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ON CONVEX OPTIMIZATION WITHOUT CONVEX REPRESENTATION
, 2011
"... We consider the convex optimization problem P: minx{f(x): x ∈ K} where f is convex continuously differentiable, and K ⊂ Rn is a compact convex set with representation {x ∈ Rn: gj(x) ≥ 0, j = 1,...,m} for some continuously differentiable functions (gj). We discuss the case where the gj ’s are not ..."
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We consider the convex optimization problem P: minx{f(x): x ∈ K} where f is convex continuously differentiable, and K ⊂ Rn is a compact convex set with representation {x ∈ Rn: gj(x) ≥ 0, j = 1,...,m} for some continuously differentiable functions (gj). We discuss the case where the gj ’s are not all concave (in contrast with convex programming where they all are). In particular, even if the gj are not concave, we consider the logbarrier function φµ with parameter µ, associated with P, usually defined for concave functions (gj). We then show that any limit point of any sequence (xµ) ⊂ K of stationary points of φµ, µ → 0, is a KarushKuhnTucker point of problem P and a global minimizer of f on K.
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"... The phrase convex optimization refers to the minimization of a convex function over a convex set. However the feasible convex set need not be always described by convex inequalities. In this article we consider a convex feasible set which are described by inequality constraints which are locally Lip ..."
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The phrase convex optimization refers to the minimization of a convex function over a convex set. However the feasible convex set need not be always described by convex inequalities. In this article we consider a convex feasible set which are described by inequality constraints which are locally Lipschitz and not necessarily convex and need not be smooth. We show that if the Slater’s constraint qualification and a simple nondegeneracy condition is satisfied then the KarushKuhnTucker type optimality condition is both necessary and sufficient. 1