Minimal Representation of Convex Regions Defined by Analytic Functions

Abstract

In this paper we are concerned with characterizing minimal representation of feasible regions defined by both linear and convex analytic constraints. We say that a representation is minimal if every other representation has either more analytic Ž . nonlinear constraints, or has the same number of analytic constraints and at least as many linear constraints. We prove necessary and sufficient conditions for the representation to be minimal. These are expressed in terms of the redundant constraints, pseudo-analytic constraints, and implicit equality constraints. In order to prove the minimal representation theorem, we present results on the facets of the convex regions defined by analytic constraints. Finally, we outline the steps of the procedure that could be used to determine a minimal representation.

On Infeasibility of Systems of Convex Analytic Inequalities

“…IIn the proof of this theorem we will use the results stated in the LEMMA 3.3[20]. Constraint kth is an implicit equality in the Jystem (1.11, following two lemmas.i f and only i fProoj Suppose first that Vx,, x 2 E 9, Therefore f k ( x 2 ) = Fk( B,x, + c,) + ( a k , x l > -d, = fk( xl) = 0, so that constraint k is an implicit equality.…”