A Simplified Treatment of Ramana's Exact Dual for Semidefinite Programming

TL;DR

This paper presents a simplified, elementary derivation of Ramana's extended dual for semidefinite programming, which guarantees no duality gap and attains optimality, enhancing understanding and application in complexity theory.

Contribution

It offers a concise, self-contained derivation of Ramana's dual using basic linear algebra, making the concept more accessible and easier to implement.

Findings

Provides a simplified derivation of Ramana's dual

Ensures duality gap is eliminated in semidefinite programming

Facilitates applications in complexity theory

Abstract

In semidefinite programming the dual may fail to attain its optimal value and there could be a duality gap, i.e., the primal and dual optimal values may differ. In a striking paper, Ramana proposed a polynomial size extended dual that does not have these deficiencies and yields a number of fundamental results in complexity theory. In this work we walk the reader through a concise and self-contained derivation of Ramana's dual, relying mostly on elementary linear algebra.