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In this paper, several existence results of strongly essential set of the solution set for Ky Fan's section problems and vector Ky Fan's point problems are obtained. Firstly, two kinds of strongly essential sets of Ky Fan's section problems are defined, and some further results on existence of the strongly essential component of solutions set of Ky Fan's section problems are proved, which generalize the conclusion in [ 22 ], and further generalize the conclusions in [ 21 , 28 ]. Secondly, based on the above results, two classes of stronger perturbations of vector-valued inequality functions are proposed respectively, and several existence results of the strongly essential component of set of vector Ky Fan's points are obtained. By comparing several metrics, we give some strong and weak relationships among the various metrics involved in the text. The main results of this paper actually generalize the relevant conclusions in the current literature. Finally, as an application, we obtain an existence result of the strongly essential component of weakly Pareto-Nash equilibrium for multiobjective games.

Non-negative least square regression (NLS) is a constrained least squares problem where the coefficients are restricted to be non-negative. It is useful for modeling non-negative responses such as time measurements, count data, histograms and so on. Existing NLS solvers are designed for cases where the predictor variables and response variables have linear relationships, and do not consider interactions among predictor variables. In this paper, we solve NLS in the complete space of power sets of variables. Such an extension is particularly useful in biology, for modeling genetic associations. Our new algorithms solve NLS problems exactly while decreasing computational burden by using an active set method. The algorithm proceeds in an iterative fashion, such that an optimal interaction term is searched by a branch-and-bound subroutine, and added to the solution set one another. The resulting large search space is efficiently restricted by novel pruning conditions and two kinds of sparsity promoting regularization; $l_{1}$ norm and non-negativity constraints. In computational experiments using HIV-1 datasets, 99% of the search space was safely pruned without losing the optimal variables. In mutagenicity datasets, the proposed method could identify long and accurate patterns compared to the original NLS. Codes are available from https://github.com/afiveithree/inlars.

The paper studies a robust mixed integer program with a single unrestricted continuous variable. The purpose of the paper is the polyhedral study of the robust solution set using submodularity. A submodular function is a set function with a diminishing returns property, and little work has been studied on the utilization of submodularity in the study of optimization problems considering data uncertainty. In this paper, we propose valid inequalities using submodularity. Valid inequalities for the robust mixed integer program are defined. A polynomial separation algorithm is proposed, and we show that the convex hull of the problem can be completely described using the proposed inequalities. In computational tests, we showed the proposed cuts are effective when they are applied to general robust discrete optimization problems with one or multiple constraints.

The multi-objective evolutionary algorithm based on decomposition (MOEA/D) has achieved remarkable success in regular optimizing multi-objective problems. In practice, there are lots of irregular problems. The projection of their Pareto fronts (PF) may not cover the whole triangle on the first octant. There is an empty PF (EPF) or complex PF area. These problems significantly reduce the performance of MOEA/D. In this paper, an improved algorithm is proposed to deal with such complex problems. The basic idea is: (1) the PBI method is used to ensure the convergence and simultaneously keep diversity over the triangle; (2) a non-dominated and maximum distance based selection is used to replace the dominated objective individuals selected by the PBI method. It is referred to selection based on the PBI and Non-dominated Maximum distance (MOEAD-PNM). First, the mathematical foundation between the penalty factor and contour line is deduced. Based on the foundation, the penalty factor setting method is proposed and the PBI-based method is used to select an individual for each reference vector. The reference vectors directing to the EPF induce “bad” solutions. Then, these solutions are gradually replaced by better individuals selected according to the non-dominated and maximum distance-based selection strategy. The proposed two-step selection method can guarantee the wideness and uniformity of the “good” solution set. Finally, the performance of the MOEAD-PNM algorithm and five classic algorithms on more than ten test problems are compared. The experimental results show the competitiveness and effectiveness of the proposed algorithm on these challenging problems.

This article deals with a new fractional nonlinear delay evolution system driven by a hemi-variational inequality in a Banach space. Utilizing the KKM theorem, a result concerned with the upper semicontinuity and measurability of the solution set of a hemivariational inequality is established. By using a fixed point theorem for a condensing set-valued map, the nonemptiness and compactness of the set of mild solutions are also obtained for such a system under mild conditions. Finally, an example is presented to illustrate our main results.

In this paper, several error bounds for the solution sets of the generalized polynomial complementarity problems (GPCPs) with explicit exponents are given. As the solution set of a GPCP is the solu...

