Necessary and sufficient optimality conditions for a class of nonsmooth minimization problems Authors A. Ben-Tal Faculty of Industrial Engineering and Management Technion-Israel Institute of Technology J. Zowe Department of Mathematics University of Bayreuth Article

Received: 16 December 1980 Revised: 21 August 1981 DOI :
10.1007/BF01585095

Cite this article as: Ben-Tal, A. & Zowe, J. Mathematical Programming (1982) 24: 70. doi:10.1007/BF01585095
Abstract The purpose of this paper is to derive, in a unified way, second order necessary and sufficient optimality criteria, for four types of nonsmooth minimization problems: thediscrete minimax problem, thediscrete l _{1} -approximation, the minimization of theexact penalty function and the minimization of theclassical exterior penalty function. Our results correct and supplement conditions obtained by various authors in recent papers.

Key words Necessary and Sufficient Second Order Conditions Nonsmooth Optimization Minimax Problem l _{1} -ApproximationPenalty Functions Directional Derivatives Download to read the full article text

References [1]

A. Auslender, “Penalty methods for computing points that satisfy second order necessary conditions”,Mathematical Programming 17 (1979) 229–238.

[2]

A. Ben-Tal and J. Zowe, “A unified theory of first and second order conditions for extremum problems in topological vector spaces”,Mathematical Programming Study 19 (1982) 39–76.

[3]

A. Ben-Tal and J. Zowe, “Discretel
_{1} -approximation and related nonlinear nondifferentiable problems”, Preprint, Institute of Mathematics, University of Bayreuth, Bayreuth (1980).

[4]

C. Charalambous, “On the condition for optimality of the non-linearl
_{1} -problem”,Mathematical Programming 19 (1980) 178–185.

[5]

T.F. Coleman and A.R. Conn, “Second order conditions for an exact penalty function”,Mathematical Programming 19 (1980) 178–185.

[6]

J.M. Danskin,The theory of max-min (Springer, Berlin, 1967).

[7]

V.F. Dem'yanov and V.N. Malozemov,Introduction to minimax (Wiley, New York, 1974).

[8]

V.F. Dem'yanov and A.B. Pevnyi, “Expansion with respect to a parameter of the extremal values of game problems”,USSR Computational Mathematics and Mathematical Physics 14 (1974) 33–45.

[9]

R.J. Duffin, “Infinite programs”, in: H.W. Kuhn and A.W. Tucker, eds.,Linear inequalities and related systems (Princeton University Press, Princeton, NH, 1956) pp. 157–171.

[10]

R. Fletcher and G.A. Watson, “First and second order conditions for a class of nondifferentiable optimization problems”,Mathematical Programming 18 (1980) 291–307.

[11]

S.P. Han and O.L. Mangasarian, “Exact penalty functions in nonlinear programming”,Mathematical Programming 17 (1979) 251–269.

[12]

W. Krabs,Optimization and approximation (Wiley, New York, 1979).

[13]

H. Maurer and J. Zowe, “First and second order necessary and sufficient optimality conditions for infinite-dimensional programming problems”,Mathematical Programming 16 (1979) 98–110.

[14]

T. Pietrzykowski, “An exact penalty method for constrained maxima”,SIAM Journal on Numerical Analysis 6 (1969) 299–304.

© The Mathematical Programming Society, Inc. 1982