Subdefinite Matrices and Quadratic Forms

Abstract

Recent results in quasiconvex and pseudoconvex programming [1], [3], [4] call the attention to these classes of functions. Little work has been done, however, in order to make easier the recognition of these types of functions. The definitions of quasiconvexity and pseudoconvexity do not offer a useful test from a practical point of view. In this paper one of the simplest problems of this kind is investigated, that of a quasiconvex, or pseudoconvex quadratic form. The connection of this problem with quadratic programming is obvious. The problem leads to a new class of real symmetric matrices, which we call subdefinite. After introducing the main definitions in the first section, we look for the characteristics of merely subdefinite (not semidefinite) matrices (? 2). The third section deals with the convexity properties of subdefinite quadratic forms. A few numerical examples are attached.