A multi-start global minimization algorithm with dynamic search trajectories

A new multi-start algorithm for global unconstrained minimization is presented in which the search trajectories are derived from the equation of motion of a particle in a conservative force field, where the function to be minimized represents the potential energy. The trajectories are modified to increase the probability of convergence to a comparatively low local minimum, thus increasing the region of convergence of the global minimum. A Bayesian argument is adopted by which, under mild assumptions, the confidence level that the global minimum has been attained may be computed. When applied to standard and other test functions, the algorithm never failed to yield the global minimum.The first author wishes to thank Prof. M. Levitt of the Department of Chemical Physics of the Weizmann Institute of Science for suggesting this line of research and also Drs. T. B. Scheffler and E. A. Evangelidis for fruitful discussions regarding Conjecture 2.1. He also acknowledges the exchange agreement award received from the National Council for Research and Development in Israel and the Council for Scientific and Industrial Research in South Africa, which made possible the visit to the Weizmann Institute where this work was initiated.     

Computer-assisted multifactor optimization of selectivity in gas chromatography

Summary A computer-assisted method is presented for the simultaneous multifactor optimization (stationary phase loading, column temperature and carrier gas flow rate) of the analytical conditions for the optimum separation of multicomponent samples in gas chromatography. The optimization of the separation over the experimental region is based on a special polynomial from twelve preliminary experiments using the resolution as the selection criterion. Computer scanning technique was used for optimum selection in three dimensions. Excellent agreement was obtained between the predicted data and the experimental results.     

Stability of efficiency evaluations in data envelopment analysis

Efficiency evaluations in data envelopment analysis are shown to be stable for arbitrary perturbations in the convex hulls of input and output data. Also, the corresponding restricted Lagrange multiplier functions are shown to be continuous. The results are proved using point-to-set mappings and a particular region of stability from input optimization.Research partly supported by National Science Foundation Grants, Office of Naval Research Grant, and by the Natural Sciences and Engineering Council of Canada.     

Scalarization of vector optimization problems

In this paper, we investigate the scalar representation of vector optimization problems in close connection with monotonic functions. We show that it is possible to construct linear, convex, and quasiconvex representations for linear, convex, and quasiconvex vector problems, respectively. Moreover, for finding all the optimal solutions of a vector problem, it suffices to solve certain scalar representations only. The question of the continuous dependence of the solution set upon the initial vector problems and monotonic functions is also discussed.The author is grateful to the two referees for many valuable comments and suggestions which led to major imporvements of the paper.     

Stochastic global optimization methods part II: Multi level methods

In Part II of our paper, two stochastic methods for global optimization are described that, with probability 1, find all relevant local minima of the objective function with the smallest possible number of local searches. The computational performance of these methods is examined both analytically and empirically.     

Image space approach to penalty methods

In this paper, we introduce a unified framework for the study of penalty concepts by means of the separation functions in the image space (see Ref. 1). Moreover, we establish new results concerning a correspondence between the solutions of the constrained problem and the limit points of the unconstrained minima. Finally, we analyze some known classes of penalty functions and some known classical results about penalization, and we show that they can be derived from our results directly.     

Variational metric and exponential penalization

Attouch and Wets have introduced recently a variational metric between closed proper convex functions. The aim of this note is to give an estimation of this metric in the case of the exponential penalties. We can therefore recover some convergence results for the exponential penalty method.The authors would like to thank the referees for their suggestions.     

Indefinite preference structures and decision analysis

It always takes time for people to digest information and make judgments. The decision maker's preference is not always clear and stable when decision analysis and decision making are performed. In this paper, we introduce a generalized preference structure to cope with indefinite preferences. We describe its general properties, its implication on value function representation, its solution concepts, and methods for obtaining the solutions.This research has been partially supported by NSF Grant No. IST-84-18863. The authors are grateful to Dr. D. J. White for his helpful comments on a previous draft.     

Second-order numerical method for domain optimization problems

This paper is concerned with a second-order numerical method for shape optimization problems. The first variation and the second variation of the objective functional are derived. These variations are discretized by introducing a set of boundary-value problems in order to derive the second-order numerical method. The boundary-value problems are solved by the conventional finite-element method.The authors would like to express their thanks to Mr. T. Masanao, who was an undergraduate student, for his cooperation and comments. They also thank Professor Y. Sakawa of Osaka University for his encouragement.A part of this paper was presented at the IFIP Conference on Control of Boundaries and Stabilization, Clermont-Ferrand, France, 1988.     

Stochastic approximation algorithm for minimax problems

A stochastic approximation algorithm for minimax optimization problems is analyzed. At each iterate, it performs one random experiment, based on which it computes a direction vector. It is shown that, under suitable conditions, it a.s. converges to the set of points satisfying necessary optimality conditions. The algorithm and its analysis bring together ideas from stochastic approximation and nondifferentiable optimization.