Sufficient optimality criteria and duality for multiobjective variational control problems with G-type I objective and constraint functions

In the paper, we introduce the concepts of G-type I and generalized G-type I functions for a new class of nonconvex multiobjective variational control problems. For such nonconvex vector optimization problems, we prove sufficient optimality conditions for weakly efficiency, efficiency and properly efficiency under assumptions that the functions constituting them are G-type I and/or generalized G-type I objective and constraint functions. Further, for the considered multiobjective variational control problem, its dual multiobjective variational control problem is given and several duality results are established under (generalized) G-type I objective and constraint functions

Duality for multiobjective variational control problems with (Φ,ρ)-invexity

In this paper, Mond-Weir and Wolfe type duals for multiobjective variational control problems are formulated. Several duality theorems are established relating efficient solutions of the primal and dual multiobjective variational control problems under TeX-invexity. The results generalize a number of duality results previously established for multiobjective variational control problems under other generalized convexity assumptions

Nondifferentiable multiobjective programming problem under strongly K-Gf-pseudoinvexity assumptions

[EN] In this paper we consider the introduction of the concept of (strongly) K-G(f)-pseudoinvex functions which enable to study a pair of nondifferentiable K-G- Mond-Weir type symmetric multiobjective programming model under such assumptions.Dubey, R.; Mishra, LN.; Sánchez Ruiz, LM.; Sarwe, DU. (2020). Nondifferentiable multiobjective programming problem under strongly K-Gf-pseudoinvexity assumptions. Mathematics. 8(5):1-11. https://doi.org/10.3390/math8050738S11185Antczak, T. (2007). New optimality conditions and duality results of type in differentiable mathematical programming. Nonlinear Analysis: Theory, Methods & Applications, 66(7), 1617-1632. doi:10.1016/j.na.2006.02.013Antczak, T. (2008). On G-invex multiobjective programming. Part I. Optimality. Journal of Global Optimization, 43(1), 97-109. doi:10.1007/s10898-008-9299-5Ferrara, M., & Viorica-Stefanescu, M. (2008). Optimality conditions and duality in multiobjective programming with invexity. YUJOR, 18(2), 153-165. doi:10.2298/yjor0802153fChen, X. (2004). Higher-order symmetric duality in nondifferentiable multiobjective programming problems. Journal of Mathematical Analysis and Applications, 290(2), 423-435. doi:10.1016/j.jmaa.2003.10.004Long, X. (2013). Sufficiency and duality for nonsmooth multiobjective programming problems involving generalized univex functions. Journal of Systems Science and Complexity, 26(6), 1002-1018. doi:10.1007/s11424-013-1089-6Dubey, R., Mishra, L. N., & Sánchez Ruiz, L. M. (2019). Nondifferentiable G-Mond–Weir Type Multiobjective Symmetric Fractional Problem and Their Duality Theorems under Generalized Assumptions. Symmetry, 11(11), 1348. doi:10.3390/sym11111348Pitea, A., & Postolache, M. (2011). Duality theorems for a new class of multitime multiobjective variational problems. Journal of Global Optimization, 54(1), 47-58. doi:10.1007/s10898-011-9740-zPitea, A., & Antczak, T. (2014). Proper efficiency and duality for a new class of nonconvex multitime multiobjective variational problems. Journal of Inequalities and Applications, 2014(1). doi:10.1186/1029-242x-2014-333Dubey, R., Deepmala, & Narayan Mishra, V. (2020). Higher-order symmetric duality in nondifferentiable multiobjective fractional programming problem over cone contraints. Statistics, Optimization & Information Computing, 8(1), 187-205. doi:10.19139/soic-2310-5070-60

