Historical Survey of Sequential Unconstrained Methods for Solving Constrained Minimization Problems

摘要

The paper gives a chronological account of significant developments in the evolution of unconstrained methods for solving mathematical programming (constrained extremization) problems. The particular methods of interest are based on the unconstrained minimization of a parametric auxiliary function for a specified sequence of values of the parameters. Under suitable conditions the minimizing sequence yields limit points that solve the original constrained problem. The general defining properties of such methods are sketched first. The common basic form of the auxiliary functions conventionally utilized is described and provides a formal synthesis of the so-called interior and exterior point 'penalty function' techniques. It is indicated in what sense the associated Lagrange multiplier technique and the conditions for optimality of the programming problem are closely associated with the considered approach. Certain key results are then briefly summarized and assessed--in the order of their appearance in the literature--in sufficient detail to indicate the degree of development and level of generality of methods based on particular auxiliary functions. A further synthesis of the sequential unconstrained penalty methods with the methods of centers is noted. The paper concludes with a brief summary of generalizations and extensions that have been obtained very recently. To recapitulate, in addition to giving a factual account of the progress made in this area, the paper endeavors to provide a perspective for certain significant developments in the approach, i.e., a basis for a synthesis of the principal techniques, and an appraisal of specific results. (Author)