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In this paper, a new pair of higher-order nondifferentiable multiobjective mixed symmetric dual programs over arbitrary cones is formulated, where each of the objective functions contains a support function of a compact convex set. Usual duality theorems are established under higher-order K- (F , α , ρ , d) -convexity assumptions. Also, the example of a higher-order dual pair, which shows that higher-order provides tighter bounds for the value of the objective function of the primal and dual problem, is given in the paper. Several known results are also discussed as special cases. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a modified Tseng's extragradient method with a new step-length rule to solve pseudo-monotone variational inequalities in real Hilbert spaces. Under suitable conditions, the sequence generated by this algorithm strongly converges to the common elements of the solution set of pseudo-monotone variational inequality problems and the fixed point set of k-demicontractive mappings. Finally, we give some numerical experiments to illustrate the effectiveness of the proposed algorithm. The main results of this paper generalize and improve some known results in the literature. [ABSTRACT FROM AUTHOR]

The growing use of optimization models to help decision making has created a demand for such tools that allow formulating and solving more models of real-world processes and systems related to human activity in which hypotheses are not verify in a way specific for classical optimization. One of the approaches for real-world extremum problems under uncertainty is interval-valued optimization. In this paper, a twice differentiable vector optimization problem with multiple interval-valued objective function and both inequality and equality constraints is considered. In this paper, the first order necessary optimality conditions of Karush-Kuhn-Tucker type are proved for differentiable interval-valued vector optimization problems under the first order constraint qualification. If the interval-valued objective function is assumed to be twice weakly differentiable and constraints functions are assumed to be twice differentiable, then two types of second order necessary optimality conditions under two various constraint qualifications are proved for such smooth interval-valued vector optimization problems. Finally, in order to illustrate the Karush-Kuhn-Tucker type necessary optimality conditions established in the paper, an example of an interval-valued optimization is given. [ABSTRACT FROM AUTHOR]

In this paper, we develop different types of strategic alliance and business cooperation between different decision-making units (DMUs) based on ratio-based data envelopment analysis (DEA-R) models. In this regard, two DMUs as partners share some of their input components in order to improve their performance in the partnership process. This is possible when a DMU has a surplus of resources in an input component, while another DMU has a shortage of resources in this component, and this adjustment of resources is done in a way to improve the performance of both DMUs. In the following, we present the inverse DEA-R models in order to determine the optimal level of some input and output components from two DMUs in the process of partnerships as partners, in order to reach the target efficiency level for each of these DMUs. Considering that DEA-R models have a larger space than weights based on the ratio of input to output components (and vice versa), therefore we expect the presented DEA-R models to have a higher discriminating power in the selection of weights compared to data envelopment analysis (DEA) models, then DEA-R models avoid pseudo-inefficiency and efficiency underestimation issues and achieve potential gains for DMUs in the process of partnerships in a competitive market. We present an application of the proposed approach to the data set of 70 banks operating in the European Union and show that these banks can improve their performance based on the optimal strategy presented. [ABSTRACT FROM AUTHOR]

Abdessamad Oussarhana and Tijani AmahroqbaLIMATI Laboratory, Poly-disciplinary Faculty, Sultan Moulay Slimane University, Beni Mellal, Morocco; bLAMAI Laboratory, Faculty of Sciences and Techniques, Cadi Ayyad University, Marrakech, Morocco We provide a counterexample to Proposition 2.3 in our paper [Optimization 20, 69(12), pp. 2611–2627] where we said that the convex set A and the element $ x_0 $ x 0 can be properly separated by a closed hyperplane if $ x_0\notin {\rm pri}(A) $ x 0 ∉ pri (A) , with $ {\rm pri}(A) $ pri (A) is the pseudo relative interior of A. In addition, we re-establish all results rely on this proposition, more precisely, [Amahroq and Oussarhan Infinite-dimensional vector optimization and a separation theorem. Optimization. 2020;69(12):2611–2627, Lemma 2.1 (ii)–(iv)], [Amahroq and Oussarhan Infinite-dimensional vector optimization and a separation theorem. Optimization. 2020;69(12):2611–2627, Theorem 3.1] and [Amahroq and Oussarhan Infinite-dimensional vector optimization and a separation theorem. Optimization. 2020;69(12):2611–2627, Corollary 5.1] (see Section 1). A revision to the proof of [Amahroq and Oussarhan Infinite-dimensional vector optimization and a separation theorem. Optimization. 2020;69(12):2611–2627, Theorem 4.1] is also done (see Section 2). We claim that, the rest of the paper (Amahroq and Oussarhan Infinite-dimensional vector optimization and a separation theorem. Optimization. 2020;69(12):2611–2627) is independent of the above incorrect statements. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the concept of approximate Benson properly efficient solutions for the set-valued equilibrium problems (in short, SVEP) and investigate its properties. Under some suitable assumptions, the linear scalarization theorems for SVEP are obtained. Two nonlinear scalarization theorems for SVEP are presented. Based on the linear scalarization results, the nonemptiness and connectedness of the approximate Benson properly efficient solution set are established under some suitable conditions in real locally convex spaces. Some examples are also given to illustrate our results. The main results of this paper improve and generalize some known results in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the solution of generalized vector bifunction variational inequalities of Stampachhia and Minty types with the concept of generalized cone. Existence results have been established over compact and non-compact sets in Banach spaces without using convexity condition on a bifunction. Some relationship between generalized vector bifunction variational inequalities and vector optimization problems have been established. [ABSTRACT FROM AUTHOR]

