Document Type : Research Article
Author
Department of Mathematics, Payame Noor University (PNU), P.O. Box 19395-4697, Tehran, Iran.
Abstract
The primary objective of this paper is to enhance several well-known geometric constraint qualifications and necessary optimality conditions for nonsmooth semi-infinite optimization problems (SIPs). We focus on defining novel algebraic Mangasarian-Fromovitz type constraint qualifications, and on presenting two Karush-Kuhn-Tucker type necessary optimality conditions for nonsmooth SIPs defined by locally Lipschitz functions. Then, by employing a new type of generalized invex functions, we present sufficient conditions for the optimality of a feasible point of the considered problems. It is noteworthy that the new class of invex functions we considered encompasses several classes of invex functions introduced previously. Our results are based on the Michel-Penot subdifferential.
Keywords
- Semi-Infinite optimization
- Constraint qualification
- Optimality conditions
- Michel-Penot subdifferential
Main Subjects
[1] Antczak, T., Mishra, S.K., Upadhyay, B.B. (2016). “First order duality results for a new class of nonconvex semi-infinite minimax fractional programming problems”, Journal of Advanced Mathematical Studies, 9, 832-162.
[2] Caristi, G., Kanzi, N. (2015). “Karush-Kuhn-Tuker type conditions for optimality of non-smooth multiobjective semi-infinite programming”, International Journal of Mathematical Analysis, 39, 1929-1938.
[3] Caristi, G., Ferrara, M., Stefanescu, A. (2006). “Mathematical programming with (ρ, Φ)-invexity”,
In Generalized Convexity and Related Topics, Lecture Notes in Economics and Mathematical Systems, Vol. 583. (I.V. Konnor, D.T. Luc, and A.M. Rubinov, eds.). Springer, Berlin-Heidelberg-New York, 167-176.
In Generalized Convexity and Related Topics, Lecture Notes in Economics and Mathematical Systems, Vol. 583. (I.V. Konnor, D.T. Luc, and A.M. Rubinov, eds.). Springer, Berlin-Heidelberg-New York, 167-176.
[4] Caristi, G., Ferrara, M., Stefanescu, A. (2010). “Semi-infinite multiobjective programming with generalized invexity”, Mathematical Reports, 62, 217-233.
[5] Giorgi, J., Gwirraggio, A., Thierselder, J. (2004). “Mathematics of optimization: Smooth and nonsmooth cases”, Elsivier.
[6] Goberna, M.A., López, M.A. (1998). “Linear semi-infinite optimization, Wiley, Chichester.
[6] Goberna, M.A., López, M.A. (1998). “Linear semi-infinite optimization, Wiley, Chichester.
[7] Habibi, S., Kanzi, N., Ebadian, A. (2020). ”Weak Slater qualification for nonconvex multiobjective semi-infinite programming”. Iranian Journal of Science and Technology, Transactions A: Science, 44, 417-424.
[8] Hanson, M.A. (1981). “On sufficiency of the Kuhn–Tucker conditions”, Journal of Mathematical Analysis and Applications, 80, 545-550.
[9] Hettich, R., Kortanek, K.O. (1993). “Semi-infinite programming: Theory, methods, and applications”, SIAM Review, 35, 380-429.
[10] Jeyakumar, V. (1988). “Equivalence of saddle-points and optima, and duality for a class of nonsmooth non-convex problems”, Journal of Mathematical Analysis and Applications, 130, 334-343.
[11] Kanzi, N. (2011). “Necessary optimality conditions for nonsmooth semi-infinite programming
problems”, Journal of Global Optimization, 49, 713-725.
problems”, Journal of Global Optimization, 49, 713-725.
[12] Kanzi, N. (2013). “Lagrange multiplier rules for non-differentiable DC generalized semi-infinite
programming problems”, Journal of Global Optimization, 56, 417-430.
programming problems”, Journal of Global Optimization, 56, 417-430.
