Characterization of Positive Definite and Semide Finite Matrices via Quadratic Programming Duality
S.-P. Han, Olvi L. Mangasarian · SIAM Journal on Algebraic and Discrete Methods · 1984
Positive definite and semidefinite matrices induce well-known duality results in quadratic programming. The converse is established here. Thus if certain duality results hold for a pair of dual quadratic programs, then the underlying matrix must be positive definite or semidefinite. For example, if a strict local minimum of a quadratic program exceeds or equals a strict global maximum of the dual, then the underlying symmetric matrix Q is positive definite. If a quadratic program has a local minimum, then the underlying matrix Q is positive semidefinite if and only if the primal minimum exceeds or equals the dual global maximum and $x^T Qx = 0$ implies $Qx = 0$. A significant implication of these results is that the Wolfe dual may not be meaningful for nonconvex quadratic programs and for nonlinear programs without locally positive definite or semidefinite Hessians, even if the primal second order sufficient optimality conditions are satisfied.