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New sequential Lagrange multiplier conditions characterizing optimality without constraint qualification for convex programs. (English) Zbl 1046.90059
Summary: In this paper a new sequential Lagrange multiplier condition characterizing optimality without a constraint qualification for an abstract nonsmooth convex program is presented in terms of the subdifferentials and the $ϵ$-subdifferentials. A sequential condition involving only the subdifferentials, but at nearby points to the minimizer for constraints, is also derived. For a smooth convex program, the sequential condition yields a limiting Kuhn-Tucker condition at nearby points without a constraint qualification. It is shown how the sequential conditions are related to the standard Lagrange multiplier condition. Applications to semidefinite programs, semi-infinite programs, and semiconvex programs are given. Several numerical examples are discussed to illustrate the significance of the sequential conditions.
MSC:
 90C25 Convex programming 52A41 Convex functions and convex programs (convex geometry) 26E15 Calculus of real functions on infinite-dimensional spaces