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Zbl 1046.90059
Jeyakumar, V.; Lee, G. M.; Dinh, N.
New sequential Lagrange multiplier conditions characterizing optimality without constraint qualification for convex programs.
(English)
[J] SIAM J. Optim. 14, No. 2, 534-547 (2003). ISSN 1052-6234; ISSN 1095-7189/e

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.
MSC 2000:
*90C25 Convex programming
52A41 Convex functions and convex programs (convex geometry)
26E15 Calculus of functions on infinite-dimensional spaces

Keywords: $\varepsilon$-subdifferential; sequential $\epsilon$-subgradient optimality conditions; necessary and sufficient conditions

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