Generic Optimality Conditions for Semialgebraic Convex Programs

Generic Optimality Conditions for Semialgebraic Convex Programs

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Article ID: iaor20113639
Volume: 36
Issue: 1
Start Page Number: 55
End Page Number: 70
Publication Date: Feb 2011
Journal: Mathematics of Operations Research
Authors: , ,
Abstract:

We consider linear optimization over a nonempty convex semialgebraic feasible region F. Semidefinite programming is an example. If F is compact, then for almost every linear objective there is a unique optimal solution, lying on a unique ‘active’ manifold, around which F is ‘partly smooth,’ and the second‐order sufficient conditions hold. Perturbing the objective results in smooth variation of the optimal solution. The active manifold consists, locally, of these perturbed optimal solutions; it is independent of the representation of F and is eventually identified by a variety of iterative algorithms such as proximal and projected gradient schemes. These results extend to unbounded sets F.

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