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

Nick Dimou

Published 2022-11-04Version 1

Strong duality for conic linear problems $(P)$ and $(D)$ generated by convex cones $S\subset X$, $T\subset Y$, a linear operator $A:X\rightarrow Y$ and a bilinear symmetric objective function $\langle\cdot,\cdot\rangle$, eventually comes down to the feasibility of the problems $min\langle z,z\rangle,\;z\in\{Ax-b:x\in S\}$ and $ min\langle w,w\rangle,\; w\in\{A^Ty-c:y\in T\}$. Under the latter condition, strong duality theorems as well as geometric and algebraic characterizations of optimal solutions are obtained via natural generalizations of the Farka's alternative lemma and related (finite real-space) linear programming theorems. Some applications of the main theory are discussed in the case of linear programming in complex space and some new results regarding special forms of continuous linear programming problems are given.