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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 $\epsilon$-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.

90C25Convex programming
52A41Convex functions and convex programs (convex geometry)
26E15Calculus of real functions on infinite-dimensional spaces
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