[13] Kanzi, N. (2015). “Regularity conditions for non-differentiable infinite programming problems
using Michel-Penot subdifferential”, Journal of Control and Optimization in Applied Mathematics,
1, 21-30.
using Michel-Penot subdifferential”, Journal of Control and Optimization in Applied Mathematics,
1, 21-30.
[14] Kanzi, N. (2015). “Karush–Kuhn–Tucker types optimality conditions for non-smooth semi-infinite
vector optimization problems”, Journal of Control and Optimization in Applied Mathematics, 1,
21-30.
[15] Kanzi, N. (2015). “Constraint qualifications in semi-infinite systems and their applications in nonsmooth semi-infinite problems with mixed constraints”, Journal of Mathematical Extension, 9, 45-56.
vector optimization problems”, Journal of Control and Optimization in Applied Mathematics, 1,
21-30.
[15] Kanzi, N. (2015). “Constraint qualifications in semi-infinite systems and their applications in nonsmooth semi-infinite problems with mixed constraints”, Journal of Mathematical Extension, 9, 45-56.
[16] Kanzi, N., Nobakhtian, S. (2008). “Nonsmooth semi-infinite programming problems with mixed constraints”, Journal of Mathematical Analysis and Applications, 351, 170-181.
[17] Kanzi, N., Nobakhtian, S. (2008). “Optimality conditions for nonsmooth semi-infinite programming”, Optimization, 59, 717-727.
[18] Kanzi, N., Soleimani-Damaneh, M. (2020).” Characterization of the weakly efficient solutions in
nonsmooth quasiconvex multiobjective optimization”. Journal of Global Optimization, 77, 627- 641.
[19] Kazemi, S., Kanzi, N., Ebadian, A. (2019). ”Estimating the Frechet normal cone in optimization problems with nonsmooth vanishing constraints”. Iranian Journal of Science and Technology, Transactions A: Science, 43, 2299-2306.
[20] Kim, D.S., Schaible, S. (2004). “Optimality and duality for invex nonsmooth multiobjective programming problems”, Optimization, 53, 165-176.
[21] Li, X.F., Dong, J.L., Liu, Q.H. (1997). “Lipschitz B-vex functions and nonsmooth programming”,
Journal of Optimization Theory and Applications, 93, 557-574.
Journal of Optimization Theory and Applications, 93, 557-574.
[22] Michel, P., Penot, J.P. (1984). “Calcul sous- differentiel pour des fonctions lipschitziennes et non
lipschitziennes”, Comptes rendus de l’Académie des Sciences Paris sér. I Mathematique, 12, 269- 272.
lipschitziennes”, Comptes rendus de l’Académie des Sciences Paris sér. I Mathematique, 12, 269- 272.
[23] Michel, P., Penot, J.P. (1992). “A Generalized derivative for calm and stable functions”, Differential and Integral Equations, 5, 433-454.
[24] Stein, O. (2004). “On constraint qualifications in nonsmooth optimization”, Journal of Optimization Theory and Applications, 121, 647-671.
[25] Upadhyay, B.B. (2022). “Nondifferentiable generalized minimax fractional programming under
(ϕ, ρ)-invexity”, Yugoslav Journal of Operations Research, 13, 32-27.
(ϕ, ρ)-invexity”, Yugoslav Journal of Operations Research, 13, 32-27.
[26] Upadhyay, B.B., Mishra, S.K. (2015). “Nonsmooth semi-infinite minmax programming involving
generalized (ϕ, ρ)-invexity”, Journal of Systems Science and Complexity, 28, 857-875.
generalized (ϕ, ρ)-invexity”, Journal of Systems Science and Complexity, 28, 857-875.
[27] Vial, J.P. (1983). “Strong and weak convexity of sets and functions”, Mathematics of Operations
Research, 8, 231-259.
Research, 8, 231-259.
[28] Zalmai, G.J., Zhang, Q. (2012). “Optimality conditions and duality in nonsmooth semi-infinite
programming”, Numerical Functional Analysis and Optimization, 33(4), 452-472.
programming”, Numerical Functional Analysis and Optimization, 33(4), 452-472.
)