Mond-Weir Duality Research Articles

Code and Data Repository for Solving Bilevel Programs Based on Lower-level Mond-Weir Duality

The goal of this repository is to provide a large number of numerical examples to illustrate the effectiveness of solving bilevel programs based on lower-level Mond-Weir duality.

Solving Bilevel Programs Based on Lower-Level Mond-Weir Duality

This paper focuses on developing effective algorithms for solving a bilevel program. The most popular approach is to replace the lower-level problem with its Karush-Kuhn-Tucker conditions to generate a mathematical program with complementarity constraints (MPCC). However, MPCC does not satisfy the Mangasarian-Fromovitz constraint qualification (MFCQ) at any feasible point. In this paper, inspired by a recent work using the lower-level Wolfe duality (WDP), we apply the lower-level Mond-Weir duality to present a new reformulation, called MDP, for bilevel program. It is shown that, under mild assumptions, they are equivalent in globally or locally optimal sense. An example is given to show that, different from MPCC, MDP may satisfy the MFCQ at its feasible points. Relations among MDP, WDP, and MPCC are investigated. On this basis, we extend the MDP reformulation to present another new reformulation (called eMDP), which has similar properties to MDP. Furthermore, to compare two new reformulations with the MPCC and WDP approaches, we design a procedure to generate 150 tested problems randomly and comprehensive numerical experiments show that MDP has quite evident advantages over MPCC and WDP in terms of feasibility to the original bilevel programs, success efficiency, and average CPU time, whereas eMDP is far superior to all other three reformulations. History: Accepted by Pascal Van Hentenryck, Area Editor for Computational Modeling: Methods & Analysis. Funding: This work was supported by the National Natural Science Foundation of China [Grants 12071280 and 11901380]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2023.0108 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2023.0108 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

Higher-order Mond-Weir duality of set-valued fractional minimax programming problems

In this paper, we consider a set-valued fractional minimax programming problem (abbreviated as SVFMPP) (MFP), in which both the objective and constraint maps are set-valued. We use the concept of higher-order ?-cone arcwisely connectivity, introduced by Das [1], as a generalization of higher-order cone arcwisely connected setvalued maps. We explore the higher-order Mond-Weir (MWD) form of duality based on the supposition of higher-order ?-cone arcwisely connectivity and prove the associated higher-order converse, strong, and weak theorems of duality between the primary (MFP) and the analogous dual problem (MWD).

Sufficient optimality conditions and Mond-Weir duality results for a fractional multiobjective optimization problem

<p style='text-indent:20px;'>In this work, we are concerned with a fractional multiobjective optimization problem <inline-formula><tex-math id="M1">\begin{document}$ (P) $\end{document}</tex-math></inline-formula> involving set-valued maps. Based on necessary optimality conditions given by Gadhi et al. [<xref ref-type="bibr" rid="b14">14</xref>], using support functions, we derive sufficient optimality conditions for <inline-formula><tex-math id="M2">\begin{document}$ \left( P\right) , $\end{document}</tex-math></inline-formula> and we establish various duality results by associating the given problem with its Mond-Weir dual problem <inline-formula><tex-math id="M3">\begin{document}$ \left( D\right) . $\end{document}</tex-math></inline-formula> The main tools we exploit are convexificators and generalized convexities. Examples that illustrates our findings are also given.</p>

Set-valued optimization in variable preference structures with new variants of generalized convexity

ABSTRACT A set-valued optimization problem with variable preferences is considered. Relations between local and global solutions, optimality conditions, and Wolfe and Mond-Weir duality properties are studied. Both minimal and nondominated solutions are discussed with general variable preferences. The results are proved for three main types of solutions in vector optimization: weak, Pareto, and strong solutions. New variants of generalized derivatives and convexity are proposed and used in all the results.

On symmetric gH-derivative: Applications to dual interval-valued optimization problems

This paper provides a complete study on properties of symmetric gH-derivative. More precisely, a necessary and sufficient condition for the symmetric gH-differentiability of interval-valued functions is presented. Further, we clarify the relationship between the symmetric gH-differentiability and gH-differentiability. Moreover, quasi-mean value theorem, chain rule and some operations of symmetric gH-differentiable interval-valued functions are established. As applications, we develop the Mond–Weir duality theory for a class of symmetric gH-differentiable interval-valued optimization problems. Weak, strong and strict converse duality theorems are formulated and proved. Also, several examples are presented in order to support the corresponding theoretical results.

