Abstract
In this paper, we use the penalty approach in order to study a class of vector inequality-constrained minimization problems on Banach spaces. A penalty function is said to have the generalized exact penalty property if there is a penalty coefficient for which approximate solutions of the unconstrained penalized problem are close enough to approximate solutions of the corresponding constrained problem. For our class of problems, we establish the generalized exact penalty property and obtain an estimation of the exact penalty.
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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. Optim. 32, 301–334 (1995)
Burachik, R.S., Iusem, A.N., Melo, J.G.: Duality and exact penalization for general augmented Lagrangians. J. Optim. Theory Appl. 147, 125–140 (2010)
Burachik, R.S., Kaya, C.Y.: An augmented penalty function method with penalty parameter updates for nonconvex optimization. Nonlinear Anal. 75, 1158–1167 (2012)
Burke, J.V.: An exact penalization viewpoint of constrained optimization . SIAM J. Control Optim. 29, 968–998 (1991)
Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)
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. Soviet. Math. Dokl. 8, 459–462 (1966)
Mordukhovich, B.S.: Penalty functions and necessary conditions for the extremum in nonsmooth and nonconvex optimization problems. Uspekhi. Math. Nauk. 36, 215–216 (1981)
Mordukhovich, B.S.: Variational analysis and generalized differentiation, I: basic theory. Springer, Berlin (2006)
Mordukhovich, B.S.: Variational analysis and generalized differentiation, II: applications. Springer, Berlin (2006)
Rubinov, A.M., Yang, X.Q., Bagirov, A.M.: Penalty functions with a small penalty parameter. Optim. Methods Softw. 17, 931–964 (2002)
Uderzo, A.: Exact penalty functions and calmness for mathematical programming under nonlinear perturbations. Nonlinear Anal. 73, 1596–1609 (2010)
Yang, X.Q., Ralph, D.: Characterizations for perturbed exact penalty functions. Nonlinear Anal. 62, 101–106 (2005)
Ye, J.J.: The exact penalty principle. Nonlinear Anal. 75, 1642–1654 (2012)
Zangwill, W.I.: Non-linear programming via penalty functions. Management Sci. 13, 344–358 (1967)
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)
Zaslavski, A.J.: Optimization on Metric and Normed Spaces. Springer, New York (2010)
Zaslavski, A.J.: An estimation of exact penalty in constrained optimization. J. Nonlinear Convex Anal. 11, 381–389 (2010)
Zaslavski, A.J.: An estimation of exact penalty for infinite-dimensional inequality-constrained minimization problems. Set-Valued Var. Anal. 19, 385–398 (2011)
Zaslavski, A.J.: An approximate exact penalty in constrained vector optimization on metric spaces. J. Optim. Theory Appl. (2013). http://dx.doi.org/10.1007/s10957-013-0288-6
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Zaslavski, A.J. An Approximate Exact Penalty for Vector Inequality-Constrained Minimization Problems. Vietnam J. Math. 42, 499–508 (2014). https://doi.org/10.1007/s10013-014-0077-z
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DOI: https://doi.org/10.1007/s10013-014-0077-z