Abstract
A unified view on constraint qualifications for nonsmooth equality and inequality constrained programs is presented. A fairly general constraint qualification for programs involving B-differential functions is given. Further specification to piecewise differentiable equality constraints and locally Lipschitz continuous inequality constraints yields a nonsmooth version of the Mangasarian-Fromovitz constraint qualification.
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References
Robinson, S. M.,Local Structure of Feasible Sets in Nonlinear Programming, Part III: Stability and Sensitivity, Mathematical Programming Study, Vol. 30, pp. 45–66, 1987.
Demyanov, V. F.,Quasidifferential Calculus, Optimization Software, Publications Division, New York, New York, 1986.
Shapiro, A.,On Concepts of Directional Differentiability, Journal of Optimization Theory and Applications, Vol. 66, pp. 477–487, 1990.
Aubin, J. P., andFrankowska, H.,On Inverse Function Theorems for Set-Valued Maps, Working Paper WP-84-68, IIASA, Laxenburg, Austria, 1984.
Kuntz, L., andScholtes, S.,Constraint Qualifications in Quasidifferentiable Programming, Mathematical Programming, Vol. 60, pp. 339–347, 1993.
Luderer, B., Rösiger, R., andWürker, U.,On Necessary Conditions in Quasidifferentiable Calculus: Independence of the Specific Choice of Quasidifferentials, Optimization, Vol. 22, pp. 643–660, 1991.
Rockafellar, R. T.,Lipschitzian Properties of Multifunctions, Nonlinear Analysis, Vol. 9, pp. 867–885, 1985.
Shapiro, A.,On Optimality Conditions in Quasidifferentiable Optimization, SIAM Journal on Control and Optimization, Vol. 22, pp. 610–617, 1984.
Ward, D. E.,A Constraint Qualification in Quasidifferentiable Programming, Optimization, Vol. 22, pp. 661–668, 1991.
Jongen, H. T., andWeber, G. W.,Nonlinear Optimization: Characterization of Structural Stability, Jorunal of Global Optimization, Vol. 1, pp. 47–64, 1991.
Bartels, S. G., Kuntz, L., andScholtes, S.,Continuous Selections of Linear Functions and Nonsmooth Critical Point Theory, Preprint No. 45, Institut für Statistik und Mathematische Wirtschafstheorie, Universität Karlsruhe, Karlsruhe, Germany, 1992, to appear in: Nonlinear Analysis, Theory, Methods, and Applications.
Weber, G. W.,Charakterisierung Struktureller Stabilität in der Nichtlinearen Optimierung, PhD Thesis, RWTH Aachen, Germany, 1992.
Clarke, F. H.,Optimization and Nonsmooth Analysis, Les Publication CRM, Montreal, Québec, Canada, 1989.
Mangasarian, O. L., andFromovitz, S.,The Fritz John Necessary Optimality Conditions in the Presence of Equality and Inequality Constraints, Journal of Mathematical Analysis and Applications, Vol. 17, pp. 37–47, 1967.
Kuntz, L., andScholtes, S.,Structural Analysis of Nonsmooth Mappings, Inverse Functions, and Metric Projections, Preprint No. 48, Institut für Statistik und Mathematische Wirtschaftstheorie, Universität Karlsruhe, Karlsruhe, Germany, 1992, to appear in: Mathematical Analysis and Applications.
Schrijver, A.,Theory of Linear and Integer Programming, John Wiley and Sons, New York, New York, 1986.
Robinson, S. M.,Mathematical Foundations of Nonsmooth Embedding Methods, Mathematical Programming, Vol. 48, pp. 221–229, 1990.
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Communicated by O. L. Mangasarian
This work was supported by the Deutsche Forschungsgemeinschaft, DFG-Grant No. Pa 219/5-1.
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Kuntz, L., Scholtes, S. A nonsmooth variant of the Mangasarian-Fromovitz constraint qualification. J Optim Theory Appl 82, 59–75 (1994). https://doi.org/10.1007/BF02191779
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DOI: https://doi.org/10.1007/BF02191779