Minimax Fractional Programming Problem Research Articles

(G, V)-invexity nondifferentiable generalized minimax fractional programming

In this paper, we study the nondifferentiable generalized minimax fractional programming problem under (G, V )-invexity. G-sufficient optimality conditions for the considered nondifferentiable generalized minimax fractional programming problem are established under the concept of (G, V )-invexity. Further, two types of dual models are formulated and G-duality theorems relating to the primal minimax fractional programming problem and dual problems are established. These results extend several known results to a wider class of programs.

Set-valued minimax fractional programming problems under ρ-cone arcwise connectedness

Abstract In this paper, we consider a set-valued minimax fractional programming problem (MFP), where the objective as well as constraint maps are set-valued. We introduce the notion of ρ-cone arcwise connectedness of set-valued maps as a generalization of cone arcwise connected set-valued maps. We establish the sufficient Karush-Kuhn-Tucker (KKT) conditions for the existence of minimizers of the problem (MFP) under ρ-cone arcwise connectedness assumption. Further, we study the Mond-Weir (MWD), Wolfe (WD), and mixed (MD) types of duality models and prove the corresponding weak, strong, and converse duality theorems between the primal (MFP) and the corresponding dual problems under ρ-cone arcwise connectedness assumption.

Sufficient optimality condition and duality of nondifferentiable minimax ratio constraint problems under (p, r)-ρ-(η, θ)-invexity

Abstract There are several classes of decision-making problems that explicitly or implicitly prompt fractional programming problems. Portfolio selection problems, agricultural planning, information transfer, numerical analysis of stochastic processes, and resource allocation problems are just a few examples. The huge number of applications of minimax fractional programming problems inspired us to work on this topic. This paper is concerned with a nondifferentiable minimax fractional programming problem. We study a parametric dual model, corresponding to the primal problem, and derive the sufficient optimality condition for an optimal solution to the considered problem. Further, we obtain the various duality results under (p, r)-ρ-(η, θ)-invexity assumptions. Also, we identify a function lying exclusively in the class of (−1, 1)-ρ-(η, θ)-invex functions but not in the class of (1, −1)-invex functions and convex function already existing in the literature. We have given a non-trivial model of nondifferentiable minimax problem and obtained its optimal solution using optimality results derived in this paper.

Nondifferentiable generalized minimax fractional programming under (Ф,ρ)-invexity

In this paper, a class of nonconvex nondifferentiable generalized minimax fractional programming problems is considered. Sufficient optimality conditions for the considered nondifferentiable generalized minimax fractional programming problem are established under the concept of (?, ?)-invexity. Further, two types of dual models are formulated and various duality theorems relating to the primal minimax fractional programming problem and dual problems are established. The results established in the paper generalize and extend several known results in the literature to a wider class of nondifferentiable minimax fractional programming problems. To the best of our knowledge, these results have not been established till now.

On Minimax Fractional Semi-Infinite Programming Problems with Applications

Using some advanced tools of variational analysis and generalized differentiation, we establish necessary conditions for locally optimal solutions of minimax fractional programming problems under the limiting constraint qualification (LCQ). Sufficient conditions for such solutions to the considered problem are also provided by introducing generalized convex functions defined in terms of the limiting subdifferential for locally Lipschitz functions. In addition, some duality results for minimax fractional programming problems are also provided. Finally, by using the obtained results, we derive necessary and sufficient conditions for weak Pareto solutions to the multiobjective semi-infinite fractional optimization problem.

Mixed Lagrange function in minimax fractional programming problems

In this paper, the aim of this work to study mixed Lagrange function for the minimax fractional programming problem with nonsmooth exponential $(p,r)$-invex functions with respect to $\eta$. We introduced a new concept of saddle point for a mixed Lagrange function. We present the equivalence between a saddle point of the mixed Lagrange function and an optimal solution in the considered minimax fractional programming problem under appropriate nonsmooth exponential $(p,r)$-invexity hypotheses.

Robust optimality and duality for minimax fractional programming problems with support functions

Robust optimality and duality for minimax fractional programming problems with support functions

Higher-Order Non-Symmetric Duality for Nondifferentiable Minimax Fractional Programs with Square Root Terms

In this paper, we emphasize on a nondifferentiable minimax fractional programming (NMFP) problem and obtain appropriate duality results for higher-order dual model under higher-order B-(p,r)-invex functions. We provide a nontrivial illustration of a function which belongs to the class of higher-order B-(p,r)-invex but not in the class of second-order B-(p,r)-invex functions already existing in literature. An example of finding a minimax solution of NMFP problem by using higher-order B-(p,r)-invex functions has also been given. Various known results are discussed as particular cases.

Duality for Unified Higher-Order Minimax Fractional Programming with Support Function under Type-I Assumptions

This article is devoted to discussing the nondifferentiable minimax fractional programming problem with type-I functions. We focus our study on a nondifferentiable minimax fractional programming problem and formulate a higher-order dual model. Next, we establish weak, strong, and strict converse duality theorems under generalized higher-order strictly pseudo ( V , α , ρ , d ) -type-I functions. In the final section, we turn our focus to study a nondifferentiable unified minimax fractional programming problem and the results obtained in this paper naturally unify. Further, we extend some previously known results on nondifferentiable minimax fractional programming in the literature.

Optimality conditions for set-valued minimax fractional programming problems

In this paper, we establish the sufficient Karush–Kuhn–Tucker (KKT) conditions for the existence of minimizers of a set-valued minimax fractional programming problem. The duals of Mond–Weir, Wolfe, and mixed types of the said problem are also formulated and the duality results are proved.