An important aspect of optimization algorithms, for instance evolutionary algorithms, are termination criteria that measure the proximity of the found solution to the optimal solution set. A frequently used approach is the numerical verification of necessary optimality conditions such as the Karush–Kuhn–Tucker (KKT) conditions. In this paper, we present a proximity measure which characterizes the violation of the KKT conditions. It can be computed easily and is continuous in every efficient solution. Hence, it can be used as an indicator for the proximity of a certain point to the set of efficient (Edgeworth-Pareto-minimal) solutions and is well suited for algorithmic use due to its continuity properties. This is especially useful within evolutionary algorithms for candidate selection and termination, which we also illustrate numerically for some test problems.

This paper aims at investigating the Painleve–Kuratowski convergence of the solution sets for set optimization problems with respect to the perturbations of the feasible set and the objective mappi...

Modified Adaptive Weight Approach (MAWA) is one of the simplest methods used for solving time-cost optimization problems. This type of optimization problem, categorized as multi-objective optimization, can be solved with metaheuristic algorithms. Here, metaheuristic algorithms evaluate a randomly generated solution set, referred to as population, within the boundary condition of a solution space. The weight factor values determined by MAWA are applied to all solutions in the population without distinguishing the solutions in that population. However, the potential solutions in the population indicate different fitness properties concerning solution space. In this paper to solve time-cost optimization problems, new adaptive weight formulations are proposed. In contrast to MAWA, the novelty of this study’s formulations is to adaptively detect the weight factor value, depending on the solution‘s fitness in the population. The results obtained from the numerical experiments examined in this study show that the proposed formulations can improve the performance of MAWA and can find identical or slightly different Pareto results for investigated multi-objective optimization problems.

In this paper, we introduce the concept of trade-off ratio function, which is closely related to the well-known Geoffrion’s proper efficiency for multi-objective optimization problems, and investigate its boundedness property. For linear multi-objective optimization problems, we show that the trade-off ratio function is bounded on the efficient solution set. For piecewise linear multi-objective optimization problems, we show that all efficient solutions are always properly efficient in Borwein’s sense, and moreover, all efficient solutions are properly efficient in Geoffrion’s sense if and only if a recession condition holds. Finally, we provide an example to illustrate that the trade-off ratio function may be unbounded on the efficient solution set to piecewise linear multi-objective optimization problems, even if the recession condition holds, while it is bounded on the supported efficient solution set.

With the advancement toward 6G technology, mobile data growth is estimated to increase many fold. There will also be an increase in the control plane load (IoT, IoE). Such problems call for the technologies that can efficiently utilize the resources to optimize the system performance, and the possible solution is cognitive radio technology. As the spectrum sensing is the key enabler of cognitive radio technology, in this paper, the multi-objective parameters defining the efficiency of spectrum sensing for a cognitive radio network (CRN), which are throughput, interference, and energy efficiency, defined in terms of sensing time, power allocation, and detection threshold are dealt. In this paper, a novel Multi-Objective Modified Grey Wolf Optimization (MOMGWO) algorithm is proposed to solve the multi-objective optimization problem in the field of spectrum sensing in a cognitive radio network which is an important paradigm in wireless communication technology. Modification in Grey Wolf Optimization (GWO) is applied to balance the trade-off between exploration and exploitation process in conventional GWO, to obtain global optima. Modification is introduced in terms of mutation in leader selection, discrimination weight, and mutation coefficient. The non-dominated solution set of the proposed algorithm is compared with the existing algorithms like Non-dominated Sorting Genetic Algorithm (NSGA-II), Multi-Objective Particle Swarm Optimization (MOPSO), Multi-Objective Cat Swarm Optimization (MOCSO), and conventional Multi-Objective Grey Wolf Optimization (MOGWO) algorithm. The simulation result shows that the proposed MOMGWO has outperformed the existing algorithms with respect to the quality of the Pareto front. Thus, the best solutions for the spectrum sensing parameters in optimizing multi-objective problems for cognitive radio network can be obtained via the proposed MOMGWO algorithm.

This paper aims at investigating connectedness of weak p -minimal solution set for set optimization problems by using the scalarization method. We make a new attempt to derive a scalarization result for weak p -minimal solution set for set optimization problems. By using the scalarization result, we establish connectedness of weak p -minimal solution set for set optimization problems.

