Abstract
In non-regular problems the classical optimality conditions are totally inapplicable. Meaningful results were obtained for problems with conic constraints by Izmailov and Solodov (SIAM J Control Optim 40(4):1280–1295, 2001). They are based on the so-called 2-regularity condition of the constraints at a feasible point. It is well known that generalized convexity notions play a very important role in optimization for establishing optimality conditions. In this paper we give the concept of Karush–Kuhn–Tucker point to rewrite the necessary optimality condition given in Izmailov and Solodov (SIAM J Control Optim 40(4):1280–1295, 2001) and the appropriate generalized convexity notions to show that the optimality condition is both necessary and sufficient to characterize optimal solutions set for non-regular problems with conic constraints. The results that exist in the literature up to now, even for the regular case, are particular instances of the ones presented here.
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References
Arutyunov A.V.: Optimality Conditions: Abnormal and Degenerate Problems. Mathematics and Its Applications. Kluwer, Dordrecht (2000)
Arutyunov A.V., Avakov E.R., Izmailov A.F.: Necessary optimality conditions for constrained optimization problems under relaxed constrained qualifications. Math. Program. 114, 37–68 (2008)
Avakov E.R.: Extremum conditions for smooth problems with equality-type constraints. USSR Comput. Math. Math. Phys. 25, 24–32 (1985)
Avakov E.R.: Necessry extremum conditions for smooth abnormal problems with equality- and inequality-type constrains. Math. Notes 45(6), 431–437 (1989)
Avakov E.R.: Necessary conditions for a minimum for nonregular problems in Banach spaces. The maximum principle for abnormal optimal control problems. Proc. Steklov Inst. Math. 185, 1–32 (1990)
Avakov E.R.: Necessary first-order conditions for abnormal variational calculus problems (in Russian). Differ. Equ. 27, 739–745 (1991)
Avakov A.V., Arutyunov E.R.: Abnormal problems with a nonclosed image. Doklady Math. 70, 924–927 (2004)
Bonnans J.F., Shapiro A.: Perturbation Analysis of Optimization Problems. Springer, New York (2000)
Craven B.D.: Invex functions and constrained local minima. Bull. Aust. Math. Soc. 24, 357–366 (1981)
Craven B.D.: Control and Optimization. Chapman and Hall Mathematics, London (1995)
Hanson M.A.: On sufficiency of the Kuhn–Tucker conditions. J. Math. Anal. Appl. 80, 545–550 (1981)
Hernández-Jiménez B., Rojas-Medar M.A., Osuna-Gómez R., Beato-Moreno A.: Generalized convexity in non-regular programming problems with inequality-type constraints. J. Math. Anal. Appl. 352, 604–613 (2009)
Hernández-Jiménez B., Osuna-Gómez R., Rojas-Medar M.A., Arana-Jiménez M.: Characterization of optimal solutions for nonlinear programming problems with conic constraints. Optimization 60(5), 619–626 (2011)
Izmailov A.F.: Optimality conditions for degenerate extremum problems with inequality-type constraints. Comput. Math. Math. Phys. 34(6), 723–736 (1994)
Izmailov A.F.: Optimality conditions in extremal problems with nonregular inequality constraints. Math. Notes 66(1), 72–81 (1999)
Izmailov A.F., Solodov M.V.: The theory of 2-regularity for mappings with Lipschitzian derivatives and its applications to optimality conditions. Math. Oper. Res. 27, 614–635 (2002)
Izmailov A.F., Solodov M.V.: Optimality conditions for irregular inequality-constrained problems. SIAM J. Control Optim. 40(4), 1280–1295 (2001)
Izmailov A.F., Tretyakov A.A.: Factor Analysis of Nonlinear Mappings. (in Russian) Nauka, Moscow (1994)
Ledzewicz U., Schättler H.: High order approximations and generalized necessary conditions for optimality. SIAM J. Control Optim. 37, 33–53 (1999)
Martin D.M.: The essence of invexity. J. Optim. Theory Appl. 17(1), 65–76 (1985)
Robinson S.M.: Stability theorems for systems of inequalities, part II: differentiable nonlinear systems. SIAM J. Numer. Anal. 13, 597–607 (1976)
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The authors have been partially supported by M.E.C. (Spain), Project MTM2010-15383 and Fondecyt-Chile, Grant No 1120260.
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Hernández-Jiménez, B., Osuna-Gómez, R., Rojas-Medar, M.A. et al. Generalized convexity for non-regular optimization problems with conic constraints. J Glob Optim 57, 649–662 (2013). https://doi.org/10.1007/s10898-012-9935-y
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DOI: https://doi.org/10.1007/s10898-012-9935-y