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Generalized convexity in non-regular programming problems with inequality-type constraints. (English) Zbl 1160.90624

Summary: Convexity plays a very important role in optimization for establishing optimality conditions. Different works have shown that the convexity property can be replaced by a weaker notion, the invexity. In particular, for problems with inequality-type constraints, Martin defined a weaker notion of invexity, the Karush-Kuhn-Tucker-invexity (hereafter KKT-invexity), that is both necessary and sufficient to obtain Karush-Kuhn-Tucker-type optimality conditions. It is well known that for this result to hold the problem has to verify a constraint qualification, i.e., it must be regular or non-degenerate. In non-regular problems, the classical optimality conditions are totally inapplicable. Meaningful results were obtained for problems with inequality-type constraints by Izmailov. They are based on the 2-regularity condition of the constraints at a feasible point. In this work, we generalize Martin’s result to non-regular problems by defining an analogous concept, the 2-KKT-invexity, and using the characterization of the tangent cone in the 2-regular case and the necessary optimality condition given by Izmailov.

MSC:

90C26 Nonconvex programming, global optimization
90C29 Multi-objective and goal programming
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References:

[1] Arutyunov, A. V., Optimality Conditions: Abnormal and Degenerate Problems, Math. Appl. (2000), Kluwer Academic Publishers · Zbl 0894.90133
[2] 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) · Zbl 1149.90140
[3] Avakov, E. R., Extremum conditions for smooth problems with equality-type constrains, Zh. Vychisl. Mat. Mat., 25, 680-693 (1985) · Zbl 0571.49016
[4] Avakov, E. R., Necessary conditions of an extremum for smooth abnormal problems with equality and inequality-type constrains, Mat. Zametki, 45, 6, 3-11 (1989) · Zbl 0699.49026
[5] Avakov, E. R.; Arutyunov, A. V.; Izmailov, A. F., Necessary conditions for an extremum in a mathematical programming problem, Proc. Steklov Inst. Math., 256, 2-25 (2007) · Zbl 1160.49016
[6] Bazaara, M. S.; Sherali, H. D.; Shetty, C. M., Nonlinear Programming: Theory and Algorithms (1993), John Wiley and Sons: John Wiley and Sons Inc., New York
[7] Belash, K. N.; Tret’yakov, A. A., Methods for solving degenerate problems, Zh. Vychisl. Mat. Mat., 28, 1097-1102 (1988) · Zbl 0653.65039
[8] Bliss, G. A., Lectures on the Calculus of Variations (1946), University of Chicago Press · Zbl 0063.00459
[9] Brezhneva, O.; Tret’yakov, A. A., Optimality conditions for degenerate extremum problems with equality constraints, SIAM J. Control Optim., 42, 2, 729-745 (2003) · Zbl 1037.49025
[10] Craven, B. D., Invex functions and constrained local minima, Bull. Austral. Math. Soc., 24, 357-366 (1981) · Zbl 0452.90066
[11] Hanson, M. A., On sufficiency of the Kuhn-Tucker conditions, J. Math. Anal. Appl., 80, 545-550 (1981) · Zbl 0463.90080
[12] Izmailov, A. F.; Tret’yakov, A. A., Lemmas on the representation of degenerate non-linear mappings, Vestnik MGU Ser. 15, 2, 59-65 (1993)
[13] Izmailov, A. F., Degenerate extremum problems with inequality-type constraints, Zh. Vychisl. Mat. Mat. Fiz., 32, 1570-1581 (1992) · Zbl 0790.49017
[14] Izmailov, A. F., Optimality conditions for degenerate extremum problems with inequality-type constraints, Comput. Math. Math. Phys., 34, 6, 723-736 (1994) · Zbl 0831.49028
[15] Izmailov, A. F.; Solodov, M., Optimality conditions for irregular inequality-constrained problems, SIAM J. Control Optim., 40, 1280-1295 (2001) · Zbl 1102.90376
[16] Kuhn, H. W.; Tucker, A. W., Nonlinear programming, (Neyman, J., Proc. 2nd Berkeley Symposium on Mathematical Statistics and Probability (1951), University of California Press: University of California Press Berkeley, CA) · Zbl 0044.05903
[17] Ledzewicz, U.; Schättler, H., Second order conditions for extremum problems with nonregular equality constraints, J. Optim. Theory Appl., 86, 113-194 (1995) · Zbl 0835.49016
[18] Ledzewicz, U.; Schättler, H., High order approximations and generalized necessary conditions for optimality, SIAM J. Control Optim., 37, 33-53 (1999) · Zbl 0941.49016
[19] Mangasarian, O. L., Nonlinear Programming (1969), McGraw-Hill · Zbl 0194.20201
[20] Martin, D. M., The essence of invexity, J. Optim. Theory Appl., 17, 1, 65-76 (1985) · Zbl 0552.90077
[21] Rangarajan, K. S., A First Course in Optimization Theory (1996), Cambridge University Press · Zbl 0885.90106
[22] Pourciau, B. H., Modern multiplier rules, Amer. Math. Monthly, 87, 6, 433-452 (1980) · Zbl 0454.90067
[23] Rockafellar, R. T., Lagrange multipliers and optimality, SIAM Rev., 35, 2, 183-238 (1993) · Zbl 0779.49024
[24] Tret’yakov, A. A., The necessary and sufficient conditions for Pth order optimality, Zh. Vychisl. Mat. Mat., 24, 203-209 (1984) · Zbl 0537.49009
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