Berge's notion of equilibrium represents a complementary alternative to the Nash equilibrium when modeling socioeconomic behavior and human interactions. While the notion of Nash equilibrium is based on self-interest, as players seek to maximize their own payoffs given the action of the other players, the idea behind Berge equilibrium is mutual support, as given the action of one of the players, all others select their actions looking for her best interest. However, because of the demanding conditions involved, the existence of Berge equilibria is rarely guaranteed. In this paper, we propose vector-valued normal-form games as a unified framework in which to study and extend the concept of Berge equilibrium. Based on the equilibria of the so-called altruistic game, we introduce new equilibrium concepts which constitute different relaxations of Berge's notion, although they still retain the underlying idea of mutual support. We establish the links between these new equilibria, Nash equilibrium, Berge equilibrium, and other related concepts already existing in the literature. Our approach has the advantage that it permits the incorporation of preference information to identify the equilibria which are consistent with different altruistic attitudes of the players. [ABSTRACT FROM AUTHOR]

In this paper, we introduce new classes of higher order generalized strong invex functions under non-differentiable settings. The optimality results are derived for higher order strict global minimizers of non-differentiable multiobjective programming problems using these functions. Numerical examples and illustrations are provided in support of new classes of functions and the optimality conditions. We also study the mixed dual problem and establish weak, strong and converse duality results. Furthermore, as an application, we present a non-differentiable case of vector variational-like inequality problem and establish the equivalence between its solutions and higher order strict global minimizers of the non-differentiable multiobjective programming problem. [ABSTRACT FROM AUTHOR]

The multi-objective integer programming problem often occurs in multi-criteria decision-making situations, where the decision variables are integers. In the present paper, we have discussed an algorithm for finding all efficient solutions of a multi-objective integer quadratic programming problem. The proposed algorithm is based on the aspect that efficient solutions of a multi-objective integer quadratic programming problem can be obtained by enumerating ranked solutions of an integer quadratic programming problem. For determining ranked solutions of an integer quadratic programming problem, we have constructed a related integer linear programming problem and from ranked solutions of this integer linear programming problem, ranked solutions of the original integer quadratic programming problem are generated. Theoretically, we have shown that the developed method generates the set of all efficient solutions in a finite number of steps, and numerically we have elaborated the working of our algorithm and compared our results with existing algorithms. Further, we have analyzed that the developed method is efficient for solving a multi-objective integer quadratic programming problem with a large number of constraints, variables and objectives. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the globally proper efficiency of set optimization problems. Firstly, we use the so-called certainly set less order relation to define a new kind of set order relation. Based on the new set order relation, we introduce the notion of the globally proper efficient solution of the set optimization problem. Secondly, we establish Lagrange multiplier rule of the set optimization problem. Finally, we obtain Lagrangian duality theorems and saddle point theorems. We also give some examples to illustrate our results. [ABSTRACT FROM AUTHOR]

In this paper, we study and develop robust nonsmooth optimality conditions and duality analysis for an uncertain multiobjective programming problem with constraints ((UCMOP), for brevity). First, we introduce the constraint qualification of the (GRSCQ) type and then establish some robust necessary optimality conditions in terms of the generalized subdifferentials for some types of minima (including robust weakly efficient and robust properly efficient) to such problem involving stable functions. Under suitable assumptions on the pseudo-convexity of objective and constraint functions, robust necessary nonsmooth optimality conditions become robust sufficient optimality conditions. An application of the obtained results for its Wolfe and Mond–Weir types dual problem is presented and some illustrative examples are also provided for our findings. [ABSTRACT FROM AUTHOR]

In this paper, we perform optimality conditions and sensitivity analysis for parametric nonconvex minimax programming problems. Our aim is to study the necessary optimality conditions by using the Mordukhovich (limiting) subdifferential and to give upper estimations for the Mordukhovich subdifferential of the optimal value function in the problem under consideration. The optimality conditions and sensitivity analysis are obtained by using upper estimates for Mordukhovich subdifferentials of the maximum function. The results on optimality conditions are then applied to parametric multiobjective optimization problems. An example is given to illustrate our results. [ABSTRACT FROM AUTHOR]

