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Regularizing the abstract convex program
, 1981
"... Characterizations of optimality for the abstract convex program fi = inf ( p(x) : g(x) E 4, x E R 1, P) where S is an arbitrary convex cone in a finite dimensional space, R is a convex set, and p and g are respectively convex and Sconvex (on a), were given in [lo]. These characterizations hold with ..."
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Cited by 41 (9 self)
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Characterizations of optimality for the abstract convex program fi = inf ( p(x) : g(x) E 4, x E R 1, P) where S is an arbitrary convex cone in a finite dimensional space, R is a convex set, and p and g are respectively convex and Sconvex (on a), were given in [lo]. These characterizations hold without any constraint qualification. They use the “minimal cone ”.S ’ of (P) and the cone of directions of constancy D;(S’). In the faithfully convex case these cones can be used to regularize (P), i.e., transform (P) into an equivalent program (P,) for which Slater’s condition holds. We present an algorithm that finds both S ’ and Db(S’). The main step of the algorithm consists in solving a particular complementarity problem. We also present a characterization of optimality for (P) in terms of the cone of directions of constancy of a convex functional D & rather than DB(S’).
Presolving for Semidefinite Programs Without Constraint Qualifications
, 1998
"... Presolving for linear programming is an essential ingredient in many commercial packages. This step eliminates redundant constraints and identically zero variables, and it identifies possible infeasibility and unboundedness. In semidefinite programming, identically zero variables corresponds to lack ..."
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Cited by 5 (0 self)
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Presolving for linear programming is an essential ingredient in many commercial packages. This step eliminates redundant constraints and identically zero variables, and it identifies possible infeasibility and unboundedness. In semidefinite programming, identically zero variables corresponds to lack of a constraint qualification which can result in both theoretical and numerical difficulties. A nonzero duality gap can exist which nullifies the elegant and powerful duality theory. Small perturbations can result in infeasibility and/or large perturbations in solutions. Such problems fall into the class of illposed problems. It is interesting to note that classes of problems where constraint qualifications fail arise from semidefinite programming relaxations of hard combinatorial problems. We look at several such classes and present two approaches to find regularized solutions. Some preliminary numerical results are included.
Numerical Decomposition of a Convex Function 1
"... Abstract. Given the n xp orthogonal matrix A and the convex function f: R" ~ R, we find two orthogonal matrices P and Q such that f is almost constant on the convex hull of ± the columns of P, f is sufficiently nonconstant on the column space of Q, and the column spaces of P and Q provide an o ..."
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Abstract. Given the n xp orthogonal matrix A and the convex function f: R" ~ R, we find two orthogonal matrices P and Q such that f is almost constant on the convex hull of ± the columns of P, f is sufficiently nonconstant on the column space of Q, and the column spaces of P and Q provide an orthogonal direct sum decomposition of the column space of A. This provides a numerically stable algorithm for calculating the cone of directions of constancy, at a point x, of a convex function. Applications to convex programming are discussed. Key Words. Convex functions, convex programming, cone of directions of constancy, numerical stability. 1.