Abstract
In this paper, some infinite alternative theorems in locally convex vector spaces are proved and some applications of these theorems in optimization theory are addressed. Extensions of constraint qualification conditions, in the absence of differentiability using Clarke’s generalized gradient and Mordukhovich’s subdifferential, in Banach and Asplund spaces are introduced; and relationships between them are established. Some of these relations, are proved using the provided alternative theorems.
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Bao, T.Q., Mordukhovich, B.S.: Relative Pareto minimizers for multiobjective problems: existence and optimality conditions. Math. Program. 122, 101–138 (2010)
Bazaraa, M.S., Sherali, H.D., Shetty, C.M.: Nonlinear Programming: Theory and Algorithms, 2nd edn. Wiley, New York (1993)
Clarke, F.H., Ledyaev, Y.S., Stern, R.J., Wolenski, P.R.: Nonsmooth Analysis and Control Theory. Springer, New York (1998)
Conway, J.B.: A Course in Functional Analysis. Springer, New York (1985)
Goberna, M.A., López, M.A.: Linear Semi-infinite Optimization. Wiley, New York (1998)
Leibowitz, G.M.: Lectures on Complex Function Algebras. Scott, Foresman and Company, Illinois (1970)
Mangasarian, O.L.: Nonlinear Programming. McGraw-Hill, New York (1969)
Mordukhovich, B.S.: Maximum principle in problems of time optimal control with nonsmooth constraints. J. Appl. Math. Mech. 40, 960–969 (1976)
Mordukhovich, B.S.: Variations Analysis and Generalized Differentiation, I: Basic Theory. Springer, Grundlehren Series (Fundamental Principles of Mathematical Sciences), vol. 330. Springer, Grundlehren (2006)
Mordukhovich, B.S.: Variations Analysis and Generalized Differentiation, II: Applications. Springer, Grundlehren Series (Fundamental Principles of Mathematical Sciences), vol. 331. Springer, Grundlehren (2006)
Rudin, W.: Functional Analysis. McGraw-Hill, Singapore (1991)
Soleimani-damaneh, M.: Characterization of nonsmooth quasiconvex and pseudoconvex functions. J. Math. Anal. Appl. 330, 1387–1392 (2007)
Soleimani-damaneh, M.: Infinite (semi-infinite) problems to characterize the optimality of nonlinear optimization problems. Eur. J. Oper. Res. 188(1), 49–56 (2008)
Soleimani-damaneh, M.: Generalized invexity in separable Hilbert spaces. Topology 48, 66–79 (2009)
Soleimani-damaneh, M.: On generalized convexity in Asplund spaces. Nonlinear Anal. 70, 3072–3075 (2009)
Soleimani-damaneh, M.: Optimality and invexity in optimization problems in Banach algebras (spaces). Nonlinear Anal. 71, 5522–5530 (2009)
Soleimani-damaneh, M.: Nonsmooth optimization using Mordukhovich’s subdifferential. SIAM J. Control Optim. 48, 3403–3432 (2010)
Soleimani-damaneh, M.: An optimization modelling for string selection in molecular biology using Pareto optimality. Appl. Math. Model. 35, 3887–3892 (2011)
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This research was in part supported by a grant from IPM (No. 90260124).
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Asadi, M.B., Soleimani-damaneh, M. Infinite Alternative Theorems and Nonsmooth Constraint Qualification Conditions. Set-Valued Var. Anal 20, 551–566 (2012). https://doi.org/10.1007/s11228-012-0211-6
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DOI: https://doi.org/10.1007/s11228-012-0211-6
Keywords
- Mordukhovich’s subdifferential
- Clarke’s generalized gradient
- Locally convex space
- Alternative theorems
- Optimization
- Nonsmooth analysis
- Banach (Asplund) space
- Weak* topology