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Perturbation analysis of secondorder cone programming problems
 Mathematical Programming
, 2005
"... We discuss first and second order optimality conditions for nonlinear secondorder cone programming problems, and their relation with semidefinite programming problems. For doing this we extend in an abstract setting the notion of optimal partition. Then we state a characterization of strong regular ..."
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Cited by 15 (0 self)
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We discuss first and second order optimality conditions for nonlinear secondorder cone programming problems, and their relation with semidefinite programming problems. For doing this we extend in an abstract setting the notion of optimal partition. Then we state a characterization of strong regularity in terms of second order optimality conditions.
Strong Duality and Minimal Representations for Cone Optimization
, 2008
"... The elegant results for strong duality and strict complementarity for linear programming, LP, can fail for cone programming over nonpolyhedral cones. One can have: unattained optimal values; nonzero duality gaps; and no primaldual optimal pair that satisfies strict complementarity. This failure is ..."
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Cited by 14 (2 self)
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The elegant results for strong duality and strict complementarity for linear programming, LP, can fail for cone programming over nonpolyhedral cones. One can have: unattained optimal values; nonzero duality gaps; and no primaldual optimal pair that satisfies strict complementarity. This failure is tied to the nonclosure of sums of nonpolyhedral closed cones. We take a fresh look at known and new results for duality, optimality, constraint qualifications, and strict complementarity, for linear cone optimization problems in finite dimensions. These results include: weakest and universal constraint qualifications, CQs; duality and characterizations of optimality that hold without any CQ; geometry of nice and devious cones; the geometric relationships between zero duality gaps, strict complementarity, and the facial structure of cones; and, the connection between theory and empirical evidence for lack of a CQand failure of strict complementarity. One theme is the notion of minimal representation of the cone and the constraints in order to regularize the problem and avoid both the theoretical and numerical difficulties that arise due to (near) loss of a CQ. We include a discussion on obtaining these representations efficiently.
Preprocessing and Regularization for Degenerate Semidefinite Programs
, 2013
"... This paper presentsa backward stable preprocessing technique for (nearly) illposed semidefinite programming, SDP, problems, i.e., programs for which the Slater constraint qualification, existence of strictly feasible points, (nearly) fails. Current popular algorithms for semidefinite programming r ..."
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Cited by 3 (0 self)
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This paper presentsa backward stable preprocessing technique for (nearly) illposed semidefinite programming, SDP, problems, i.e., programs for which the Slater constraint qualification, existence of strictly feasible points, (nearly) fails. Current popular algorithms for semidefinite programming rely on primaldual interiorpoint, pd ip methods. These algorithms require the Slater constraint qualification for both the primal and dual problems. This assumption guarantees the existence of Lagrange multipliers, wellposedness of the problem, and stability of algorithms. However, there are many instances of SDPs where the Slater constraint qualification fails or nearly fails. Our backward stable preprocessing technique is based on applying the BorweinWolkowicz facial reduction process to find a finite number, k, of rankrevealing orthogonal rotations of the problem. After an appropriate truncation, this results in a smaller, wellposed, nearby problem that satisfies the Robinson constraint qualification, and one that can be solved by standard SDP solvers. The
A Variational Approach to Lagrange Multipliers
"... Abstract We discuss Lagrange multiplier rules from a variational perspective. This allows us to highlight many of the issues involved and also to illustrate how broadly an abstract version can be applied. ..."
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Abstract We discuss Lagrange multiplier rules from a variational perspective. This allows us to highlight many of the issues involved and also to illustrate how broadly an abstract version can be applied.