Optimality conditions and duality for minimax fractional programming problems with data uncertainty

In this paper, we consider minimax nondifferentiable fractional programming problems with data uncertainty in both the objective and constraints. Via robust optimization, we establish the necessary and sufficient optimality conditions for an uncertain minimax convex-concave fractional programming problem under the robust subdifferentiable constraint qualification. Making use of these optimality conditions, we further obtain strong duality results between the robust counterpart of this programming problem and the optimistic counterpart of its conventional Wolf type and Mond-Weir type dual problems. We also show that the optimistic counterpart of the Wolf type dual of an uncertain minimax linear fractional programming problem with scenario uncertainty (or interval uncertainty) in objective function and constraints is a simple linear programming, and show that the robust strong duality results in sense of Wolf type always hold for this linear minimax fractional programming problem.

Semi-infinite minimax fractional programming under (Φ, <i>ρ</i>)-<i>V</i>-invexity and generalised (Φ, <i>ρ</i>)-<i>V</i>-invexity. Optimality

Semi–infinite minimax fractional programming problems with both inequality and equality constraints are considered. The sets of parametric sufficient optimality conditions for a new class of non–convex differentiable semi–infinite minimax fractional programming problems are obtained under (generalised) (Φ, ρ)–V–invexity assumptions imposed on the objective and constraint functions. With the reference to the said functions, we extend some results of optimality for a larger class of non–convex semi–infinite minimax fractional programming problems in comparison to those ones, previously established in the literature under other generalised convex notions.

Remarks on duality for second order minimax fractional programming with B-(p,r)-invex functions

In this paper, we have pointed out that the proofs of the Theorems 3.1 and 3.3 in the recent paper (Sonali et al. in Ann Oper Res 244:603–617, 2016) are erroneous. A modified dual to the second order minimax fractional programming problem has been formulated and the rectified proofs of these results under generalized B-(p,r)-invexity have been established.

Necessary optimality conditions for minimax programming problems with mathematical constraints

In this paper, necessary optimality conditions in terms of upper and/or lower subdifferentials of both cost and constraint functions are derived for minimax optimization problems with inequality, equality and geometric constraints in the setting of non-differentiatiable and non-Lipschitz functions in Asplund spaces. Necessary optimality conditions in the fuzzy form are also presented. An application of the fuzzy necessary optimality condition is shown by considering minimax fractional programming problem.

A new branch and bound algorithm for minimax ratios problems

Abstract This study presents an efficient branch and bound algorithm for globally solving the minimax fractional programming problem (MFP). By introducing an auxiliary variable, an equivalent problem is firstly constructed and the convex relaxation programming problem is then established by utilizing convexity and concavity of functions in the problem. Other than usual branch and bound algorithm, an adapted partition skill and a practical reduction technique performed only in an unidimensional interval are incorporated into the algorithm scheme to significantly improve the computational performance. The global convergence is proved. Finally, some comparative experiments and a randomized numerical test are carried out to demonstrate the efficiency and robustness of the proposed algorithm.

Saddle Point Criteria in Nonsmooth Semi-Infinite Minimax Fractional Programming Problems

This paper considers a nonsmooth semi-infinite minimax fractional programming problem (SIMFP) involving locally Lipschitz invex functions. The authors establish necessary optimality conditions for SIMFP. The authors establish the relationship between an optimal solution of SIMFP and saddle point of scalar Lagrange function for SIMFP. Further, the authors study saddle point criteria of a vector Lagrange function defined for SIMFP.

Duality for Semi-Infinite Minimax Fractional Programming Problem Involving Higher-Order (Φ, ρ)-V-Invexity

ABSTRACTIn this paper, we introduce the concept of higher-order (Φ, ρ)-V-invexity and present two types of higher-order dual models for a semi-infinite minimax fractional programming problem. Weak, strong, and strict converse duality theorems are discussed under the assumptions of higher-order (Φ, ρ)-V-invexity to establish a relation between the primal and dual problems.

Duality for nondifferentiable minimax fractional programming problem involving higher order $$(\varvec{C},\varvec{\alpha}, \varvec{\rho}, \varvec{d})$$ ( C , α , ρ , d ) -convexity

In this paper, we present new class of higher-order \((C, \alpha , \rho , d)\)-convexity and formulate two types of higher-order duality for a nondifferentiable minimax fractional programming problem. Based on the higher-order \((C, \alpha , \rho , d)\)-convexity, we establish appropriate higher-order duality results. These results extend several known results to a wider class of programs.

Saddle point criteria in semi-infinite minimax fractional programming under (Φ,ρ)-invexity

Semi-infinite minimax fractional programming problems with both inequality and equality constraints are considered. The sets of parametric saddle point conditions are established for a new class of nonconvex differentiable semi-infinite minimax fractional programming problems under(?,?)-invexity assumptions. With the reference to the said concept of generalized convexity, we extend some results of saddle point criteria for a larger class of nonconvex semi-infinite minimax fractional programming problems in comparison to those ones previously established in the literature.

OPTIMALITY AND PARAMETRIC DUALITY FOR NONSMOOTH MINIMAX FRACTIONAL PROGRAMMING PROBLEMS INVOLVING L-INVEX-INFINE FUNCTIONS

The Karush-Kuhn-Tucker type necessary optimality conditions are given for the nonsmooth minimax fractional programming problem with inequality and equality constraints. Subsequently, based on the idea of L-invex-infine functions defined in terms of the limiting/Mordukhovich subdifferential of locally Lipschitz functions, we obtain sufficient optimality conditions for the considered nonsmooth minimax fractional programming problem and also we provide an example to justify the existence of sufficient optimality conditions. Furthermore, we propose a parametric type dual problem and explore duality results.