We consider the Cauchy problem for a differential inclusion with a Caputo fractional derivative of order $$ \alpha \in (0, 1)$$ . It is assumed that the set-valued mapping defining the right-hand side of the inclusion has nonempty, convex, and compact values, is upper semicontinuous, and satisfies the sublinear growth condition. We prove that the solution set of the Cauchy problem is nonempty and compact and establish the nature of the dependence of the solution set on the initial data, the right-hand side of the inclusion, and the order of differentiation. The semigroup property of the solution set is established.

As a complement to two recent papers by Toan and Yao (Mordukhovich subgradients of the value function to a parametric discrete optimal control problem. J Global Optim. 2014;58:595–612), and by An a...

In this paper, we present a Frank–Wolfe-type theorem for nonconvex cubic programming problems. This result is a direct extension of the previous ones by Andronov et al. (Izvestija Akadem. Nauk SSSR, Tekhnicheskaja Kibernetika 4:194–197, 1982) and Flores-Bazan et al. (Math. Program. 145:263–290, 2014). Under suitable conditions, we characterize the compactness of the solution set of cubic programming problems. Sufficient conditions for the existence of solutions of quadratic variational inequalities are proposed. We also provide several numerical examples, which not only illustrate the obtained results but also show that the existing results cannot apply.

In this paper we investigate the number partitioning problem, using the tropical semiring (max-plus algebra). We show that the problem is reduced to deciding whether one of integers is a solution of a tropical analogue of algebraic equations with coefficients composed of other integers. For n up to 6 we derive concretely and explicitly the equation and its solution set. The derivation requires only routine algebraic computations, so can be applied for n larger than 6. Our approach based on max-plus algebra reveals the mathematical structure of the problem and provides a new view point for the P versus NP problem, since the problem is well-known to be NP-complete.

We study geometric programming with a single-term exponent subject to bipolar max-product fuzzy relation equation constraints in the area of economics and the covering problem. The structure of its feasible domain is characterized and the lower and upper bound vectors on its solution set are determined. It is shown that each component of one of its optimal solutions is the corresponding component of either the lower bound or the upper bound vector. This interesting property helps us to create a value matrix and present some necessary and sufficient conditions for its consistency checking. Moreover, some sufficient conditions are proposed to detect one of its optimal solutions without solving the problem. A modified branch-and-bound method is extended to solve the problem, in a general case, with use of the value matrix. An efficient algorithm is then designed to solve the problem using the sufficient conditions and the modified branch-and-bound method. Its computational complexity is also analyzed. Finally, some examples are provided to illustrate its importance and the steps of the algorithm.

The bipolar system of fuzzy relation equations with the max-Łukasiewicz composition is investigated in this work. A bipolar path approach is proposed for such a system. It is found that the complete solution set of the bipolar system is fully determined by its conservative bipolar paths, which are finite. Our proposed resolution approach is performed using the so-called path-based algorithm, step by step, and illustrated with numerical examples. Moreover, the global and local minimal (or maximal) solutions are discussed in this paper with a comparison to those of a classical unipolar system.

The computation of hypervolume is a key issue in multiobjective optimization, particularly, multiobjective evolutionary optimization. However, it is NP-hard to compute the exact hypervolume value. Monte Carlo methods have been widely used for approximating the hypervolume. Observing that the basic Monte Carlo method and the fully polynomial-time randomized approximation scheme (FPRAS) suit different solution sets, we propose a combination of these two methods and show that it performs very well on a number of solution sets.

This paper concerns matrix decompositions in which the factors are restricted to lie in a closed subvariety of a matrix group. Such decompositions are of relevance in control theory: given a target matrix in the group, can it be decomposed as a product of elements in the subvarieties, in a given order? And if so, what can be said about the solution set to this problem? Can an irreducible curve of target matrices be lifted to an irreducible curve of factorisations? We show that under certain conditions, for a sufficiently long and complicated such sequence, the solution set is always irreducible, and we show that every connected matrix group has a sequence of one-parameter subgroups that satisfies these conditions, where the sequence has length less than 1.5 times the dimension of the group., 11 pages

