×

Optimality conditions for max-type nonsmooth minimization problems. (English) Zbl 0892.49009

Consider a closed subset \(D\) of a Hilbert space \(X\), and a function \(f:X\to\mathbb{R}\) having the following structure \[ f(x)= \sum^m_{i=1} h_i(\max\{f_{ij}(x): j=1,\dots,n_i\}). \] The purpose of this paper is to establish first-order and second-order optimality conditions for the problem \[ \text{Minimize}\quad \{f(x): x\in D\}. \] The strategy followed by the authors is to reformulate this minimization problem as a standard nonlinear program with a finite number of equality and inequality constraints. Then they work out the general optimality conditions derived by S. Di [SIAM J. Optim. 6, No. 1, 178-197 (1996; Zbl 0846.49007)].
Reviewer: A.Seeger (Avignon)

MSC:

49J52 Nonsmooth analysis
90C30 Nonlinear programming

Citations:

Zbl 0846.49007
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Auslender A., Mathematical Programming 17 pp 229– (1979) · Zbl 0497.90061 · doi:10.1007/BF01588245
[2] Ben-Tal A., Mathematical Programming 24 pp 70– (1982) · Zbl 0488.90059 · doi:10.1007/BF01585095
[3] Charalambous C., Mathematical Programming 19 pp 178– (1980)
[4] Coleman T.F., Mathematical Programming 19 pp 178– (1980) · Zbl 0441.65053 · doi:10.1007/BF01581639
[5] Danskin J.M., The theory of max-min (1967) · Zbl 0154.20009
[6] Dem’yanov V.F., Introduction to minimax (1974)
[7] Di S., SIAM Journal on Optimization 6 pp 178– (1996) · doi:10.1137/0806010
[8] Di S., Journal of Optimization Theory and Applications 81 pp 469– (1994) · Zbl 0803.49019 · doi:10.1007/BF02193096
[9] Guignard M., SIAM Journal of Control 7 pp 232– (1969) · Zbl 0182.53101 · doi:10.1137/0307016
[10] Halkin H., SIAM Journal of Control 12 pp 229– (1974) · Zbl 0314.90087 · doi:10.1137/0312017
[11] Pietrzykowski T., SIAM Journal on Numerical Analysis 6 pp 299– (1969) · Zbl 0181.46501 · doi:10.1137/0706028
[12] Rockafellar R.T., Transactions of the American Mathematical Society 307 pp 75– (1988) · doi:10.1090/S0002-9947-1988-0936806-9
[13] Studniarski M., SIAM Journal of Control and Optimization 24 pp 1044– (1986) · Zbl 0604.49017 · doi:10.1137/0324061
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.