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CONSTRAINT QUALIFICATIONS FOR EXTENDED FARKAS’S LEMMAS AND LAGRANGIAN DUALITIES IN CONVEX INFINITE PROGRAMMING
"... Abstract. For an inequality system defined by a possibly infinite family of proper functions (not necessarily lower semicontinuous), we introduce some new notions of constraint qualifications in terms of the epigraphs of the conjugates of these functions. Under the new constraint qualifications, we ..."
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Abstract. For an inequality system defined by a possibly infinite family of proper functions (not necessarily lower semicontinuous), we introduce some new notions of constraint qualifications in terms of the epigraphs of the conjugates of these functions. Under the new constraint qualifications, we obtain characterizations of those reverseconvex inequalities which are consequence of the constrained system, and we provide necessary and/or sufficient conditions for a stable Farkas lemma to hold. Similarly, we provide characterizations for constrained minimization problems to have the strong or strong stable Lagrangian dualities. Several known results in the conic programming problem are extended and improved. Key words. convex inequality system, Farkas lemma, strong Lagrangian duality, conic programming
Stationarity and Regularity of Infinite Collections of Sets. Applications to Infinitely Constrained Optimization
 J OPTIM THEORY APPL
, 2011
"... This article continues the investigation of stationarity and regularity properties of infinite collections of sets in a Banach space started in Kruger and López: J Optim. Theory Appl. 154(2) (2012), and is mainly focused on the application of the stationarity criteria to infinitely constrained optim ..."
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This article continues the investigation of stationarity and regularity properties of infinite collections of sets in a Banach space started in Kruger and López: J Optim. Theory Appl. 154(2) (2012), and is mainly focused on the application of the stationarity criteria to infinitely constrained optimization problems. We consider several settings of optimization problems, which involve (explicitly or implicitly) infinite collections of sets and deduce for them necessary conditions characterizing stationarity in terms of dual space elements – normals and/or subdifferentials.
Robust Least Square Semidefinite Programming with Applications
, 2013
"... In this paper, we consider a least square semidefinite programming problem under ellipsoidal data uncertainty. We show that the robustification of this uncertain problem can be reformulated as a semidefinite linear programming problem with an additional secondorder cone constraint. We then provide ..."
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In this paper, we consider a least square semidefinite programming problem under ellipsoidal data uncertainty. We show that the robustification of this uncertain problem can be reformulated as a semidefinite linear programming problem with an additional secondorder cone constraint. We then provide an explicit quantitative sensitivity analysis on how the solution under the robustification depends on the size/shape of the ellipsoidal data uncertainty set. Next, we prove that, under suitable constraint qualifications, the reformulation has zero duality gap with its dual problem, even when the primal problem itself is infeasible. The dual problem is equivalent to minimizing a smooth objective function over the Cartesian product of secondorder cones and the Euclidean space, which is easy to project onto. Thus, we propose a simple variant of the spectral projected gradient method [7] to solve the dual problem. While it is wellknown that any accumulation point of the sequence generated from the algorithm is a dual optimal solution, we show in addition that the dual objective value along the sequence generated converges to a finite value if and only if the primal problem is feasible, again under suitable constraint qualifications. This latter fact leads to a simple certificate for primal
H.: About [q]regularity properties of collections of sets
 J. Math. Anal. Appl
, 1016
"... Abstract We examine three primal space local Hölder type regularity properties of finite collections of sets, namely, ..."
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Abstract We examine three primal space local Hölder type regularity properties of finite collections of sets, namely,
On extension of Fenchel duality and its application
 SIAM Journal on Optimization
, 2008
"... Abstract. By considering the epigraphs of conjugate functions, we extend the Fenchel duality, applicable to a (possibly infinite) family of proper lower semicontinuous convex functions on a Banach space. Applications are given in providing fuzzy KKT conditions for semiinfinite programming. Key word ..."
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Abstract. By considering the epigraphs of conjugate functions, we extend the Fenchel duality, applicable to a (possibly infinite) family of proper lower semicontinuous convex functions on a Banach space. Applications are given in providing fuzzy KKT conditions for semiinfinite programming. Key words. Fenchel duality, epigraph, KKT conditions, semiinfinite programming. AMS subject classifications. Primary, 90C34; 90C25 Secondary, 52A07; 41A29; 90C46
Penalty methods for constrained nonLipschitz optimization
, 2015
"... We consider a class of constrained optimization problems with a possibly nonconvex nonLipschitz objective and a convex feasible set being the intersection of a polyhedron and a possibly degenerated ellipsoid. Such a problem has a wide range of applications in data science, where the objective is us ..."
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We consider a class of constrained optimization problems with a possibly nonconvex nonLipschitz objective and a convex feasible set being the intersection of a polyhedron and a possibly degenerated ellipsoid. Such a problem has a wide range of applications in data science, where the objective is used for inducing sparsity in the solutions while the constraint set models the noise tolerance and incorporates other prior information for data fitting. To solve this kind of constrained optimization problems, a common approach is the penalty method. However, there is little theory on exact penalization for problems with nonconvex nonLipschitz objectives. In this paper, we study the existence of exact penalty parameters regarding local minimizers, stationary points and minimizers under suitable assumptions. Moreover, we discuss a penalty method whose subproblems are solved via a nonmonotone proximal gradient method with a suitable update scheme for the penalty parameters, and prove the convergence of the algorithm to a KKT point of the constrained problem. Preliminary numerical results demonstrate the efficiency of the penalty method for finding sparse solutions.