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A Conic Duality Frank–Wolfe-Type Theorem via Exact Penalization in Quadratic Optimization

Department of Statistics and Decision Support Systems, University of Vienna, A-1210 Vienna, Austria
Department of Statistics and Decision Support Systems, University of Vienna, A-1210 Vienna, Austria
werner.schachinger{at}univie.ac.at
immanuel.bomze{at}univie.ac.at

The famous Frank–Wolfe theorem ensures attainability of the optimal value for quadratic objective functions over a (possibly unbounded) polyhedron if the feasible values are bounded. This theorem does not hold in general for conic programs where linear constraints are replaced by more general convex constraints like positive semidefiniteness or copositivity conditions, despite the fact that the objective can be even linear. This paper studies exact penalizations of (classical) quadratic programs, i.e., optimization of quadratic functions over a polyhedron, and applies the results to establish a Frank–Wolfe-type theorem for the primal-dual pair of a class of conic programs that frequently arises in applications. One result is that uniqueness of the solution of the primal ensures dual attainability, i.e., existence of the solution of the dual.

Key Words: copositive programming; dual attainability
History: Received: February 28, 2007; revision received: March 25, 2008;