Duality in Nonlinear Programming: A Simplified Applications-Oriented Development

TLDR

Nonlinear duality theory has few computational or theoretical applications relative to its theoretical literature over the past decade. The study aims to rework and extend convex duality theory to reduce obstacles to its practical application. The authors use a perturbation‑function approach in finite‑dimensional spaces, bypassing conjugate functions and fixed‑point theorems, to simplify proofs and apply the theory to convex systems. New results address previously neglected computational questions about solving programs via their duals.

Abstract

The number of computational or theoretical applications of nonlinear duality theory is small compared to the number of theoretical papers on this subject over the last decade. This study attempts to rework and extend the fundamental results of convex duality theory so as to diminish the existing obstacles to successful application. New results are also given having to do with important but usually neglected questions concerning the computational solution of a program via its dual. Several applications are made to the general theory of convex systems. The general approach is to exploit the powerful concept of a perturbation function, thus permitting simplified proofs (no conjugate functions or fixed-point theorems are needed) and useful geometric and mathematical insights. Consideration is limited to finite-dimensional spaces.An extended summary is given in the Introduction.