Abstract
We use the penalty approach in order to study inequality-constrained minimization problems in infinite dimensional spaces. A penalty function is said to have the exact penalty property if there is a penalty coefficient for which a solution of an unconstrained penalized problem is a solution of the corresponding constrained problem. In this paper we consider a large class of inequality-constrained minimization problems for which a constraint is a mapping with values in a normed ordered space. For this class of problems we introduce a new type of penalty functions, establish the exact penalty property and obtain an estimation of the exact penalty. Using this exact penalty property we obtain necessary and sufficient optimality conditions for the constrained minimization problems.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Auslender, A.: Penalty and barrier methods: a unified framework. SIAM J. Optim. 10, 211–230 (1999)
Auslender, A.: Asymptotic analysis for penalty and barrier methods in variational inequalities. SIAM J. Control Optim. 37, 653–671 (1999)
Boukari, D., Fiacco, A.V.: Survey of penalty, exact-penalty and multiplier methods from 1968 to 1993. Optimization 32, 301–334 (1995)
Burke, J.V.: An exact penalization viewpoint of constrained optimization. SIAM J. Control Optim. 29, 968–998 (1991)
Clarke, F.H.: A new approach to Lagrange multipliers. Math. Oper. Res. 1, 165–174 (1976)
Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley Interscience (1983)
Demyanov, V.F.: Exact penalty functions and problems of the calculus of variations. Autom. Remote Control 65, 280–290 (2004)
Demyanov, V.F., Vasiliev, L.V.: Nondifferentiable Optimization. Optimization Software, Inc., New York (1985)
Di Pillo, G., Grippo, L.: Exact penalty functions in constrained optimization. SIAM J. Control Optim. 27, 1333–1360 (1989)
Ekeland, I.: On the variational principle. J. Math. Anal. Appl. 47, 324–353 (1974)
Eremin, I.I.: The penalty method in convex programming. Sov. Math. Dokl. 8, 459–462 (1966)
Eremin, I.I.: The penalty method in convex programming. Cybernetics 3, 53–56 (1971)
Han, S.-P., Mangasarian, O.L.: Exact penalty function in nonlinear programming. Math. Program. 17, 251–269 (1979)
Hiriart-Urruty, J.-B., Lemarechal, C.: Convex Analysis and Minimization Algorithms. Springer, Berlin (1993)
Kutateladze, S.S.: Convex operators. Usp. Mat. Nauk 34, 167–196 (1979)
Mordukhovich, B.S.: Penalty functions and necessary conditions for the extremum in nonsmooth and nonconvex optimization problems. Usp. Mat. Nauk. 36, 215–216 (1981)
Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, II: Applications. Springer, Berlin (2006)
Zangwill, W.I.: Nonlinear programming via penalty functions. Manage. Sci. 13, 344–358 (1967)
Zaslavski, A.J.: Existence of solutions of minimization problems with an increasing cost function and porosity. Abstr. Appl. Anal. 2003, 651–670 (2003)
Zaslavski, A.J.: Generic existence of solutions of minimization problems with an increasing cost function. Nonlinear Funct. Anal. Appl. 8, 181–213 (2003)
Zaslavski, A.J.: A sufficient condition for exact penalty in constrained optimization. SIAM J. Optim. 16, 250–262 (2005)
Zaslavski, A.J.: Existence of approximate exact penalty in constrained optimization. Math. Oper. Res. 32, 484–495 (2007)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Zaslavski, A.J. An Estimation of Exact Penalty for Infinite-Dimensional Inequality-Constrained Minimization Problems. Set-Valued Anal 19, 385–398 (2011). https://doi.org/10.1007/s11228-010-0144-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11228-010-0144-x
Keywords
- Approximate solution
- Ekeland’s variational principle
- Minimization problem
- Ordered vector space
- Penalty function