propose a computational strategy which allowed us to rediscover most putative global optima known in the literature for clusters of up to 80 atoms and for other larger clusters, including the most difficult cluster conformations. The main feature of the proposed approach is the definition of a special purpose local optimization procedure aimed at enlarging the region of attraction of the best atomic configurations. This effect is attained by performing first an optimization of a modified potential function and using the resulting local optimum as a starting point for local optimization of the Lennard Jones potential.Extensive numerical experimentation is presented and discussed, from which it can be immediately inferred that the approach presented in this paper is extremely efficient when applied to the most challenging cluster conformations. Some attempts have also been carried out on larger clusters, which resulted in the discovery of the difficult optimum for the 102 atom cluster and for the very recently discovered new putative optimum for the 98 atom cluster.

Local optima smoothing for global optimization

It is widely believed that in order to solve large-scale global optimization problems, an appropriate mixture of local approximation and global exploration is necessary. Local approximation, if first-order information on the objective function is available, is efficiently performed by means of local optimization methods. Unfortunately, global exploration, in absence of some kind of global information on the problem, is a ‘blind’ procedure, aimed at placing observations as evenly as possible in the search domain. Often, this procedure reduces to uniform random sampling (like in Multistart algorithms or in techniques based on clustering). In this paper, we propose a new framework for global exploration which tries to guide random exploration towards the region of attraction of low-level local optima. The main idea originated by the use of smoothing techniques (based on Gaussian convolutions): the possibility of applying a smoothing transformation not to the objective function but to the result of local searches seems to have never been explored yet. Although an exact smoothing of the results of local searches is impossible to implement, in this paper we propose a computational approximation scheme which has proven to be very efficient and (maybe more important) extremely robust in solving large-scale global optimization problems with huge numbers of local optima and, in particular, for problems displaying a ‘funnel’ structure.

Efficient Algorithms for Large Scale Global Optimization: Lennard-Jones Clusters

A stochastic global optimization method is applied to the challenging problem of finding the minimum energy conformation of a cluster of identical atoms interacting through the Lennard-Jones potential. The method proposed incorporates within an already existing and quite successful method, monotonic basin hopping, a two-phase local search procedure which is capable of significantly enlarging the basin of attraction of the global optimum. The experiments reported confirm the considerable advantages of this approach, in particular for all those cases which are considered in the literature as the most challenging ones, namely 75, 98, 102 atoms. While being capable of discovering all putative global optima in the range considered, the method proposed improves by more than two orders of magnitude the speed and the percentage of success in finding the global optima of clusters of 75, 98, 102 atoms.

Disk Packing in a Square: A New Global Optimization Approach

We present a new computational approach to the problem of placing n identical nonoverlapping disks in the unit square in such a way that their radii are maximized. The problem has been studied in a large number of papers, from both a theoretical and a computational point of view. In this paper, we conjecture that the problem possesses a so-called funneling landscape, a feature that is commonly found in molecular conformation problems. Based on this conjecture, we develop a stochastic search algorithm that displays excellent numerical performance. Thanks to this algorithm, we could improve over previously known putative optima in the range n ≤ 130 in as many as 32 instances, the smallest of which is n = 53.

Efficiently packing unequal disks in a circle

Placing non-overlapping circles in a smallest container is a hard task. In this paper we present our strategy for optimally placing circles in a smallest circle which led us to win an international competition by properly mixing local and global optimization strategies with random search and local moves.

Random Linkage: a family of acceptance/rejection algorithms for global optimisation

This paper introduces a new, infinite, class of stochastic algorithms for the optimization of multimodal functions defined over a compact set. The main idea is that of generalizing some well known methods inspired by clustering techniques which make use of local search routines started at selected points in a random sample. It is shown that there exist interesting classes of global optimization algorithms all possessing the following theoretical properties: convergence to the global optimum with probability 1; detection of the global optimum in a finite number of iterations w.p.1; probability of starting a local search decreasing to 0; total number of local searches performed even if the algorithm is run forever, finite w.p.1.

Solving molecular distance geometry problems by global optimization algorithms

Abstract In this paper we consider global optimization algorithms based on multiple local searches for the Molecular Distance Geometry Problem (MDGP). Three distinct approaches (Multistart, Monotonic Basin Hopping, Population Basin Hopping) are presented and for each of them a computational analysis is performed. The results are also compared with those of two other approaches in the literature, the DGSOL approach (Moré, Wu in J. Glob. Optim. 15:219–234, 1999) and a SDP based approach (Biswas et al. in An SDP based approach for anchor-free 3D graph realization, Technical Report, Operations Research, Stanford University, 2005).