Immanuel M. Bomze

The Maximum Clique Problem

In this paper we present a survey of results concerning algorithms, complexity, and applications of the maximum clique problem. We discuss enumerative and exact algorithms, heuristics, and a variety of other proposed methods. An up to date bibliography on the maximum clique and related problems is also provided.

Solving Standard Quadratic Optimization Problems via Linear, Semidefinite and Copositive Programming

The problem of minimizing a (non-convex) quadratic function over the simplex (the standard quadratic optimization problem) has an exact convex reformulation as a copositive programming problem. In this paper we show how to approximate the optimal solution by approximating the cone of copositive matrices via systems of linear inequalities, and, more refined, linear matrix inequalities (LMIs). In particular, we show that our approach leads to a polynomial-time approximation scheme for the standard quadratic optimzation problem. This is an improvement on the previous complexity result by Nesterov who showed that a 2/3-approximation is always possible. Numerical examples from various applications are provided to illustrate our approach.

Evolution towards the Maximum Clique

As is well known, the problem of finding a maximum clique in a graph isNP-hard. Nevertheless, NP-hard problems may have easy instances. This paperproposes a new, global optimization algorithm which tries to exploit favourabledata constellations, focussing on the continuous problem formulation: maximizea quadratic form over the standard simplex. Some general connections of thelatter problem with dynamic principles of evolutionary game theory areestablished. As an immediate consequence, one obtains a procedure whichconsists (a) of an iterative part similar to interior-path methods based on theso-called replicator dynamics; and (b) a routine to escape from inefficient,locally optimal solutions. For the special case of finding a maximum clique ina graph where the quadratic form arises from a regularization of the adjacencematrix, part (b), i.e. escaping from maximal cliques not of maximal size, isaccomplished with block pivoting methods based on (large) independent sets,i.e. cliques of the complementary graph. A simulation study is included whichindicates that the resulting procedure indeed has some merits.

On Standard Quadratic Optimization Problems

A standard quadratic optimization problem (QP) consists of finding (global) maximizers of a quadratic form over the standard simplex. Standard QPs arise quite naturally in copositivity-based procedures which enable an escape from local solutions. Furthermore, several important applications yield optimization problems which can be cast into a standard QP in a straightforward way. As an example, a new continuous reformulation of the maximum weight clique problem in undirected graphs is presented which considerably improves previous attacks both as numerical stability and interpretation of the results are concerned. Apparently also for the first time, an equivalence between standard QPs and QPs on the positive orthant is established. Also, a recently presented global optimization procedure (GENF - genetical engineering via negative fitness) is shortly reviewed.

On Copositive Programming and Standard Quadratic Optimization Problems

A standard quadratic problem consists of finding global maximizers of a quadratic form over the standard simplex. In this paper, the usual semidefinite programming relaxation is strengthened by replacing the cone of positive semidefinite matrices by the cone of completely positive matrices (the positive semidefinite matrices which allow a factorization FFT where F is some non-negative matrix). The dual of this cone is the cone of copositive matrices (i.e., those matrices which yield a non-negative quadratic form on the positive orthant). This conic formulation allows us to employ primal-dual affine-scaling directions. Furthermore, these approaches are combined with an evolutionary dynamics algorithm which generates primal-feasible paths along which the objective is monotonically improved until a local solution is reached. In particular, the primal-dual affine scaling directions are used to escape from local maxima encountered during the evolutionary dynamics phase.

Lotka-Volterra equation and replicator dynamics: A two-dimensional classification

The replicator equation arises if one equips a certain game theoretical model for the evolution of behaviour in animal conflicts with dynamics. It serves to model many biological processes not only in sociobiology but also in population genetics, mathematical ecology and even in prebiotic evolution. After a short survey of these applications, a complete classification of the two-dimensional phase flows is presented. The methods are also used to obtain a classification of phase portraits of the well-known generalized Lotka-Volterra equation in the plane.

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.

Non-cooperative two-person games in biology: A classification

This article compares evolutionary equilibrium notions with solution concepts in rational game theory. Both static and dynamic evolutionary game theory are treated. The methods employed by dynamic theory, so-called “game dynamics”, could be discovered to be relevant for rational game theory also.

Copositive optimization — recent developments and applications

Due to its versatility, copositive optimization receives increasing interest in the Operational Research community, and is a rapidly expanding and fertile field of research. It is a special case of conic optimization, which consists of minimizing a linear function over a cone subject to linear constraints. The diversity of copositive formulations in different domains of optimization is impressive, since problem classes both in the continuous and discrete world, as well as both deterministic and stochastic models are covered. Copositivity appears in local and global optimality conditions for quadratic optimization, but can also yield tighter bounds for NP-hard combinatorial optimization problems. Here some of the recent success stories are told, along with principles, algorithms and applications.

Dynamical aspects of evolutionary stability

Selection is often viewed as a process that maximizes the average fitness of a population. However, there are often constraints even on the phenotypic level which may prevent fitness optimization. Consequently, in evolutionary game theory, models of frequency dependent selection are investigated, which focus on equilibrium states that are characterized by stability (or uninvadability) rather than by optimality. The aim of this article is to relate these stability notions with asymptotic stability in the so-called “replicator dynamics”, by generalizing results, which are well-known for elementary situations, to a fairly general setting applicable, e.g. to complex populations. Moreover, a purely dynamical characterization of evolutionary stability and uninvadability is presented.