Search

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]

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]

Application problems can often not be solved adequately by numerical algorithms as several difficulties might arise at the same time. When developing and improving algorithms which hopefully allow to handle those difficulties in the future, good test instances are required. These can then be used to detect the strengths and weaknesses of different algorithmic approaches. In this paper we present a generator for test instances to evaluate solvers for multiobjective mixed-integer linear and nonlinear optimization problems. Based on test instances for purely continuous and purely integer problems with known efficient solutions and known nondominated points, suitable multiobjective mixed-integer test instances can be generated. The special structure allows to construct instances scalable in the number of variables and objective functions. Moreover, it allows to control the resulting efficient and nondominated sets as well as the number of efficient integer assignments. [ABSTRACT FROM AUTHOR]

In multi-objective mixed-integer convex optimization, multiple convex objective functions need to be optimized simultaneously while some of the variables are restricted to take integer values. In this paper, we present a new algorithm to compute an enclosure of the nondominated set of such optimization problems. More precisely, we decompose the multi-objective mixed-integer convex optimization problem into several multi-objective continuous convex optimization problems, which we refer to as patches. We then dynamically compute and improve coverages of the nondominated sets of those patches to finally combine them to obtain an enclosure of the nondominated set of the multi-objective mixed-integer convex optimization problem. Additionally, we introduce a mechanism to reduce the number of patches that need to be considered in total. Our new algorithm is the first of its kind and guaranteed to return an enclosure of prescribed quality within a finite number of iterations. For selected numerical test instances we compare our new criterion space based approach to other algorithms from the literature and show that much larger instances can be solved with our new algorithm. [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]

This paper is devoted to the study of a class of multiobjective semi-infinite programming problems on Hadamard manifolds (in short, (MOSIP-HM)). We derive some alternative theorems analogous to Tucker's theorem, Tucker's first and second existence theorem, and Motzkin's theorem of alternative in the framework of Hadamard manifolds. We employ Motzkin's theorem of alternative to establish necessary and sufficient conditions that characterize KKT pseudoconvex functions using strong KKT vector critical points and efficient solutions of (MOSIP-HM). Moreover, we formulate the Mond-Weir and Wolfe-type dual problems related to (MOSIP-HM) and derive the weak and converse duality theorems relating (MOSIP-HM) and the dual problems. Several non-trivial numerical examples are provided to illustrate the significance of the derived results. The results deduced in the paper extend and generalize several notable works existing 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 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]

Bilevel programming is characterized by the existence of two optimization problems in which the constraint region of the upper level problem is implicitly determined by the lower level optimization problem. This hierarchical design of optimization is suitable to model a large number of real-life applications. However, when dealing with a non linear multi-objective optimization context, new complexities arise due to conflicting objectives. In this paper, an exact method is described to solve an integer indefinite quadratic bilevel maximization problem with multiple objectives at the upper level, where the objective functions at both levels are the product of two linear functions. The algorithm suggested aims to produce a set of efficient solutions by employing a branch and cut approach. It optimizes the indefinite quadratic problem of the upper level within the feasible region of the original problem in an iterative manner. Then, it introduces the Dantzig cut technique to identify the optimal solution for the integer indefinite quadratic bilevel programming problem. Additionally, the algorithm utilizes an efficient cut that reduces the search process for obtaining the set of efficient solutions of the main problem, along with a branching constraint for the integer decision variables. The algorithm was implemented and tested on instances generated randomly, yielding positive outcomes. [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 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]

