Document Type : Research Article
Authors
Department of Mathematics‎, ‎Payame Noor University (PNU)‎,‎ P.O‎. ‎BOX 19395-4697‎, ‎Tehran‎, ‎Iran‎.
10.30473/coam.2023.67823.1234
Abstract
This paper aims to establish first-order necessary optimality conditions for non-smooth multi-objective generalized semi-infinite programming problems. These problems involve inequality constraints whose index set depends on the decision vector, and all emerging functions are assumed to be locally Lipschitz. We introduce a new constraint qualification for these problems. Building upon this qualification, we derive an upper estimate for the Clarke sub-differential of the value function of the problem. Furthermore, we demonstrate the necessary optimality conditions for properly efficient solutions to the problem.
Keywords
- Constraint qualification
- Generalized semi-infinite optimization
- Clarke subdifferential
- Marginal function
Main Subjects
[1] Aubin, J.P.(1998).“Optimaandequilibria, anintroductiontononlinearanalysis”, Springer, Berlin.
[2] Bank, B., Guddat, J., Klatte, D., Kummer, B., Tammer, K. (1982). “Non-Linear parametric optimization”, Birkhauser.
[3] Barilla D., Caristi G., Kanzi N. (2022). “Optimality and duality in nonsmooth semi-infinite optimization, using a weak constraint qualification”, Decisions in Economics and Finance, 45(2), 503-519.
[4] Borwein, J.M., Lewis, A.S. (2000). “Convex analysis and nonlinear optimization: theory and examples”, Springer, New York.
[5] Burachik, R.S., Iusem, A.N. (2008). “Set-valued mapping and enlargements of monotone operators”, Springer, New York.
[6] Clarke, F.H. (1983). “Optimization and nonsmooth analysis”, Wiley.
[7] Dempe, S. (2002). “Foundations of bilevel programming”, Kluwer.
[8] Demyanov, V.F., Rubinov, A.M. (1998). “Constructive nonsmooth analysis”, Springer.
[9] Ehrgott, M. (2005). “Multicriteria optimization”, Springer, Berlin.
[10] Gauvin, J., Dubeau, F. (1982). “Differential properties of the marginal function in mathematical programming”, Mathematical Programming Study, 19, 101-119.
[11] Graettinger, T.J., Krogh, B.H. (1998). “The acceleration radius: A global performance measure for robotic manipulators”, IEEE Jornal of Robotics and Automation, 4, 60-69.
[12] Habibi S., Kanzi N., Ebadian A. (2020). “Weak Slater qualification for nonconvex multiobjective semi-infinite programming”, Iranian Journal of Science and Technology, Transaction A: Science, 44, 417-424.
[13] Hiriart- Urruty, J.B., Lemarechal, C. (1991). “Convex analysis and minimization algorithms, I”, Springer, Berlin, Heidelberg.
[14] Hoffman, A., Reinhardt, R. (1994). “On reverse Chebyshev approximation problems”, Preprint M 08/94, Facualty of Mathematics and Natural Sciences, Technical University of Ilmenau, 14, 42-56.
[15] Kanzi, N. (2013). “Lagrange multiplier rules for non-differentiable DC generalized semi-infinite programming problems”, Journal of Global Optimization, 56, 417-430.
[16] Kanzi, N. (2017). “Two-level optimization problems with infinite number of convex lower level constraints”, Control and Optimization in Applied Mathematics, 2, 35-46.
[17] Kanzi, N., Nobakhtian, S. (2008). “Optimality conditions for nonsmooth semi-infinite programming”, Optimization, 59, 717-727.
[18] Kanzi, N., Nobakhtian, S. (2010). “Necessary optimality conditions for nonsmooth generalized semi-infinite programming problems”, European Journal of Operational Research, 205, 253-261.
[19] Li, W., Nahak, C., Singer, I. (2000). “Constraint qualifications in semi-infinite systems of convex inequalities”, SIAM Journal of Optimization, 11, 31-52.
[20] Mordukhovich, B.S. (2006). “Variational analysis and generalized differentiation. I. Basic theory”, Springer, Berlin.
[21] Mordukhovich, B.S., Nam, N.M. (2005). “Variational stability and marginal functions via generalized differentiation”, Mathematics of Operations Research, 30, 800-816.
[22] Mordukhovich, B.S., Nam, N.M., Yen, N.D. (2009). “Subgradients of marginal functions in parametric mathematical programming”, Mathematical Programming, 116, 369-396.
[23] Robinson, S.M. (1982). “Generalized equations and their solutions, Part II: Applications to non-linear programming”, Mathematical Programming Study, 19, 200-221.
[24] Rockafellar, R.T., Wets, J.B. (1998). “Variational analysis”, Springer, New York.
[25] Rückman, J.J., Shapiro, A. (1999). “First-order optimality conditions in generalized semi-infinite programming”, Journal of Optimization Theory and Applications, 3, 677-691.
[26] Rückmann, J.J., Stein, O. (2001). “On convex lower level problems in generalized semi-infinite optimization”, in Semi-infinite Programming-Recent Advances, M. A. Goberna and M. A. López, Eds, Kluwer, Boston, 121-134.
[27] Soroush, H. (2023). “A new weak Slater constraint qualification for non-smooth multiobjective semi-infinite programming”, Control and Optimization in Applied Mathematics, 8, 49-61.
[28] Stein, O. (2003). “Bi-level strategies in semi-infinite programming”, Kluwer.
[29] Stein, O., Still, G. (2000). “On optimality conditions for generalized semi-infinite programming problems”, Journal of Optimization Theory and Applications, 104, 443-458.
[30] Still, G. (1999).“Generalized semi-infinite programming: Theory and methods”, European Journal of Operational Research, 119, 301-313.
[31] Vazquez, F.G., Rebollar, L.A.H., Rückmann, J.J. (2018). “On vector generalized semi-infinite programming”, Revista Investigacion Operacional, 39, 341-352.
[32] Ye, J.J., Wu, S.Y. (2008). “First order optimality conditions for generalized semi-infinite programming problems”, Journal of Optimization Theory and Applications, 137, 419-434.
)