Gradient States for Some Dualities with the C2 Extremal Principle ††This research was partly supported by Project NR047-021, CNR Contract N00014-75-C-0569 with the Center for Cybernetic Studies, The University of Texas.

Abstract

Charnes and Cooper developed the duality state characterizations for constrained problems involving the information theoretic functional in the primal and an unconstrained functional in the dual problem. The unconstrained problem involved only exponential and linear functions. This chapter discusses functions that are convex and differentiable on an open convex set in finite dimensional Euclidean space. It reviews the influence of properties of the range of the gradient mapping and of the decoupling equality system on the possible duality states for the dual problem obtained via the extremal principle for dualities. The C 2 extremal principle for dualities is an approach to derive dual optimization problems with proper duality inequality that simplifies and generalizes the Fenchel–Rockafellar scheme. The derivation is accomplished in two stages. The first is the achievement of the duality inequality. The second is the decoupling of the primal and dual variables.