عنوان مقاله [English]
In this paper, we present a method for charaterizing the solution set of nonconvex optimization problems via their dual problems. In fact, the constrainted optimization problem which is considerd has pseudoconvex and locally Lipschitz functions, which are not necessarily convex and smooth, and include a wide class of non-convex non-smooth functions. In the proposed method, a dual problem is formulated to characterizations of the solution set of the primal problem in a mixed type of Wolfe type and Mond-Weir type. First, we introduce some of the properties of the Lagrangian functions associated to these problems and then we explain the proof of the characterization of their solution sets.
Burke, J. V., & Ferris, M. C. (1991). Characterization of solution sets of convex programs. Operations research letters, 10(1), 57-60.
Clarke, F. H. (1990). Optimization and nonsmooth analysis (Vol. 5). Siam.
Deng, S. (2009). Characterizations of the nonemptiness and boundedness of weakly efficient solution sets of convex vector optimization problems in real reflexive Banach spaces. Journal of optimization theory and applications, 140(1), 1.
Dinh, N., Goberna, M. A., & López Cerdá, M. A. (2006). From linear to convex systems: Consistency, Farkas' lemma and applications. J. Convex Anal. 13, 113–133.
Dinh, N., Jeyakumar, V., & Lee, G. M. (2006). Lagrange multiplier characterizations of solution sets of constrained pseudolinear optimization problems. Optimization, 55(3), 241-250.
Goberna, M. A., & López, M. A. (1998). A comprehensive survey of linear semi-infinite optimization theory. In Semi-infinite programming (pp. 3-27). Springer, Boston, MA.
Hanson, B. (1998). Nonsmooth analysis and control theory. The American mathematical monthly, 105(7), 691.
Hiriart-Urruty, J. B. (1979). New concepts in nondifferentiable programming. Bull. Soc. Math. France, 60, 57-85.
Ivanov, V. (2001). First order characterizations of pseudoconvex functions. Serdica mathematical journal, 27(3), 203p-218p.
Jeyakumar, V., & Yang, X. Q. (1995). On characterizing the solution sets of pseudolinear programs. Journal of optimization theory and applications, 87(3), 747-755.
Jeyakumar, V., Lee, G. M., & Dinh, N. (2004). Lagrange multiplier conditions characterizing the optimal solution sets of cone-constrained convex programs. Journal of optimization theory and applications, 123(1), 83-103.
Jeyakumar, V., Lee, G. M., & Dinh, N. (2006). Characterizations of solution sets of convex vector minimization problems. European journal of operational research, 174(3), 1380-1395.
Kim, D. S., & Son, T. Q. (2011). Characterizations of solution sets of a class of nonconvex semi-infinite programming problems. J. Nonlinear Convex Anal, 12, 429-440.
Lalitha, C. S., & Mehta, M. (2009). Characterizations of solution sets of mathematical programs in terms of Lagrange multipliers. Optimisation, 58(8), 995-1007.
Liu, C., Yang, X. M., & Lee, H. (2011). Characterizations the of solution sets of pseudoinvex programs and variational inequalities. Journal of inequalities and applications.
Mangasarian, O. L. (1975). Pseudo-convex functions. In Stochastic optimization models in finance (pp. 23-32). Academic Press.
Mangasarian, O. L. (1988). A simple characterization of solution sets of convex programs. Operations research letters, 7(1), 21-26.
Penot, J. P. (2003). Characterization of solution sets of quasiconvex programs. Journal of optimization theory and applications, 117(3), 627.
Pini, R., & Singh, C. (1997). A survey of recent [1985-1995] advances in generalized convexity with applications to duality theory and optimality conditions. Optimization, 39(4), 311-360.
Son, T. Q., & Dinh, N. (2008). Characterizations of optimal solution sets of convex infinite programs. Top, 16(1), 147-163.
Son, T. Q., & Kim, D. S. (2014). A new approach to characterize the solution set of a pseudoconvex programming problem. Journal of computational and applied mathematics, 261, 333-340.
Son, T. Q., Strodiot, J. J., & Nguyen, V. H. (2009). ε-optimality and ε-Lagrangian duality for a nonconvex programming problem with an infinite number of constraints. Journal of optimization theory and applications, 141(2), 389-409.
Yang, X. M. (2009). On characterizing the solution sets of pseudoinvex extremum problems. Journal of optimization theory and applications, 140(3), 537-542.