Abstract
Using the steepest-descent method combined with the Armijo stepsize rule, we give an algorithm for finding a solution to the inclusion 0∈F(x), whereF is a set-valued map with smooth support function. As an example, we consider the special caseF(x)=g(x)+K, withK being a convex cone andg a single-valued function. The relation between the present algorithm and that given by Burke and Han is also discussed.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Burke, J. V.,Methods for Solving Generalized Systems of Inequalities with Applications to Nonlinear Programming, PhD Thesis, University of Illinois at Urbana-Champaign, 1983.
Burke, J., andHan, S. P.,A Gauss-Newton Approach to Solving Generalized Inequalities, Mathematics of Operations Research, Vol. 11, pp. 632–643, 1986.
Lemarechal, C., andZowe, J.,Approximation to Multi-Valued Mapping: Existence, Uniqueness, Characterization, Schwerpunktprogram der Deutschen Forschungsgemeinschaft “Anwendungskezogene Optimierung und Steuerung,” Report No. 5, 1987.
Dien, P. H.,Locally Lipschitzian Set-Valued Maps and Generalized Extremal Problems, Acta Mathematica Vietnamica, Vol. 8, pp. 109–122, 1983.
Clarke, F. H.,Optimization and Nonsmooth Analysis, Wiley-Interscience, New York, New York, 1983.
Zowe, J., andKurcyusz, S.,Regularity and Stability for the Mathematical Programming Problem in Banach Spaces, Applied Mathematics and Optimization, Vol. 5, pp. 49–62, 1979.
Maurer, H., andZowe, J.,First-Order and Second-Order Necessary and Sufficient Optimality Condition for Infinite-Dimensional Programming Problems, Mathematical Programming, Vol. 16, pp. 98–110, 1979.
Robinson, S. M.,Regularity and Stability for Convex Multi-Valued Functions, Mathematics of Operations Research, Vol. 1, pp. 130–143, 1976.
Robinson, S. M.,Stability Theory for Systems of Inequalities, Part 1: Linear Systems, SIAM Journal on Numerical Analysis, Vol. 12, pp. 754–768, 1975.
Robinson, S. M.,Stability Theory for Systems of Inequalities, Part 2: Differentiable Nonlinear Systems, SIAM Journal on Numerical Analysis, Vol. 13, pp. 497–513, 1976.
Duong, P. C., andTuy, H.,Stability, Surjectivity, and Local Invertibility of Nondifferentiable Mappings, Acta Mathematica Vietnamica, Vol. 3, pp. 89–105, 1978.
Garcia-Palomares, U., andRestuccia, A.,Application of the Armijo Stepsize Rule to the Solution of a Nonlinear System of Equalities and Inequalities, Journal of Optimization Theory and Applications, Vol. 41, pp. 405–415, 1983.
Garcia-Palomares, U., andRestuccia, A.,A Global Quadratic Algorithm for Solving a System of Mixed Equalities and Inequalities, Mathematical Programming, Vol. 21, pp. 290–300, 1981.
Phung, H. T., andDien, P. H.,Solving Generalized Inequalities with Application to Systems of Mixed Equalities and Inequalities, Proceedings of the Symposium on Numerical Methods and Optimization, Hanoi, Vietnam, 1989.
Author information
Authors and Affiliations
Additional information
Communicated by O. L. Mangasarian
The valuable comments and helpful suggestions of the referee are gratefully acknowledged. Sincere thanks are due to Dr. J. Burke for submitting the necessary material and to Dr. C. Lemarechal for advices and encouragement.
Rights and permissions
About this article
Cite this article
Dien, P.H., Phung, H.T. Algorithm for finding a solution to the inclusion 0∈F(x). J Optim Theory Appl 67, 509–531 (1990). https://doi.org/10.1007/BF00939647
Issue Date:
DOI: https://doi.org/10.1007/BF00939647