This paper deals with a certain class of multiobjective semi-infinite programming problems with switching constraints (in short, MSIPSC) in the framework of Hadamard manifolds. We introduce Abadie constraint qualification (in short, ACQ) for MSIPSC in the Hadamard manifold setting. Necessary criteria of weak Pareto efficiency for MSIPSC are derived by employing ACQ. Further, sufficient criteria of weak Pareto efficiency for MSIPSC are deduced by using geodesic quasiconvexity and pseudoconvexity assumptions. Subsequently, Mond–Weir type and Wolfe type dual models are formulated related to the primal problem MSIPSC, and thereafter, several duality results are established that relate MSIPSC and the corresponding dual models. Several non-trivial examples are furnished in the framework of well-known Hadamard manifolds, such as the set consisting of all symmetric positive definite matrices and the Poincaré half plane, to illustrate the importance of the results derived in this article. To the best of our knowledge, this is the first time that optimality conditions and duality results for MSIPSC have been studied in the setting of Hadamard manifolds. [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]

Multiobjective integer programs (MOIPs) simultaneously optimize multiple objective functions over a set of linear constraints and integer variables. In this paper, we present continuous, convex hull and Lagrangian relaxations for MOIPs and examine the relationship among them. The convex hull relaxation is tight at supported solutions, i.e., those that can be derived via a weighted-sum scalarization of the MOIP. At unsupported solutions, the convex hull relaxation is not tight and a Lagrangian relaxation may provide a tighter bound. Using the Lagrangian relaxation, we define a Lagrangian dual of an MOIP that satisfies weak duality and is strong at supported solutions under certain conditions on the primal feasible region. We include a numerical experiment to illustrate that bound sets obtained via Lagrangian duality may yield tighter bounds than those from a convex hull relaxation. Subsequently, we generalize the integer programming value function to MOIPs and use its properties to motivate a set-valued superadditive dual that is strong at supported solutions. We also define a simpler vector-valued superadditive dual that exhibits weak duality but is strongly dual if and only if the primal has a unique nondominated point. [ABSTRACT FROM AUTHOR]

In this paper, by revisiting coderivative calculus rules for convex multifunctions in finite-dimensional spaces, we derive formulae for estimating/computing the basic subdifferential and the coderivative of the efficient point multifunction of parametric convex vector optimization problems. These results are then applied to a broad class of conventional convex vector optimization problems with the presence of operator constraints and equilibrium ones. Examples are also designed to analyze and illustrate the obtained results. [ABSTRACT FROM AUTHOR]

It is possible to solve unbounded convex vector optimization problems (CVOPs) in two phases: (1) computing or approximating the recession cone of the upper image and (2) solving the equivalent bounded CVOP where the ordering cone is extended based on the first phase. In this paper, we consider unbounded CVOPs and propose an alternative solution methodology to compute or approximate the recession cone of the upper image. In particular, we relate the dual of the recession cone with the Lagrange dual of weighted sum scalarization problems whenever the dual problem can be written explicitly. Computing this set requires solving a convex (or polyhedral) projection problem. We show that this methodology can be applied to semidefinite, quadratic, and linear vector optimization problems and provide some numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, we propose and analyze an away-step Frank–Wolfe algorithm designed for solving multiobjective optimization problems over polytopes. We prove that each limit point of the sequence generated by the algorithm is a weak Pareto optimal solution. Furthermore, under additional conditions, we show linear convergence of the whole sequence to a Pareto optimal solution. Numerical examples illustrate a promising performance of the proposed algorithm in problems where the multiobjective Frank–Wolfe convergence rate is only sublinear. [ABSTRACT FROM AUTHOR]

This paper investigates nonsmooth multiobjective fractional programming (NMFP) with inequalities and equalities constraints in real reflexive Banach spaces. It derives a quotient calculus rule for computing the first- and second-order Clarke derivatives of fractional functions involving locally Lipschitz functions. A novel second-order Abadie-type regularity condition is presented, defined with the help of the Clarke directional derivative and the Páles–Zeidan second-order directional derivative. We establish both first- and second-order strong necessary optimality conditions, which contain some new information on multipliers and imply the strong KKT necessary conditions, for a Borwein-type properly efficient solution of NMFP by utilizing generalized directional derivatives. Moreover, it derives second-order sufficient optimality conditions for NMFP under a second-order generalized convexity assumption. Additionally, we derive duality results between NMFP and its second-order dual problem under some appropriate conditions [ABSTRACT FROM AUTHOR]

