Necessary conditions

for constrained optimization problems

with semicontinuous and continuous data

Authors:
Jonathan M. Borwein, Jay S. Treiman and Qiji J. Zhu

Journal:
Trans. Amer. Math. Soc. **350** (1998), 2409-2429

MSC (1991):
Primary 49J52; Secondary 49J40, 49J50, 58C20

DOI:
https://doi.org/10.1090/S0002-9947-98-01984-9

MathSciNet review:
1433112

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

**1.**J. M. Borwein and A. Ioffe, Proximal analysis in smooth spaces,*Set-valued Analysis***4**(1996), 1-24. MR**96m:49028****2.**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), 517-527. MR**88k:49013****3.**J. M. Borwein and H. M. Strojwas, Proximal analysis and boundaries of closed sets in Banach spaces, Part 1: Theory,*Canad. J. Math.***38**(1986), 431-452. Part 2: Applications,*Canad. J. Math.***39**(1987), 428-472. MR**87h:90258**; MR**88f:46034****4.**J. M. Borwein and Q. J. Zhu, Variational Analysis in Non-reflexive Spaces and Applications to Control Problems with Perturbations,*Nonlinear Analysis***28**(1997), 889-915. CMP**97:05****5.**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. CMP**96:17****6.**F. H. Clarke,*Optimization and Nonsmooth Analysis*, John Wiley & Sons, Inc., New York 1983. MR**85m:49002****7.**F. H. Clarke,*Methods of dynamic and nonsmooth optimization*, CBMS-NSF Regional conference series in applied mathematics, SIAM, Philadelphia, 1989. MR**91j:49001****8.**R. Deville, G. Godefroy and V. Zizler, A smooth variational principle with applications to Hamilton-Jacobi equations in infinite dimensions,*J. Funct. Anal.***111**(1993), 197-212. MR**94b:49010****9.**R. Deville, G. Godefroy and V. Zizler,*Smoothness and renormings in Banach spaces*, Pitman Monographs and Surveys in Pure and Applied Mathematics, No. 64, John Wiley & Sons, Inc., New York 1993. MR**94d:46012****10.**J. Diestel,*Geometry of Banach spaces-Selected topics*, Lecture Notes in Math.,**485**, Springer-Verlag, 1975. MR**57:1079****11.**M. Fabian, J. H. Whitfield and V. Zizler, Norms with locally Lipschitz derivatives,*Israel J. Math.***44**(1983), 262-276. MR**84i:46028****12.**A. D. Ioffe, Approximate subdifferentials and applications, Part 3,*Mathematica***36**(1989), 1-38. MR**90g:49012****13.**A. D. Ioffe, Proximal analysis and approximate subdifferentials,*J. London Math. Soc.***41**(1990), 175-192. MR**91i:46045****14.**A. Y. Kruger, Generalized differentials of nonsmooth functions and necessary conditions for an extremum, Siberian Math. J.**26**(1985), 370-379. MR**86j:49038****15.**A. Y. Kruger and B. S. Mordukhovich, Extremal points and the Euler equation in nonsmooth optimization, Dokl. Akad. Nauk BSSR**24**(1980), 684-687. MR**82b:90127****16.**Y. Li and S. Shi, A generalization of Ekeland's -variational principle and of its Borwein-Preiss smooth variant, to appear in*J. Math. Anal. Appl.***17.**P. D. Loewen,*Optimal Control via Nonsmooth Analysis*, CRM Lecture Notes Series, Amer. Math. Soc., Summer School on Control, CRM, Université de Montréal, (1992), Amer. Math. Soc., Providence, 1993. MR**94h:49003****18.**B. S. Mordukhovich, Maximum principle in problems of time optimal control with nonsmooth constraints,*J. Appl. Math. Mech.***40**(1976), 960-969. MR**58:7284****19.**B. S. Mordukhovich, Metric approximations and necessary optimality conditions for general classes of nonsmooth extremal problems, Soviet Math. Dokl.**22**(1980), 526-530. MR**82b:90104****20.**B. S. Mordukhovich,*``Approximation methods in problems of optimization and control''*, Nauka, Moscow, 1988. [English transl., Wiley/Interscience, to appear]. MR**89m:49001****21.**B. S. Mordukhovich and Y. Shao, Nonsmooth sequential analysis in Asplund spaces,*Trans. Amer. Math. Soc.***348**(1996), 1235-1280. MR**96h:49036****22.**R. R. Phelps,*Convex functions, monotone operators and differentiability*, Lecture Notes in Mathematics, No. 1364, Springer Verlag, N.Y., Berlin, Tokyo, 1988, Second Edition 1993. MR**90g:46063**; MR**94f:46055****23.**G. Pisier, Martingales with values in uniformly convex spaces,*Israel J. Math.***20**(1975), 326-350. MR**52:14940****24.**R. T. Rockafellar, Proximal subgradients, marginal values and augmented Lagrangians in nonconvex optimization,*Math. Oper. Res.***6**(1981), 424-436. MR**83m:90088****25.**R. T. Rockafellar, Extensions of subgradient calculus with applications to optimization,*Nonlinear Analysis, TMA***9**(1985), 665-698. MR**87a:90148****26.**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.**27.**J. S. Treiman, Clarke's gradients and epsilon-subgradients in Banach spaces,*Trans. Amer. Math. Soc.***294**(1986), 65-78. MR**87d:90188****28.**J. S. Treiman, Lagrange multipliers for nonconvex generalized gradients with equality, inequality and set constraints, preprint.**29.**Q. J. Zhu, Calculus rules for subderivatives in smooth Banach spaces, preprint.

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (1991):
49J52,
49J40,
49J50,
58C20

Retrieve articles in all journals with MSC (1991): 49J52, 49J40, 49J50, 58C20

Additional Information

**Jonathan M. Borwein**

Affiliation:
Department of Mathematics and Statistics, Simon Fraser University, Burnaby, BC V5A 1S6, Canada

Email:
jborwein@cecm.sfu.ca

**Jay S. Treiman**

Affiliation:
Department of Mathematics and Statistics, Western Michigan University, Kalamazoo, Michigan 49008

Email:
treiman@math-stat.wmich.edu

**Qiji J. Zhu**

Affiliation:
Department of Mathematics and Statistics, Western Michigan University, Kalamazoo, Michigan 49008

Email:
zhu@math-stat.wmich.edu

DOI:
https://doi.org/10.1090/S0002-9947-98-01984-9

Keywords:
Constrained optimization problems,
multipliers,
nonsmooth analysis,
subderivatives,
normals,
fuzzy calculus

Received by editor(s):
September 14, 1995

Received by editor(s) in revised form:
August 9, 1996

Additional Notes:
Research of the first author was supported by NSERC and by the Shrum Endowment at Simon Fraser University

Article copyright:
© Copyright 1998
American Mathematical Society