Optimality conditions and Mond–Weir duality for a class of differentiable semi-infinite multiobjective programming problems with vanishing constraints

In this paper, the class of differentiable semi-infinite multiobjective programming problems with vanishing constraints is considered. Both Karush–Kuhn–Tucker necessary optimality conditions and, under appropriate invexity hypotheses, sufficient optimality conditions are proved for such nonconvex smooth vector optimization problems. Further, vector duals in the sense of Mond–Weir are defined for the considered differentiable semi-infinite multiobjective programming problems with vanishing constraints and several duality results are established also under invexity hypotheses.

Mond-Weir duality under ρ -(η, θ) -invexity in Banach space

The weak, the strong and the converse duality theorems of a pair of nonlinear programming problems are established under ρ -(η, θ) -invexity, generalized ρ -(η, θ) -invexity, strongly ρ -pseudo-invex functions in Banach spaces. The proposed work deals with suitable numerical examples.

Optimality conditions and duality with new variants of generalized derivatives and convexity

In this paper, a general vector optimization problem with inequality constraints is considered, this topic is very popular and important model with a long research history in optimization. The generality of setting is mainly expressed in the following three factors. The underlying spaces being linear spaces without topology (except the decision space being additionally equipped with this structure in some results). The “orderings” in both objective and constraint spaces are defined by arbitrary nonempty sets (not necessarily convex cones). The problem data are nonsmooth mappings, i.e., they are not Fréchet differentiable. For this problem, the optimality conditions and Wolfe and Mond-Weir duality properties are investigated , which lie at the heart of optimization theory. These results are established for the three main and typical optimal solutions: (Pareto) minimal, weak minimal, and strong minimal solutions in both local and global considerations. The research define a type of Gateaux variation to play the role of a derivative. For optimality conditions, and introduce the concepts of on-set differentiable quasiconvexity for global solutions and sequential differentiable quasiconvexity for local ones. Furthermore, each of them is separated into type 1 for sufficient optimality conditions and type 2 for necessary ones. After obtaining optimality conditions, applying them to derive weak and strong duality relations for the above types of solutions following our duality schemes of the Wolfe and Mon-Weir types. Due to the complexity of the research subject: considerations of duality are different from that of optimality conditions, we have to design two more appropriate types of generalized quasiconvexity: scalar quasiconvexity for the weak solution and scalar strict convexity for the Pareto solution. So all the results are in terms of the aforementioned Gateaux variation and various types of generalized quasiconvexity. The results are remarkably different from the related known ones with some clear advantages in particular cases of applications.

Điều kiện tối ưu và đối ngẫu cho bài toán tối ưu đa trị sử dụng đạo hàm đa trị Clarke theo hướng nón

This paper is to deal with Mond-Weir duality and Wolfe duality for constrained set-valued optimization problems in terms of cone-directed Clarke derivatives. Firstly, necessary and sufficient optimality conditions for constrained set-valued optimizations in terms of cone-directed Clarke derivatives for the cone-semilocally convex like maps are investigated. Then, the Mond-Weir duality and Wolfe duality for a constrained set-valued optimization and their weak duality, strong duality and converse duality are considered.

Sufficient optimality conditions and duality results for a bilevel multiobjective optimization problem via a Ψ reformulation

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.

On duality of fuzzy multiobjective optimisation problems: application to a multiplicative search technique

In this paper, we present a new study to minimise the expected value of the first meeting time between one of the searchers and the randomly moving target in the space (case of the multiplicative search technique) from a viewpoint of computational optimisation. We introduce the dual problem for the fuzzy multiobjective nonlinear programming problem (FMONLP) with inequality constraints which is obtained in El-Hadidy [International Journal of Computational Methods, Vol. 13, No. 6 (2016) 1650038 (38 pages)]. For this problem, we generalise the following classes of the vector-valued functions for fuzzy sets: weak strictly pseudo-quasi type I, strong pseudo-quasi type I, weak quasi strictly type I and weak strictly pseudo type I. We provide the fuzzy forms of Mond-Weir and general Mond-Weir duals for the involved problem and consequently various duality results are derived.

