Optimization Methods Based On Projected Variable Metric Search

As a first step to the realization of a new computer program to solve general nonlinear optimization problems, as a possible replacement of M.A.P. (Method of approximate programming, see Griffith and Stewart P[21, we have developed a computer code which minimizes a nonlinear objective function subjected to a set of linear equality and/or inequality constraints. The method we have chosen is a generalization to linearly constrained problems of the variable metric technique upon which the well known algorithms for unconstrained optimization of Davidon [2], Fletcher and Powell [4], NmAlen[1] and many others are based. This choice was based on the fact that quasi Newton (=variable metric) techniques compared very favorably with the optimization rethods used in the past. We therefore expect the new algorithm to be faster and more robust than the algorithms dealing with uncorrected gradient information. Part I of this report describes the mathematics and theoretical backgrounds behind our new linearly constrained optimization code, which we have called : VLICO ( a Variable metric method for Linearly Constrained Optimization.) In part II we discuss the extension of the linearly constrained optimization code VLICO, to the case of the general nonlinear programming problem. We have used for this extension the two phaseynethod described by J.B. Rosen [22]. The resulting algorithm has been implemented in a computer program called VANOP (VAriable metric Nonlinear OPtimization) which has shown a fast and robust convergence behaviour on a broad class of test problems, and therefore may be a possible replacement for the MAP code which is now often used.