Sufficient Conditions and Duality Results in Bilevel Optimization Using Tangential Subdifferentials

ABSTRACTIn this paper, we are concerned with a bilevel multiobjective optimization problem . Using the function Ψ introduced by Gadhi and Dempe [Necessary optimality conditions and a new approach to multiobjective bilevel optimization problems. J Optim Theory Appl. 2012;155:100–114], we reformulate as a single level mathematical programming problem and establish/exhibit the global equivalence between the two problems and . Using a generalized convexity introduced by Dutta and Chandra [Convexificator, generalized convexity and vector optimization. Optimization. 2004;53:77–94], we derive sufficient optimality conditions for the problem and establish Mond-Weir duality results. To illustrate the obtained results some examples are given.

Fractional Calculus Approach for Variational Problems: Characterization of Sufficient Optimality Conditions and Duality

A popular way for developing optimality conditions of bilevel programs is to express them as (single-level) constrained optimization problems involving optimal value functions of lower programs, in which the computation of first/second-order directional derivatives for the value functions or solution mappings of the lower level problems is a challenging task. This type of directional derivatives based optimality conditions are not suitable for analysing theoretical properties of numerical algorithms for bilevel programs, especially when lower programs are non-convex optimization problems. This paper focuses on bilevel programs whose lower level optimization problems are non-convex. We introduce the concept of bi-local solutions and prove that, under the Jacobian uniqueness conditions on the lower level problem, the bi-local solutions are just the local solutions of the optimization problem constrained by Karush–Kuhn–Tucker (KKT) conditions of the lower level problem. Importantly, under suitable conditions, we prove that the KKT constrained optimization problem satisfies the Mangasarian–Fromovitz constraint qualification (MFCQ). Based on the equivalence of bi-local solutions and local solutions of the KKT constrained optimization problem, we establish the second-order necessary and sufficient optimality conditions for the bi-local optimal solutions. Different from second-order optimality conditions in terms of second-order directional derivatives of value functions, the second-order optimality conditions in this paper rely only on the second-order derivatives of the problem functions of the bilevel program. Moreover, the second-order sufficient optimality conditions can be applied for analysing convergence properties of numerical algorithms for the bilevel program.

Strong duality for a trust-region type relaxation of the quadratic assignment problem

In this paper we introduce the notion of \(\rho -(\eta , \theta )\)-B-invex function and generalized \(\rho -(\eta ,\theta )\)-B-invex function between Banach spaces. By considering these functions, sufficient optimality conditions are obtained for a single objective optimization problem in Banach space. Duality results (i.e. weak duality, strong duality and converse duality of Mond–Weir type and similar to Mixed type duals) are established under \(\rho -(\eta , \theta )\)-B-invexity and weak and strong duality of Mond–Weir type dual are also established under generalized \(\rho -(\eta ,\theta )\)-B-invexity in Banach space.

In this paper, a strong nonlinear Lagrangian duality result is established for an inequality constrained multiobjective optimization problem. This duality result improves and unifies existing strong nonlinear Lagrangian duality results in the literature. As a direct consequence, a strong nonlinear Lagrangian duality result for an inequality constrained scalar optimization problem is obtained. Also, a variant set of conditions is used to derive another version of the strong duality result via nonlinear Lagrangian for an inequality constrained multiobjective optimization problem.

In the present paper, we focus our study on an interval-valued variational problem and derive sufficient optimality conditions by using the notion of invexity. In order to relate the primal interval-valued variational problem and its dual, several duality results, viz., weak, strong and converse duality results are established. Further, the Lagrangian function for the considered interval-valued variational problem is defined and we present some relations between an optimal solution of the considered interval-valued variational problem and a saddle point of the Lagrangian function. In order to illustrate the results proved in the paper, some examples of interval-valued variational problems have been formulated.

Fritz-John Optimality and Duality for Non-convex Programs

The importance of quasi efficiency lies in its versatile nature as it permits a definite tolerable error that depend on the decision variables. This has been a motivating factor for us to introduce the notion of quasi e cient solution for the non-smooth multiobjective continuous time programming problem. Necessary optimality conditions are derived for this problem. To derive sufficient optimality conditions, the concept of approximate convexity has been extended to continuous case in this paper. A mixed dual is proposed for which weak and strong duality results are proved.

Duality and penalty methods are popular in optimization. The study on duality and penalty methods for nonconvex multiobjective optimization problems is very limited. In this paper, we introduce vector-valued nonlinear Lagrangian and penalty functions and formulate nonlinear Lagrangian dual problems and nonlinear penalty problems for multiobjective constrained optimization problems. We establish strong duality and exact penalization results. The strong duality is an inclusion between the set of infimum points of the original multiobjective constrained optimization problem and that of the nonlinear Lagrangian dual problem. Exact penalization is established via a generalized calmness-type condition.

This paper introduces some new generalizations of the concept of (Φ,ρ)-invexity for nondifferentiable locally Lipschitz functions using the tools of Clarke subdifferential. These functions are used to derive the necessary and sufficient optimality conditions for a class of nonsmooth semi-infinite minmax programming problems, where set of restrictions are indexed in a compact set. Utilizing the sufficient optimality conditions, the authors formulate three types of dual models and establish weak and strong duality results. The results of the paper extend and unify naturally some earlier results from the literature.

In this paper, we consider new sufficient conditions of optimality of the second-order for equality constrained optimization problems, which essentially enhance and complement the classical ones and are constructive. For example, they establish equivalence between sufficient conditions in the equality constrained optimization problems and sufficient conditions for optimality in equality constrained problems by reducing the latter to equalities with the help of introducing slack variables. Previously, when using the classical sufficient optimality conditions, this fact was not considered to be true, that is, the existing classical sufficient conditions were not complete, so the proposed optimality conditions complement the classical ones and close the question of the equivalence of the problems with inequalities and the problems with equalities when reducing the first to the second by introducing slack variables.

Both parametric and nonparametric necessary and sufficient optimality conditions are established for a class of nonsmooth continuous-time programming problems. Subsequently, the structure and contents of these optimality results are utilized as a basis for constructing two parametric and four parameter-free duality models, and for proving weak, strong, and strict converse duality theorems. Furthermore, it is briefly pointed out how similar optimality and duality results can be derived for two special cases of the main problem containing arbitrary norms and square roots of positive semidefinite quadratic forms. These results generalize a number of similar results in the area of continuous-time programming, and subsume a great variety of cognate results originally developed for finite-dimensional nonlinear and fractional programming problems.

A Co-evolutionary Decomposition-based Chemical Reaction Algorithm for Bi-level Combinatorial Optimization Problems

In this chapter, we present a unified formulation of generalized convex functions. Based on these concepts, sufficient optimality conditions for a nondifferentiable multiobjective programming problem are presented. We also introduce a general Mond-Weir type dual problem of the problem and establish weak duality theorem under generalized convexity assumptions. Strong duality result is derived using a constraint qualification for nondifferentiable multiobjective programming problems.