On duality of fuzzy multiobjective optimisation problems: application to a multiplicative search technique

In this paper, we present a new study to minimise the expected value of the first meeting time between one of the searchers and the randomly moving target in the space (case of the multiplicative search technique) from a viewpoint of computational optimisation. We introduce the dual problem for the fuzzy multiobjective nonlinear programming problem (FMONLP) with inequality constraints which is obtained in El-Hadidy [International Journal of Computational Methods, Vol. 13, No. 6 (2016) 1650038 (38 pages)]. For this problem, we generalise the following classes of the vector-valued functions for fuzzy sets: weak strictly pseudo-quasi type I, strong pseudo-quasi type I, weak quasi strictly type I and weak strictly pseudo type I. We provide the fuzzy forms of Mond-Weir and general Mond-Weir duals for the involved problem and consequently various duality results are derived.

Mond-Weir Duality for Multiobjective Programming with Invexity

Mond-Weir Duality for Multiobjective Programming with Invexity

On optimality conditions and duality results in a class of nonconvex quasidifferentiable optimization problems

In the paper, the class of nonconvex nonsmooth optimization problems with the quasidifferentiable functions is considered. Further, a new notion of nonsmooth generalized convexity, namely, the concept of r-invexity with respect to a convex compact set is introduced. Several conditions for quasidifferentiable r-invexity with respect to a convex compact set are given. Furthermore, the sufficient optimality conditions and several Mond–Weir duality results are established for the considered nonconvex quasidifferentiable optimization problem under assumption that the functions constituting it are r-invex with respect to the same function eta and with respect to convex compact sets which are equal to Minkowski sum of their subdifferentials and superdifferentials. It is also illustrated that, for such nonsmooth extremum problems, the Lagrange multipliers may not be constant.

Sufficiency and Duality for Nonsmooth Multiobjective Programming Problems Involving Generalized (Φ, ρ)-V-Type I Functions

In this paper, a class of nonsmooth multiobjective programming problems is considered. We introduce the new concepts of (Φ, ρ)-V-type I, (pseudo, quasi) (Φ, ρ)-V-type I and (quasi, pseudo) (Φ, ρ)-V-type I functions, in which the involved functions are locally Lipschitz. Based upon these generalized (Φ, ρ)-V-type I functions, the sufficient optimality conditions for weak efficiency, efficiency and proper efficiency are derived. Mond-Weir duality results are also established under the aforesaid functions.

Interval-valued optimization problems involving (α, ρ)-right upper-Dini-derivative functions.

We consider an interval-valued multiobjective problem. Some necessary and sufficient optimality conditions for weak efficient solutions are established under new generalized convexities with the tool-right upper-Dini-derivative, which is an extension of directional derivative. Also some duality results are proved for Wolfe and Mond-Weir duals.

Higher Order Mond-Weir Duality for Super Efficient Solutions of Set-Valued Optimization

In real normed linear spaces,by virtue of the cone-directed higher order generalized adjacent derivatives,a higher order Mond-Weir dual problem for a constrained set-valued optimization is considered in the sense of super efficient solutions.Under the assumption of generalized cone-convexity,with the help of the properties of cone-directed higher order generalized adjacent derivatives by applying separate theorem for convex sets,a strong duality theorem is established.By taking advantage of the scalarization theorem for a super efficient point,a converse duality theorem is obtained.

Proper efficiency conditions and duality results for nonsmooth vector optimization in Banach spaces under [formula omitted]-invexity

In this paper, a new class of nonsmooth nonconvex multiobjective programming problems in Banach spaces is considered. The sufficient optimality conditions for efficiency and proper efficiency are established under nonsmooth (Φ,ρ)-invexity and generalized (Φ,ρ)-invexity notions, after extending these concepts to the functions between Banach spaces. Further, Mond–Weir duality theorems are also established under (Φ,ρ)-invexity for the considered vector optimization problems in Banach spaces.

Sufficient Conditions and Duality Theorems for Nondifferentiable Minimax Fractional Programming

We consider nondifferentiable minimax fractional programming problems involving B-(p, r)-invex functions with respect to η and b. Sufficient optimality conditions and duality results for a class of nondifferentiable minimax fractional programming problems are obtained undr B-(p, r)-invexity assumption on objective and constraint functions. Parametric duality, Mond-Weir duality, and Wolfe duality problems may be formulated, and duality results are derived under B-(p, r)-invex functions.