This paper proposes a line search technique to solve a special class of multi-objective optimization problems in which the objective functions are supposed to be convex but need not be differentiable. This is an iterative process to determine Pareto critical points. A suitable sub-problem is proposed at every iteration of the iterative process to determine the direction vector using the sub-differential of every objective function at that point. The proposed method is verified in numerical examples. This methodology does not bear any burden of selecting suitable parameters like the scalarization methods. [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 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]

In this paper, in order to characterize optimality of a class of nonconvex differentiable multiobjective programming problems, we introduce two new types of vector variational-like inequalities, namely, a weak variational E-inequality and a vector variational E-inequality. Namely, under (strictly) E-convexity, we prove relationships between the aforesaid vector variational-like inequalities and differentiable vector optimization problems. Further, under (strictly) pseudo-E-convexity, we are in position to identify vector critical E-points, weak E-Pareto (E-Pareto) solutions of differentiable vector optimization problems and solutions of these introduced vector variational-like inequalities. [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 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]

In this paper, we propose a spectral projected subgradient method with a 1-memory momentum term for solving constrained convex multiobjective optimization problem. This method combines the subgradient-type algorithm for multiobjective optimization problems with the idea of the spectral projected algorithm to accelerate the convergence process. Additionally, a 1-memory momentum term is added to the subgradient direction in the early iterations. The 1-memory momentum term incorporates, in the present iteration, some of the influence of the past iterations, and this can help to improve the search direction. Under suitable assumptions, we show that the sequence generated by the method converges to a weakly Pareto efficient solution and derive the sublinear convergence rates for the proposed method. Finally, computational experiments are given to demonstrate the effectiveness of the proposed method. [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]

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 consider multi-objective optimization problems with a sparsity constraint on the vector of variables. For this class of problems, inspired by the homonymous necessary optimality condition for sparse single-objective optimization, we define the concept of L-stationarity and we analyze its relationships with other existing conditions and Pareto optimality concepts. We then propose two novel algorithmic approaches: the first one is an iterative hard thresholding method aiming to find a single L-stationary solution, while the second one is a two-stage algorithm designed to construct an approximation of the whole Pareto front. Both methods are characterized by theoretical properties of convergence to points satisfying necessary conditions for Pareto optimality. Moreover, we report numerical results establishing the practical effectiveness of the proposed methodologies. [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]

This paper aims to obtain new Karush–Kuhn–Tucker optimality conditions for solutions to semi-infinite interval equilibrium problems with interval-valued objective functions. The Karush–Kuhn–Tucker conditions for the semi-infinite interval programming problem are particular cases of those found in this paper for constrained equilibrium problem. We illustrate this with some examples. In addition, we obtain solutions to the interval equilibrium problem in the unconstrained case. The results presented in this paper extend the corresponding results in the literature. [ABSTRACT FROM AUTHOR]

This paper gives verifiable conditions for the existence of solutions of some set quasi-optimization problems involving Minkowski difference, where the objective maps are defined in complete metric spaces. Our proof method is not based on any scalarizing approach, and our existence results are written in terms of the given data of the problems. Several examples are provided. As applications of the main results of this paper, new results on the existence of robust solutions for uncertain multi-objective quasi-optimization problems with set-valued maps are formulated. [ABSTRACT FROM AUTHOR]

