Abstract
This paper is concerned with first-order optimality conditions for nonsmooth extremal problems. Local approximations are obtained in terms of positively homogeneous functions representable as the sum of sublinear and superlinear functions or, equivalently, as the difference of two sublinear functions (d.s.l. functions). The resulting optimality conditions are expressed in the form of set inclusions. The idea of such approximations is exploited through the detailed study of d.s.l. functions and the cones corresponding to nonpositive values of d.s.l. functions.
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References
F.H. Clarke, “Generalized gradients and applications”, Transactions of the American Mathematical Society 205 (1975) 247–262.
F.H. Clarke, “A new approach to Lagrange multipliers”, Mathematics of Operations Research 1 (1976) 165–174.
F.H. Clarke, Optimization and nonsmooth analysis (Wiley & Sons, New York, 1983).
V.F. Demyanov and V.N. Malozemov, Introduction to minimax (Wiley & Sons, New York, 1974).
V.F. Demyanov and A.M. Rubinov, “On quasidifferentiable functionals”, Soviet Mathematics Doklady 21 (1980) 14–17.
V.F. Demyanov and L.N. Polyakova, “Minimization of a quasidifferentiable function on a quasidifferentiable set”, USSR Computational Mathematics and Mathematical Physics 20 (1980) 34–43.
V.F. Demyanov and L.V. Vasiliev, Nondifferentiable optimization (Nauka, Moscow, 1981) (in Russian).
V.F. Demyanov and A.M. Rubinov, “On some approaches to the nonsmooth optimization problem” (in Russian), Ekonomika i Matematicheskie Metody 17 (1981) 1153–1174.
V.F. Demyanov, “Quasidifferentiable functions: necessary conditions and descent directions”, Working paper WP-83-64, International Institute for Applied Systems Analysis (Laxenburg, Austria, 1983).
P. Hartman, “On functions representable as a difference of convex functions”, Pacific Journal of Mathematics 9 (1959) 707–713.
A.D. Ioffe, “Necessary and sufficient conditions for a local minimum. 1: A reduction theorem and first-order conditions”, SIAM Journal on Control and Optimization 17 (1979) 245–250.
L. N. Polyakova, “Necessary conditions for an extremum of quasidifferentiable functions”, Vestnik Leningrad University Mathematics 13 (1981) 241–247.
B.N. Polyakova, “Necessary conditions for an extremum of quasidifferentiable functions”, Marcel Dekker, New York, 1971).
H. Radström, “An embedding theorem for spaces of convex sets”, Proceedings of the American Mathematical Society 3 (1952) 165–169.
A.W. Roberts and D.E. Varberg, Convex functions (Academic Press, New York, 1973).
R.T. Rockafellar, Convex analysis (Princeton University Press, Princeton, NJ, 1970).
R.T. Rockafellar, “Favorable classes of Lipschitz-continuous functions in subgradient optimization”, in: E. Nurminski, ed., Progress in nondifferentiable optimization (International Institute for Applied Systems Analysis (Laxenburg, Austria, 1982).
A. Shapiro, “On optimality conditions in quasidifferentiable optimization”, SIAM Journal on Control and Optimization 22 (1984) 610–617.
Y. Yomdin, “On functions representable as a supremum of a family of smooth functions”, SIAM Journal on Mathematical Analysis 14 (1983) 239–246.
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© 1986 The Mathematical Programming Society, Inc.
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Shapiro, A. (1986). Quasidifferential calculus and first-order optimality conditions in nonsmooth optimization. In: Demyanov, V.F., Dixon, L.C.W. (eds) Quasidifferential Calculus. Mathematical Programming Studies, vol 29. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0121136
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DOI: https://doi.org/10.1007/BFb0121136
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