A simplified treatment of Ramana’s exact dual for semidefinite programming

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 (Math. Program. Ser. B 77, 129–162, 1997) 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.