Dominance robustness (DR) has been proposed for assessing the ability of the Pareto-optimal solutions to remain to be nondominated when the decision variables are subject to noise. There are two main challenges in search for dominance robust optimal solutions in dominance robust multiobjective optimization (MOP), namely, accurate estimation of the DR measure and a good balance between convergence and DR in the presence of uncertainty. In this article, a novel robust MOP evolutionary algorithm based on decision variable assortment (DVA) is proposed to tackle these challenges. To be specific, an indicator, termed as dominance robust indicator, is proposed to measure the DR based on the dominance level and dominance relationship of the sampled points. Then, the decision variables are divided into low DR-related variables and high DR-related variables based on the DVA strategy. Finally, low and high DR related variables are optimized separately to obtain the dominance robust optimal solutions. In addition, performance indicators to quantify the performance of dominance robust optimal solution set obtained by robust MOP algorithm are proposed. Experimental results have demonstrated that the proposed algorithm is competitive in search for dominance robust optimal solutions.

Мақолада интервал алгоритмлар учун ҳисоблаш дастурларини яратиш муаммолари тадқиқ қилинган. Бунда интервал ҳисоблашларни ташкил қилишнинг ўзига хос жиҳатлари, соҳада мавжуд дастурий маҳсулотларнинг имкониятлари асосан иккита йўналишда, яъни интервал ҳисоблашлар учун яратилган махсус дастурлаш тиллари ва мавжуд дастурлаш муҳитлари учун интервал кутубхоналар яратиш соҳасида ўрганилди. Шунингдек, чизиқли интервал тенгламалар системаларининг мумкин бўлган ечимлар тўпламини 3D ҳамда 2D визуаллаштириш дастурий таъминоти имкониятлари аниқ мисолларда кўрсатиб берилди., The article examines the problems of creating computational programs for interval algorithms. At the same time, the features of the organization of interval calculations, the possibilities of existing software products created in the field of interval analysis were studied, mainly in two directions: special programming languages developed for interval calculations and interval libraries for existing software environments. Also, the possibilities of software products for 3D and 2D visualization of tolerance sets of solutions of interval linear systems of equations in specific examples are shown.

In the present paper, the problem of estimating the contingent cone to the solution set associated with certain set-valued inclusions is addressed by variational analysis methods and tools. As a main result, inner (resp. outer) approximations, which are expressed in terms of outer (resp. inner) prederivatives of the set-valued term appearing in the inclusion problem, are provided. For the analysis of inner approximations, the evidence arises that the metric increase property for set-valued mappings turns out to play a crucial role. Some of the results obtained in this context are then exploited for formulating necessary optimality conditions for constrained problems, whose feasible region is defined by a set-valued inclusion.

For targeted motion control tasks of manipulators, it is frequently necessary to make use of full levels of joint actuation to guarantee successful motion planning and path tracking. Such way of motion planning and control may keep the joint actuation in a non-sparse manner during motion control process. In order to improve sparsity of joint actuation for manipulator systems, a novel motion planning scheme which can optimally and sparsely adopt joint actuation is proposed in this paper. The proposed motion planning strategy is formulated as a constrained L1 norm optimization problem, and an equivalent enhanced optimization solution dealing with bounded joint velocity is proposed as well. A new primal dual neural network with a new solution set division is further proposed and applied to solve such bounded optimization which can sparsely adopt joint actuation for motion control. Simulation and experiment results demonstrate the efficiency, accuracy and superiority of the proposed method for optimally and sparsely adopting joint actuation. The average sparsity (i.e., -||˙θ||p where θ denotes the joint angle) of the joint motion of the manipulator can be increased by 39.22% and 51.30% for path tracking tasks in X-Y and X-Z planes respectively, indicating that the sparsity of joint actuation can be enhanced.

This paper presents a Markov sampling-based framework, called Asymptotic Bayesian Optimization, for solving a class of constrained design optimization problems. The optimization problem is converted into a unified two-phase sample generation problem which is solved by an effective Markov chain Monte Carlo simulation scheme. First, an exploration phase generates designs distributed over the feasible design space. Based on this information, an exploitation phase obtains a set of designs lying in the vicinity of the optimal solution set. The proposed formulation can handle continuous, discrete, or mixed discrete-continuous design variables. Appropriate adaptive proposal distributions for the continuous and discrete design variables are suggested. The set of optimal solutions provides valuable sensitivity information of the different quantities involved in the problem with respect to the design variables. Representative examples including an analytical problem involving nonlinear benchmark functions, a classical engineering design problem, and a performance-based design optimization problem of a structural system under stochastic excitation are presented to show the effectiveness and potentiality of the proposed optimization scheme. Validation calculations show that the scheme is a flexible, efficient and competitive choice for solving a wide range of classical and complex engineering design problems.

