Hostname: page-component-8448b6f56d-t5pn6 Total loading time: 0 Render date: 2024-04-17T20:41:11.026Z Has data issue: false hasContentIssue false

Lagrangian conditions for a nonsmooth vector-valued minimax

Published online by Cambridge University Press:  09 April 2009

B. D. Craven
Department of Mathematics and Statistics, University of Melbourne, Parkville, VIC 3052, Australia
D. V. Luu
Institute of Mathematics, P.O. Box 631, Bo Ho 10000 Hanoi, Vietnam
Rights & Permissions [Opens in a new window]


Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Lagrangian necessary and sufficient conditions for a nonsmooth vector-valued minimax in terms of Clarke's generalized Jacobians are established under suitable invexity hypotheses.

Research Article
Copyright © Australian Mathematical Society 1998


[1]Arthurs, A. M., Complementary variational principles, (Clarendon Press, Oxford, England, 1979).Google Scholar
[2]Aubin, J. P., ‘Lipschitz behaviour of solutions to convex minimization problems’, Math. Oper. Res. 9 (1984), 87111.CrossRefGoogle Scholar
[3]Clarke, F. H., Optimization and nonsmooth analysis, (Wiley, New York, 1983).Google Scholar
[4]Craven, B. D., ‘Strong vector minimization and duality’, Z. Angew. Math. Mech. 60 (1980), 15.CrossRefGoogle Scholar
[5]Craven, B. D., ‘Vector-valued optimization’, in Generalized concavity in optimization and economics (eds. Schaible, S. and Ziemba, W. T.), (Academic Press, New York, 1981), 661687.Google Scholar
[6]Craven, B. D., ‘Nondifferentiable optimization by smooth approximation’, Optimization 17 (1986), 317.CrossRefGoogle Scholar
[7]Craven, B. D., ‘A modified Wolfe dual for weak vector minimization’, Numer. Funct. Anal. Optim. 10 (1989), 899907.CrossRefGoogle Scholar
[8]Craven, B. D., ‘Quasimin and quasisaddlepoint for vector optimization’, Numer. Funct. Anal. Optim. 11 (1970), 4554.CrossRefGoogle Scholar
[9]Craven, B. D., ‘Convergence of discrete approximations for constrained minimization’, J. Austral. Math. Soc. (Series B) 35 (1994), 5059.CrossRefGoogle Scholar
[10]Craven, B. D. and Luu, D. V., ‘An approach to optimality conditions for nonsmooth minimax problems’, University of Melbourne, Department of Mathematics, Preprint Series No. 8 (1993).Google Scholar
[11]Craven, B. D. and Luu, D. V., ‘Constrained minimax for a vector-valued function’, Optimization 31 (1994), 199208.CrossRefGoogle Scholar
[12]Jeyakumar, V., ‘Convexlike alternative theorem and mathematical programming’, Optimization 16 (1985), 643652.CrossRefGoogle Scholar
[13]Mordukhovich, B., ‘Stability theory for parametric generalized equations and variational inequalities via nonsmooth analysis’, Trans. Amer. Math. Soc. 343 (1994), 609657.CrossRefGoogle Scholar
[14]Robinson, S. M., ‘Stability theory for systems of inequalities, Part II: Differentiable nonlinear systems’, SIAM J. Numer. Anal. 13 (1976), 497513.CrossRefGoogle Scholar
[15]Rockafellar, R. T., ‘Lipschitzian properties of multifunctions’, Nonlinear Anal. 9 (1985), 867885.CrossRefGoogle Scholar
[16]Yen, N. D., ‘Stability of the solution set of perturbed nonsmooth inequality systems’, Preprint. (International Centre for Theoretical Physics, Trieste, Italy, 1990).Google Scholar