exact duality in semidefinite programming based on elementary reformulations

• DOI: 10.17615/fk92-1x19
• Publication Date: 2015-01-01
• Authors: Minghui Liu; Gábor Pataki
• Keywords: Optimization and Control (math.OC); 0211 other engineering and technologies; FOS: Mathematics; 90C46 (Primary), 49N15, 52A40 (Secondary); 02 engineering and technology; 0101 mathematics; Mathematics - Optimization and Control; 01 natural sciences

To appear, SIAM Journal on Optimization<br/>In semidefinite programming (SDP), unlike in linear programming, Farkas' lemma may fail to prove infeasibility. Here we obtain an exact, short certificate of infeasibility in SDP by an elementary approach: we reformulate any semidefinite system of the form Ai*X = bi (i=1,...,m) (P) X >= 0 using only elementary row operations, and rotations. When (P) is infeasible, the reformulated system is trivially infeasible. When (P) is feasible, the reformulated system has strong duality with its Lagrange dual for all objective functions. As a corollary, we obtain algorithms to generate the constraints of {\em all} infeasible SDPs and the constraints of {\em all} feasible SDPs with a fixed rank maximal solution.<br/>