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 which 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.

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View A conic duality Frank--Wolfe type theorem via exact penalization in quadratic optimization