On representations of the feasible set in convex optimization |
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Authors: | Jean Bernard Lasserre |
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Institution: | (3) Univ. Roma ‘La Sapienza’, DIS, Roma, Italy |
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Abstract: | We consider the convex optimization problem \({\min_{\mathbf{x}} \{f(\mathbf{x}): g_j(\mathbf{x})\leq 0, j=1,\ldots,m\}}\) where f is convex, the feasible set \({\mathbf{K}}\) is convex and Slater’s condition holds, but the functions g j ’s are not necessarily convex. We show that for any representation of \({\mathbf{K}}\) that satisfies a mild nondegeneracy assumption, every minimizer is a Karush-Kuhn-Tucker (KKT) point and conversely every KKT point is a minimizer. That is, the KKT optimality conditions are necessary and sufficient as in convex programming where one assumes that the g j ’s are convex. So in convex optimization, and as far as one is concerned with KKT points, what really matters is the geometry of \({\mathbf{K}}\) and not so much its representation. |
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