Duality theory and characterizations of optimal solutions for a class of conic linear problems

TL;DR

This paper develops duality theory and characterizations of optimal solutions for a class of conic linear problems using generalized Farkas' Lemma, with applications to linear programming and complex space problems.

Contribution

It introduces new duality results and solution characterizations for conic linear problems without closure conditions, extending classical Farkas' Lemma.

Findings

Strong duality theorems established for the class of problems.

Geometric and algebraic characterizations of optimal solutions derived.

Applications demonstrated in continuous linear programming and complex space LP.

Abstract

For a primal-dual pair of conic linear problems that are described by convex cones $S\subset X$, $T\subset Y$, bilinear symmetric objective functions $\langle\cdot,\cdot\rangle_X$, $\langle\cdot,\cdot\rangle_Y$ and a linear operator $A:X\rightarrow Y$, we show that the existence of optimal solutions $x^*\in S$, $y^*\in T$ that satisfy $Ax^*=b$ and $A^Ty^*=c$ eventually comes down to the consistency and solvability of the problems $min\langle z,z\rangle_Y,\;z\in\{Ax-b:x\in S\}$ and $ min\langle w,w\rangle_X,\; w\in\{A^Ty-c:y\in T\}$. Assuming that these two problems are consistent and solvable, strong duality theorems as well as geometric and algebraic characterizations of optimal solutions are obtained via natural generalizations of the Farkas' Lemma without a closure condition. Some applications of the main theory are discussed in the cases of continuous linear programming and linear programming in complex space.