This paper addresses a linear programming problem with interval right-hand sides, forming a family of linear programs associated with each realization of the interval data. The paper focuses on the outcome range problem, which seeks the range of an additional function—termed the outcome function—over all possible optimal solutions of such linear programs. We explore the problem's applicability in diverse contexts, discuss its connections to certain existing problems, and analyze its computational complexity and theoretical foundations. Given the inherent computational challenges, we propose three heuristics to solve the problem. The first heuristic employs a reformulation–linearization technique (RLT) to obtain an outer approximation of the range of the outcome function. We also present two algorithms—a gradient-restoration-based approach (GI) and a bases inspection method (BI)—for computing an inner approximation of the range. Computational experiments illustrate the competitive advantage of our proposed approaches versus off-the-shelf solvers. The GI and BI methods present promising results in finding a cheap but tight inner approximation, while the performance of the RLT technique decreases as problem size and uncertainty increase. [ABSTRACT FROM AUTHOR]

This paper is devoted to second-order optimality conditions of constrained vector optimization problems with continuously Fréchet differentiable data. Different from conventional approaches, we employ a new one, image space analysis, to establish suitable regularity conditions and then present new second-order necessary optimality conditions that depend on one single KKT multiplier for all critical directions. Moreover, we propose one more regularity condition for the multiobjective case and compare it with the previous one. Besides, we compare the main regularity conditions with the existing ones in the literature for the scalar case. [ABSTRACT FROM AUTHOR]

This paper presents an accelerated proximal gradient method for multiobjective optimization, in which each objective function is the sum of a continuously differentiable, convex function and a closed, proper, convex function. Extending first-order methods for multiobjective problems without scalarization has been widely studied, but providing accelerated methods with accurate proofs of convergence rates remains an open problem. Our proposed method is a multiobjective generalization of the accelerated proximal gradient method, also known as the Fast Iterative Shrinkage-Thresholding Algorithm, for scalar optimization. The key to this successful extension is solving a subproblem with terms exclusive to the multiobjective case. This approach allows us to demonstrate the global convergence rate of the proposed method ( O (1 / k 2) ), using a merit function to measure the complexity. Furthermore, we present an efficient way to solve the subproblem via its dual representation, and we confirm the validity of the proposed method through some numerical experiments. [ABSTRACT FROM AUTHOR]

In this paper, we are mainly concerned with a rule for efficient (Pareto) approximate subdifferential, concerning the sum of two cone-convex set-valued vector mappings, taking values in finite or infinite-dimensional preordred spaces. The obtained formula is exact and holds under the connectedness or Attouch-Brézis qualification conditions and the regular subdifferentiability. This formula is applied to establish approximate necessary and sufficient optimality conditions for the existence of the approximate Pareto (weak or proper) efficient solutions of a set-valued vector optimization problem. [ABSTRACT FROM AUTHOR]

Necessary optimality conditions for a bilevel optimization problem are given in the paper by Kohli (J Optim Theory Appl 152: 632–651, 2012). Recently, the same author corrected his results in the note (J Optim Theory Appl 181:706–707, 2019). In this work, we have pointed out that some of the new modifications are wrong. We correct the flaws and present an alternative proof for the main result. [ABSTRACT FROM AUTHOR]

In this paper, we deal with a new class of SOS-convex (sum of squares convex) polynomial optimization problems with spectrahedral uncertainty data in both the objective and constraints. By using robust optimization and a weighted-sum scalarization methodology, we first present the relationship between robust solutions of this uncertain SOS-convex polynomial optimization problem and that of its corresponding scalar optimization problem. Then, by using a normal cone constraint qualification condition, we establish necessary and sufficient optimality conditions for robust weakly efficient solutions of this uncertain SOS-convex polynomial optimization problem based on scaled diagonally dominant sums of squares conditions and linear matrix inequalities. Moreover, we introduce a semidefinite programming relaxation problem of its weighted-sum scalar optimization problem, and show that robust weakly efficient solutions of the uncertain SOS-convex polynomial optimization problem can be found by solving the corresponding semidefinite programming relaxation problem. [ABSTRACT FROM AUTHOR]