Some Aspects Of Duality In Variational Problems And Optimal Control

This thesis is divided into six chapters. In the Ist chapter we present a brief survey of related work done in the area of multiobjective mathematical programming, optimal control and game theory. Chapter Two: In this chapter sufficient optimality criteria are derived for a control problem under generalized invexity. A Mond-Weir type dual to the control problem is proposed and various duality theorems are validated under generalized invexity assumptions on functionals appearing in the problems. It is pointed out that these results can be applied to the control problem with free boundary conditions and have linkage with results for nonlinear programming problems in the presence of inequality and equality constraints already established in the literature. Chapter Three: In this chapter a mixed type dual to the control problem in order to unify Wolfe and Mond-Weir type dual control problem is presented in various duality results are validated and the generalized invexity assumptions. It is pointed out that our results can be extended to the control problems with free boundary conditions. The duality results for nonlinear programming problems already existing in the literature are deduced as special cases of our results. Chapter Four: In this chapter two types of duals are considered for a class of variational problems involving higher order derivative. The duality results are derived without any use of optimality conditions. One set of results is based on Mond-Weir type dual that has the same objective functional as the primal problem but different constraints. The second set of results is based on a dual of an auxiliary primal with single objective function. Under various convexity and generalized convexity assumptions, duality relationships between primal and its various duals are established. Problems with natural boundary values are considered and the analogues of our results in nonlinear programming are also indicated. Chapter Five: In this chapter a certain constrained dynamic game is shown to be equivalent to a pair of symmetric dual variational problems which have more general formulation than those already existing in the literature. Various duality results are proved under convexity and generalized convexity assumptions on the appropriate functional. The dynamic game is also viewed as equivalent to a pair of dual variational problems without the condition of fixed points. It is also indicated that our equivalent formulation of a pair of symmetric dual variational problems as dynamic generalization of those already studied in the literature. Chapter Six: In this chapter a mixed type second-order dual to a variational problem is formulated as a unification of Wolfe and Mond-Weir type dual problems already treated in the literature and various duality results are validated under generalized second order invexity. Problems with natural boundary values are formulated and it also is pointed out that our duality results can be regarded as dynamic generalizations of those of (static) nonlinear programming. The subject matter of the present research thesis is fully published in the form of the following research papers written by the author: (1) Sufficiency and Duality In Control Problems with Generalized Invexity, Journal of Applied Analysis,Vol, 14 No. 1 (2008),pp.27-42. (2) Mixed Type Duality for Control Problems with Generalized Invexity, Journal of Applied Mathematics and Informatics,Vol. 26(2008), No.5-6 , pp. 819-837. (3) On Multiobjective Duality for Variational Problems, The Open Operational Research Journal,2012, 6, 1-8. (4) Constrained Dynamic Game and Symmetric Duality For Variational Problems, Journal of Mathematics and System Science 2(2012), 171-178. (5) Mixed Type Second – Order Dulaity For Variational Problems, Journal of Informatics and Mathematical Sciences , Vol5,No.1, pp.1-13,(2013)

The continuous-time problem with interval-valued functions: applications to economic equilibrium

The aim of this paper is to define the Continuous-Time Problem in an interval context and to obtain optimality conditions for this problem. In addition, we will find relationships between solutions of Interval Continuous-Time Problem (ICTP) and Interval Variationallike Inequality Problems, both Stampacchia and Minty type. Pseudo invex monotonicity condition ensures the existence of solutions of the (ICTP) problem. These results generalize similar conclusions obtained in Euclidean or Banach spaces inside classical mathematical programming problems or Continuous-Time Problems. We will finish generalizing the existence of Walrasarian equilibrium price model and the Wardrop’s principle for traffic equilibrium problem to an environment of interval-valued functions.The research in this paper has been partially supported by Ministerio de Economía y Competitividad, Spain, through grant MTM2015-66185-P and Proyectos I+D 2015 MTM2015-66185-P (MINECO/FEDER) and Fondecyt, Chile, grant 1151154

Some contributions to optimality criteria and duality in Multiobjective mathematical programming.

This thesis entitled, “some contributions to optimality criteria and duality in multiobjective mathematical programming”, offers an extensive study on optimality, duality and mixed duality in a variety of multiobjective mathematical programming that includes nondifferentiable nonlinear programming, variational problems containing square roots of a certain quadratic forms and support functions which are prominent nondifferentiable convex functions. This thesis also deals with optimality, duality and mixed duality for differentiable and nondifferentiable variational problems involving higher order derivatives, and presents a close relationship between the results of continuous programming problems through the problems with natural boundary conditions between results of their counter parts in nonlinear programming. Finally it formulates a pair of mixed symmetric and self dual differentiable variational problems and gives the validation of various duality results under appropriate invexity and generalized invexity hypotheses. These results are further extended to a nondifferentiable case that involves support functions.Digital copy of Thesis.University of Kashmir

Duality in mathematical programming.

In this thesis entitled, “Duality in Mathematical Programming”, the emphasis is given on formulation and conceptualization of the concepts of second-order duality, second-order mixed duality, second-order symmetric duality in a variety of nondifferentiable nonlinear programming under suitable second-order convexity/second-order invexity and generalized second-order convexity / generalized second-order invexity. Throughout the thesis nondifferentiablity occurs due to square root function and support functions. A support function which is more general than square root of a positive definite quadratic form. This thesis also addresses second-order duality in variational problems under suitable second-order invexity/secondorder generalized invexity. The duality results obtained for the variational problems are shown to be a dynamic generalization for thesis of nonlinear programming problem.Digital copy of Thesis.University of Kashmir

Solutions of Optimization Problems on Hadamard Manifolds with Lipschitz Functions

The aims of this paper are twofold. First, it is shown, for the first time, which types of nonsmooth functions are characterized by all vector critical points as being efficient or weakly efficient solutions of vector optimization problems in constrained and unconstrained scenarios on Hadamard manifolds. This implies the need to extend different concepts, such as the Karush-Kuhn-Tucker vector critical points and generalized invexity functions, to Hadamard manifolds. The relationships between these quantities are clarified through a great number of explanatory examples. Second, we present an economic application proving that Nash's critical and equilibrium points coincide in the case of invex payoff functions. This is done on Hadamard manifolds, a particular case of noncompact Riemannian symmetric spaces