In this paper, we investigate the problem of solving strongly monotone variational inequality problems over the solution set of a split variational inequality and fixed point problem. Strong convergence of the iterative process is proved. In particular, the problem of finding a common solution to a variational inequality with pseudomonotone mapping and a fixed point problem involving demicontractive mapping is also studied. Besides, we get a strongly convergent algorithm for finding the minimum-norm solution to the split feasibility problem, which requires only two projections at each step. A simple numerical example is given to illustrate the proposed algorithm.

(max, +)-linear functions are functions which can be expressed as the maximum of a finite number of linear functions of one variable having the form f(x1, …, xh) = max(aj + xj), where aj, j = 1, …, h, are real numbers. Similarly (min, +)-linear functions are defined. We will consider optimization problems in which the set of feasible solutions is the solution set of a finite inequality system, where the inequalities have (max, +)-linear functions of variables x on one side and (min, +)-linear functions of variables y on the other side. Such systems can be applied e.g. to operations research problems in which we need to coordinate or synchronize release and completion times of operations or departure and arrival times of passengers. A motivation example is presented and the proposed solution method is demonstrated on a small numerical example.

Boundary representation (B-rep) is one of the most common techniques for representing three-dimensional (3D) shapes. In this method, shapes are described mathematically based on the vertices, edges, and faces present. B-rep models are generated using computer-aided design systems and are employed for various purposes in downstream applications, including computer-aided engineering, manufacturing, and inspection. Therefore, the level of detail of the 3D model should be adjusted based on the purpose of use. In this study, we performed multiobjective optimization for the feature-based simplification of B-rep models. The two objectives of the general model simplification process are to minimize the data size and limit the differences in appearance with respect to the original model. To simultaneously meet these objectives, model simplification was performed to minimize both the difference in the volume and the number of faces with respect to those of the original model. For optimization, we used the simple genetic algorithm (SGA) and the non-dominated sorting genetic algorithm II (NSGA-II). The experimental results confirmed that solution set obtained using NSGA-II was of higher quality as compared with those for SGA and the conventional method.

In this paper, we propose a new inertial extrapolation method for solving the generalized split feasibility problems over the solution set of monotone variational inclusion problems in real Hilbert...

In this paper, the minimum and maximum principle sufficiency properties for a nonsmooth variational inequality problem (NVIP) are studied. We discuss the relationship among the solution set of an NVIP and those defined by its dual problem and relevant gap functions. For a pseudomonotone NVIP, the weaker sharpness of its solution set has been shown to be sufficient for it to have minimum principle sufficiency property. As special cases, pseudomonotonicity $$ _{*} $$ and pseudomonotonicity $$ ^{+} $$ of the relevant bifunction have been characterized, from which the minimum and maximum principle sufficiency properties have also been characterized.

Two main results (A) and (B) are presented in algebraic closed forms. (A) Regarding the convex quadratic equation, an analytical equivalent solvability condition and parameterization of all solutions are formulated, for the first time in the literature and in a unified framework. The philosophy is based on the matrix algebra, while facilitated by a novel equivalence/coordinate transformation (with respect to the much more challenging case of rank-deficient Hessian matrix). In addition, the parameter-solution bijection is verified. From the perspective via (A), a major application is re-examined that accounts for the other main result (B), which deals with both the infinite and finite-time horizon nonlinear optimal control. By virtue of (A), the underlying convex quadratic equations associated with the Hamilton–Jacobi equation, Hamilton–Jacobi inequality, and Hamilton–Jacobi–Bellman equation are explicitly solved, respectively. Therefore, the long quest for the constituent of the optimal controller, gradient of the associated value function, can be captured in each solution set. Moving forward, a preliminary to exactly locate the optimality using the state-dependent (resp., differential) Riccati equation scheme is prepared for the remaining symmetry condition.

