A finite algorithm to maximize certain pseudoconcave functions on polytopes

Abstract

This paper develops and proves an algorithm that finds the exact maximum of certain nonlinear functions on polytopes by performing a finite number of logical and arithmetic operations. Permissible objective functions need to be pseudoconcave and allow the closed-form solution of sets of equations$\partial f(Dy + \hat x^k )/\partial y = 0$, which are first order conditions associated with the unconstrained, but affinely transformed objective function. Examples are pseudoconcave quadratics and especially the homogeneous functioncx +m(xVx)^1/2,m < 0, V positive definite, for which sofar no finite algorithm existed.

In distinction to most available methods, this algorithm uses the internal representation [6]|of the feasible set to selectively decompose it into simplices of varying dimensions; linear programming and a gradient criterion are used to select a sequence of these simplices, which contain a corresponding sequence of strictly increasing, relative and relatively interior maxima, the greatest of which is shown to be the global maximum on the feasible set. To find the interior maxima on these simplices in a finite way, calculus maximizations on the affine hulls of subsets of their vertices are necessary; thus the above requirement that$\partial f(Dy + \hat x^k )/\partial y = 0$ be explicitly solvable.

The paper presents a flow structure of the algorithm, its supporting theory, its decision-theoretic use, and an example, computed by an APL-version of the method.