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

MathSciNet review:
1433112

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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.**Jonathan M. Borwein and Alexander Ioffe,*Proximal analysis in smooth spaces*, Set-Valued Anal.**4**(1996), no. 1, 1–24. MR**1384247**, 10.1007/BF00419371**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), no. 2, 517–527. MR**902782**, 10.1090/S0002-9947-1987-0902782-7**3.**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**, 10.4153/CJM-1986-022-4

J. M. Borwein and H. M. Strójwas,*Proximal analysis and boundaries of closed sets in Banach space. II. Applications*, Canad. J. Math.**39**(1987), no. 2, 428–472. MR**899844**, 10.4153/CJM-1987-019-4**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.**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****7.**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****8.**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**, 10.1006/jfan.1993.1009**9.**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****10.**Joseph Diestel,*Geometry of Banach spaces—selected topics*, Lecture Notes in Mathematics, Vol. 485, Springer-Verlag, Berlin-New York, 1975. MR**0461094****11.**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**, 10.1007/BF02760975**12.**A. D. Ioffe,*Approximate subdifferentials and applications. III. The metric theory*, Mathematika**36**(1989), no. 1, 1–38. MR**1014198**, 10.1112/S0025579300013541**13.**A. D. Ioffe,*Proximal analysis and approximate subdifferentials*, J. London Math. Soc. (2)**41**(1990), no. 1, 175–192. MR**1063554**, 10.1112/jlms/s2-41.1.175**14.**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****15.**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****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.**Philip D. Loewen,*Optimal control via nonsmooth analysis*, CRM Proceedings & Lecture Notes, vol. 2, American Mathematical Society, Providence, RI, 1993. MR**1232864****18.**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****19.**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****20.**B. Sh. Mordukhovich,*Metody approksimatsii v zadachakh optimizatsii i upravleniya*, “Nauka”, Moscow, 1988 (Russian). MR**945143****21.**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**, 10.1090/S0002-9947-96-01543-7**22.**Robert R. Phelps,*Convex functions, monotone operators and differentiability*, Lecture Notes in Mathematics, vol. 1364, Springer-Verlag, Berlin, 1989. MR**984602**

Robert R. Phelps,*Convex functions, monotone operators and differentiability*, 2nd ed., Lecture Notes in Mathematics, vol. 1364, Springer-Verlag, Berlin, 1993. MR**1238715****23.**Gilles Pisier,*Martingales with values in uniformly convex spaces*, Israel J. Math.**20**(1975), no. 3-4, 326–350. MR**0394135****24.**R. T. Rockafellar,*Proximal subgradients, marginal values, and augmented Lagrangians in nonconvex optimization*, Math. Oper. Res.**6**(1981), no. 3, 424–436. MR**629642**, 10.1287/moor.6.3.424**25.**R. T. Rockafellar,*Extensions of subgradient calculus with applications to optimization*, Nonlinear Anal.**9**(1985), no. 7, 665–698. MR**796082**, 10.1016/0362-546X(85)90012-4**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.**Jay S. Treiman,*Clarke’s gradients and epsilon-subgradients in Banach spaces*, Trans. Amer. Math. Soc.**294**(1986), no. 1, 65–78. MR**819935**, 10.1090/S0002-9947-1986-0819935-8**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.

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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