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Optimality conditions, approximate stationarity, and applications – a story beyond Lipschitzness. (English) Zbl 1494.49005

Summary: Approximate necessary optimality conditions in terms of Frechet subgradients and normals for a rather general optimization problem with a potentially non-Lipschitzian objective function are established with the aid of Ekeland’s variational principle, the fuzzy Frechet subdifferential sum rule, and a novel notion of lower semicontinuity relative to a set-valued mapping or set. Feasible points satisfying these optimality conditions are referred to as approximately stationary. As applications, we derive a new general version of the extremal principle. Furthermore, we study approximate stationarity conditions for an optimization problem with a composite objective function and geometric constraints, a qualification condition guaranteeing that approximately stationary points of such a problem are M-stationary, and a multiplier-penalty-method which naturally computes approximately stationary points of the underlying problem. Finally, necessary optimality conditions for an optimal control problem with a non-Lipschitzian sparsity-promoting term in the objective function are established.

MSC:

49J45 Methods involving semicontinuity and convergence; relaxation
49J50 Fréchet and Gateaux differentiability in optimization
49K27 Optimality conditions for problems in abstract spaces
90C30 Nonlinear programming
90C48 Programming in abstract spaces
58E30 Variational principles in infinite-dimensional spaces

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References:

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