Necessary conditions for constrained optimization problems with semicontinuous and continuous data

Borwein, Jonathan; Treiman, Jay; Zhu, Qiji

doi:10.1090/S0002-9947-98-01984-9

Necessary conditions for constrained optimization problems with semicontinuous and continuous data
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by Jonathan M. Borwein, Jay S. Treiman and Qiji J. Zhu PDF

Trans. Amer. Math. Soc. 350 (1998), 2409-2429 Request permission

Abstract:

We consider nonsmooth constrained optimization problems with semicontinuous and continuous data in Banach space and derive necessary conditions without constraint qualification in terms of smooth subderivatives and normal cones. These results, in different versions, are set in reflexive and smooth Banach spaces.

References

Jonathan M. Borwein and Alexander Ioffe, Proximal analysis in smooth spaces, Set-Valued Anal. 4 (1996), no. 1, 1–24. MR 1384247, DOI 10.1007/BF00419371
J. M. Borwein and D. Preiss, A smooth variational principle with applications to subdifferentiability and to differentiability of convex functions, Trans. Amer. Math. Soc. 303 (1987), no. 2, 517–527. MR 902782, DOI 10.1090/S0002-9947-1987-0902782-7
J. M. Borwein and H. M. Strójwas, Proximal analysis and boundaries of closed sets in Banach space. I. Theory, Canad. J. Math. 38 (1986), no. 2, 431–452. MR 833578, DOI 10.4153/CJM-1986-022-4
J. M. Borwein and Q. J. Zhu, Variational Analysis in Non-reflexive Spaces and Applications to Control Problems with $L^1$ Perturbations, Nonlinear Analysis 28 (1997), 889–915.
J. M. Borwein and Q. J. Zhu, Viscosity solutions and viscosity subderivatives in smooth Banach spaces with applications to metric regularity, SIAM J. Control and Optimization 34 (1996), 1568–1591.
Frank H. Clarke, Optimization and nonsmooth analysis, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1983. A Wiley-Interscience Publication. MR 709590
Frank H. Clarke, Methods of dynamic and nonsmooth optimization, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 57, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1989. MR 1085948, DOI 10.1137/1.9781611970142
Robert Deville, Gilles Godefroy, and Václav Zizler, A smooth variational principle with applications to Hamilton-Jacobi equations in infinite dimensions, J. Funct. Anal. 111 (1993), no. 1, 197–212. MR 1200641, DOI 10.1006/jfan.1993.1009
Robert Deville, Gilles Godefroy, and Václav Zizler, Smoothness and renormings in Banach spaces, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 64, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1993. MR 1211634
Joseph Diestel, Geometry of Banach spaces—selected topics, Lecture Notes in Mathematics, Vol. 485, Springer-Verlag, Berlin-New York, 1975. MR 0461094
M. Fabián, J. H. M. Whitfield, and V. Zizler, Norms with locally Lipschitzian derivatives, Israel J. Math. 44 (1983), no. 3, 262–276. MR 693663, DOI 10.1007/BF02760975
A. D. Ioffe, Approximate subdifferentials and applications. III. The metric theory, Mathematika 36 (1989), no. 1, 1–38. MR 1014198, DOI 10.1112/S0025579300013541
A. D. Ioffe, Proximal analysis and approximate subdifferentials, J. London Math. Soc. (2) 41 (1990), no. 1, 175–192. MR 1063554, DOI 10.1112/jlms/s2-41.1.175
A. Ya. Kruger, Generalized differentials of nonsmooth functions and necessary conditions for an extremum, Sibirsk. Mat. Zh. 26 (1985), no. 3, 78–90, 224 (Russian). MR 792057
A. Ja. Kruger and B. Š. Morduhovič, Extremal points and the Euler equation in nonsmooth optimization problems, Dokl. Akad. Nauk BSSR 24 (1980), no. 8, 684–687, 763 (Russian, with English summary). MR 587714
Y. Li and S. Shi, A generalization of Ekeland’s $\varepsilon$-variational principle and of its Borwein-Preiss smooth variant, to appear in J. Math. Anal. Appl.
Philip D. Loewen, Optimal control via nonsmooth analysis, CRM Proceedings & Lecture Notes, vol. 2, American Mathematical Society, Providence, RI, 1993. MR 1232864, DOI 10.1090/crmp/002
B. Sh. Mordukhovich, Maximum principle in the problem of time optimal response with nonsmooth constraints, Prikl. Mat. Meh. 40 (1976), no. 6, 1014–1023 (Russian); English transl., J. Appl. Math. Mech. 40 (1976), no. 6, 960–969 (1977). MR 0487669, DOI 10.1016/0021-8928(76)90136-2
B. Š. Morduhovič, Metric approximations and necessary conditions for optimality for general classes of nonsmooth extremal problems, Dokl. Akad. Nauk SSSR 254 (1980), no. 5, 1072–1076 (Russian). MR 592682
B. Sh. Mordukhovich, Metody approksimatsiĭ v zadachakh optimizatsii i upravleniya, “Nauka”, Moscow, 1988 (Russian). MR 945143
Boris S. Mordukhovich and Yong Heng Shao, Nonsmooth sequential analysis in Asplund spaces, Trans. Amer. Math. Soc. 348 (1996), no. 4, 1235–1280. MR 1333396, DOI 10.1090/S0002-9947-96-01543-7
Robert R. Phelps, Convex functions, monotone operators and differentiability, Lecture Notes in Mathematics, vol. 1364, Springer-Verlag, Berlin, 1989. MR 984602, DOI 10.1007/BFb0089089
Gilles Pisier, Martingales with values in uniformly convex spaces, Israel J. Math. 20 (1975), no. 3-4, 326–350. MR 394135, DOI 10.1007/BF02760337
R. T. Rockafellar, Proximal subgradients, marginal values, and augmented Lagrangians in nonconvex optimization, Math. Oper. Res. 6 (1981), no. 3, 424–436. MR 629642, DOI 10.1287/moor.6.3.424
R. T. Rockafellar, Extensions of subgradient calculus with applications to optimization, Nonlinear Anal. 9 (1985), no. 7, 665–698. MR 796082, DOI 10.1016/0362-546X(85)90012-4
J. S. Treiman, A new characterization of Clarke’s tangent cone and its applications to subgradient analysis and optimization, Ph. D. Thesis, Univ. of Washington, 1983.
Jay S. Treiman, Clarke’s gradients and epsilon-subgradients in Banach spaces, Trans. Amer. Math. Soc. 294 (1986), no. 1, 65–78. MR 819935, DOI 10.1090/S0002-9947-1986-0819935-8
J. S. Treiman, Lagrange multipliers for nonconvex generalized gradients with equality, inequality and set constraints, preprint.
Q. J. Zhu, Calculus rules for subderivatives in smooth Banach spaces, preprint.