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Zbl 0376.90081
Mifflin, Robert
Semismooth and semiconvex functions in constrained optimization. (English)
[J] SIAM J. Control Optim. 15, 959-972 (1977). ISSN 0363-0129; ISSN 1095-7138/e

See also the booklet of the author with the same title [RR-76-21. Laxenburg, Austria (1976; Zbl 0364.90091)]. This paper introduces semismooth and semiconvex functions and discusses their properties with respect to nonsmooth nonconvex constrained optimization problems. These functions are locally Lipschitz, and hence have generalized gradients. In a recent paper [Math. Oper. Res. 2, 191--207 (1977; Zbl 0395.90069)] the author has given an optimization algorithm that uses generalized gradients of the problem functions and converges to stationary points if the functions are semismooth. If the functions are semiconvex and a constraint qualification is satisfied, then we show that a stationary point is an optimal point. The paper also shows that the pointwise maximum or minimum over a compact family of continuously differentiable functions is a semismooth function and that the pointwise maximum over a compact family of semiconvex functions is a semiconvex function. Furthermore, it is shown that a semismooth composition of semismooth functions is semismooth and a type of chain rule for generalized gradients is given.

Display scanned Zentralblatt-MATH page with this review.
[Robert Mifflin]

MSC 2000:: *90C26 Nonconvex programming
90C30 Nonlinear programming
49J52 Nonsmooth analysis (other weak concepts of optimality)

Citations: Zbl 0364.90091; Zbl 0395.90069

Cited in: Zbl 1133.65043 Zbl 1153.90560 Zbl 0635.49013 Zbl 0415.90073

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