In this paper, we focus on a class of robust vector polynomial optimization problems (RVPOP in short) without any convex assumptions. By combining/improving the utopia point method (a nonlinear scalarization) for vector optimization and "joint+marginal" relaxation method for polynomial optimization, we solve the RVPOP successfully. Both theoratical and computational aspects are considered. [ABSTRACT FROM AUTHOR]

In this paper, by using the inexact scalarization proximal methods, we solve quasiconvex multiobjective optimization problems on Hadamard manifolds. Under some assumptions on the multifunction of problem and vector fields, our methods are proved to be convergent to a Pareto critical point of the problem. For the convex case, the sequences generated by the methods converge to a weak Pareto solution. [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]

In this paper we consider multi-objective optimization problems over a box. Several computational approaches to solve these problems have been proposed in the literature, that broadly fall into two main classes: evolutionary methods, which are usually very good at exploring the feasible region and retrieving good solutions even in the nonconvex case, and descent methods, which excel in efficiently approximating good quality solutions. In this paper, first we confirm, through numerical experiments, the advantages and disadvantages of these approaches. Then we propose a new method which combines the good features of both. The resulting algorithm, which we call Non-dominated Sorting Memetic Algorithm, besides enjoying interesting theoretical properties, excels in all of the numerical tests we performed on several, widely employed, test functions. [ABSTRACT FROM AUTHOR]

This paper provides a study of multiobjective fractional variational programs involving support functions. It then explains the concept of higher-order K– η convex. The paper's motivation is to study the duality results for the value of primal and dual programs. The numerical example of functional is discussed, which is higher-order K– η convex but not first-order K– η convex. A real-world example is considered to verify the results of the weak duality theorem. [ABSTRACT FROM AUTHOR]

In this paper, a class of directionally differentiable multiobjective programming problems with inequality, equality and vanishing constraints is considered. Under both the Abadie constraint qualification and the modified Abadie constraint qualification, the Karush–Kuhn–Tucker type necessary optimality conditions are established for such nondifferentiable vector optimization problems by using the nonlinear version Gordan theorem of the alternative for convex functions. Further, the sufficient optimality conditions for such directionally differentiable multiobjective programming problems with vanishing constraints are proved under convexity hypotheses. Furthermore, vector Wolfe dual problem is defined for the considered directionally differentiable multiobjective programming problem vanishing constraints and several duality theorems are established also under appropriate convexity hypotheses. [ABSTRACT FROM AUTHOR]

In this paper, we consider nonsmooth multiobjective optimization problems with equilibrium constraints. Necessary/sufficient conditions for optimality in terms of the Michel-Penot subdifferential are established. Then, we propose Wolfe- and Mond–Weir-types of dual problems and investigate duality relations under generalized convexity assumptions. Some examples are provided to illustrate our results. [ABSTRACT FROM AUTHOR]

In this paper, we consider nonsmooth multiobjective optimization problems with equilibrium constraints. Necessary/sufficient conditions for optimality in terms of the Michel-Penot subdifferential are established. Then, we propose Wolfe- and Mond–Weir-types of dual problems and investigate duality relations under generalized convexity assumptions. Some examples are provided to illustrate our results. [ABSTRACT FROM AUTHOR]

In this paper, we consider nonsmooth multiobjective optimization problems with equilibrium constraints. Necessary/sufficient conditions for optimality in terms of the Michel-Penot subdifferential are established. Then, we propose Wolfe- and Mond–Weir-types of dual problems and investigate duality relations under generalized convexity assumptions. Some examples are provided to illustrate our results. [ABSTRACT FROM AUTHOR]

The aim of this paper is to provide a general description of the Pareto subdifferential (weak and proper) of the sum of two cone-convex set-valued mappings in terms of sequences without any constraint qualifications. As an application, we derive sequential Lagrange multipliers optimality conditions for general set-valued optimization problem in terms of sequential Lagrange multipliers at nearby points for the Pareto efficient solutions, where no constraint qualification is assumed. [ABSTRACT FROM AUTHOR]