In this paper, a numerical approximation method is developed to find approximate solutions to a class of constrained multi-objective optimization problems. All the functions of the problem are not necessarily convex functions. At each iteration of the method, a particular type of subproblem is solved using the trust region technique, and the step is evaluated using the notions of actual reduction and predicted reduction. A non-differentiable $ l_{\infty } $ l∞ penalty function restricts the constraint violations. An adaptive BFGS update formula is introduced. Global convergence of the proposed algorithm is established under the Mangasarian-Fromovitz constraint qualification and some mild assumptions. Furthermore, it is justified that the proposed algorithm displays a super-linear convergence rate. Numerical results are provided to show the efficiency of the algorithm in the quality of the approximated Pareto front. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to investigate Lagrange saddle points and calmness properties for a nonconvex multiobjective optimization problem by virtue of image regularity conditions. Following along with the image space analysis, the quasi-interior, quasi-relative interior and linear image regularity conditions are presented. Simultaneously, some equivalent characterizations to Fritz John and Karush/Kuhn-Tucker saddle points are established respectively by virtue of the proposed image regularity conditions. Moreover, some calmness properties are also obtained by means of local conic image regularity conditions in the image space of the multiobjective optimization problem. [ABSTRACT FROM AUTHOR]

In response to the growing maturation of technological level in the modern medical industry, this paper proposes the PZLMVF-VIKOR method to tackle uncertain multi-attribute group decision-making in smart healthcare. First, a novel Pythagorean Z-linguistic modified value function based on regret theory is defined to consider the impact of DMs' emotions on decision outcomes. Then, a combined weighting method is used to obtain the final attribute weight vector by the closeness degree, with the expert weight determination based on consensus measures. Subsequently, we propose an extended VIKOR method in Pythagorean Z-linguistic environment to solve the evaluation problem of the development in smart healthcare. Finally, a practical case and some comparative analysis is conducted to illustrate the justifiability and effectiveness of this method. [ABSTRACT FROM AUTHOR]

Population games are games with a finite set of available strategies and an infinite number of players, in which the reward for choosing a given strategy is a function of the distribution of players over strategies. The paper shows that, in a certain class of maxmin optimization problems, it is possible to associate a population game to a given maxmin problem in such a way that solutions to the optimization problem are found from Nash equilibria of the associated game. Iterative solution methods for maxmin optimization problems can then be derived from systems of differential equations whose trajectories are known to converge to Nash equilibria. In particular, we use a discrete-time version of the celebrated replicator equation of evolutionary game theory, also known in machine learning as the exponential multiplicative weights algorithm. The resulting algorithm can be viewed as a generalization of the Iteratively Reweighted Least Squares (IRLS) method, which is well known in numerical analysis as a useful technique for solving Chebyshev function approximation problems on a finite grid. Examples are provided to show the use of the generalized IRLS method in collective investment and in decision making under model uncertainty. [ABSTRACT FROM AUTHOR]

In this paper, we derive primal and dual second-order necessary and sufficient optimality conditions for a vector optimization problem with equality and inequality constraints where the functions involved are locally Lipschitz. We introduce a weaker notion of second-order Abadie constraint qualification to derive second-order necessary conditions for weak local Pareto minima and strict local Pareto minima of order two in terms of Páles and Zeidan's second-order upper directional derivatives. Dual necessary conditions are derived for both types of minimal solutions in finite-dimensional spaces assuming the functions to be first-order continuously Fréchet differentiable. In the same setting, we derive dual and primal second-order sufficient optimality conditions for strict local Pareto minima of order two in terms of second-order lower directional derivatives. [ABSTRACT FROM AUTHOR]

In this paper, we study a nonlinear scalarization function for a variable domination structure in an arbitrary linear space without assuming any particular topology. Conditions are provided under which the nonlinear scalarization function possesses several useful properties such as finiteness, properness, positive homogeneity, subadditivity, (strict) monotonicity, convexity or continuity. These properties are employed to characterize approximate efficiency in linear spaces. [ABSTRACT FROM AUTHOR]

In this paper, we propose an adaptive sampling stochastic multigradient algorithm for solving stochastic multiobjective optimization problems. Instead of requiring additional storage or computation of full gradients, the proposed method reduces variance by adaptively controlling the sample size used. Without the convexity assumption on the objective functions, we obtain that the proposed algorithm converges to Pareto stationary points in almost surely. We then analyze the convergence rates of the proposed algorithm. Numerical experiments are presented to demonstrate the effectiveness of the proposed algorithm. [ABSTRACT FROM AUTHOR]