In this paper, we first introduce a new class of structured tensors, named strictly positive semidefinite tensors, and show that a strictly positive semidefinite tensor is not an $$R_0$$ tensor. We focus on the stochastic tensor complementarity problem (STCP), where the expectation of the involved tensor is a strictly positive semidefinite tensor. We denote such an STCP as a monotone STCP. Based on three popular NCP functions, the min function, the Fischer–Burmeister (FB) function and the penalized FB function, as well as the special structure of the monotone STCP, we introduce three new NCP functions and establish the expected residual minimization (ERM) formulation of the monotone STCP. We show that the solution set of the ERM problem is nonempty and bounded if the solution set of the expected value (EV) formulation for such an STCP is nonempty and bounded. Moreover, an approximate regularized model is proposed to weaken the conditions for nonemptiness and boundedness of the solution set of the ERM problem in practice. Numerical results indicate that the performance of the ERM method is better than that of the EV method.

Linear differential systems, in Willems’ behavioral system theory, are defined to be the solution sets to systems of linear constant coefficient PDEs, and they are naturally parameterized in a bijective way by means of polynomial modules. In this article, introducing appropriate topologies, this parametrization is made continuous in both directions. Moreover, the space of linear differential systems with a given complexity polynomial is embedded into a Grassmannian.

Distributionally robust optimization is a dominant paradigm for decision-making problems where the distribution of random variables is unknown. We investigate a distributionally robust optimization problem with ambiguities in the objective function and countably infinite constraints. The ambiguity set is defined as a Wasserstein ball centered at the empirical distribution. Based on the concentration inequality of Wasserstein distance, we establish the asymptotic convergence property of the data-driven distributionally robust optimization problem when the sample size goes to infinity. We show that with probability 1, the optimal value and the optimal solution set of the data-driven distributionally robust problem converge to those of the stochastic optimization problem with true distribution. Finally, we provide numerical evidences for the established theoretical results.

Multiobjective evolutionary algorithms based on decomposition (MOEA/D) have attracted tremendous attention and achieved great success in the fields of optimization and decision-making. MOEA/Ds work by decomposing the target multiobjective optimization problem (MOP) into multiple single-objective subproblems based on a set of weight vectors. The subproblems are solved cooperatively in an evolutionary algorithm framework. Since weight vectors define the search directions and, to a certain extent, the distribution of the final solution set, the configuration of weight vectors is pivotal to the success of MOEA/Ds. The most straightforward method is to use predefined and uniformly distributed weight vectors. However, it usually leads to the deteriorated performance of MOEA/Ds on solving MOPs with irregular Pareto fronts. To deal with this issue, many weight vector adjustment methods have been proposed by periodically adjusting the weight vectors in a random, predefined, or adaptive way. This article focuses on weight vector adjustment on a simplex and presents a comprehensive survey of these weight vector adjustment methods covering the weight vector adaptation strategies, theoretical analyses, benchmark test problems, and applications. The current limitations, new challenges, and future directions of weight vector adjustment are also discussed.

In the present work, an inverse problem for the heat equation in two dimensional space with Robin boundary condition that involving a new fractional derivative, namely, Atangana-Baleanu approach with non-local and non-singular kernel is considered. An explicit solution set { u ( x , y , t ) , a ( t ) } of the given inverse problem is obtained by using the eigenfunctions expansion method and the integral overdetermination condition. Under some assumptions the existence, uniqueness of the suggested solution, and its continuous dependence on the data are proved.

The ancient concept of circumcenter has recently given birth to the Circumcentered-Reflection method (CRM). CRM was first employed to solve best approximation problems involving affine subspaces. In this setting, it was shown to outperform the most prestigious projection based schemes, namely, the Douglas-Rachford method (DRM) and the method of alternating projections (MAP). We now prove convergence of CRM for finding a point in the intersection of a finite number of closed convex sets. This striking result is derived under a suitable product space reformulation in which a point in the intersection of a closed convex set with an affine subspace is sought. It turns out that CRM skillfully tackles the reformulated problem. We also show that for a point in the affine set the CRM iteration is always closer to the solution set than both the MAP and DRM iterations. Our theoretical results and numerical experiments, showing outstanding performance, establish CRM as a powerful tool for solving general convex feasibility problems.

In this study, we propose nonemptiness and boundedness of the solution set for a bifunction variational inequality problem with P-strict feasibility in reflexive Banach spaces. As a novel concept, P-strict feasibility for a bifunction variational inequality is introduced. Moreover, it is proved that a pseudomonotone bifunction variational inequality has a nonempty and bounded solution set if and only if it is P-strictly feasible in reflexive Banach spaces. Finally, some examples are given to illustrate the validity of the main results.

This paper proposes an improved kinetic-molecular theory optimization algorithm (OKMTOA) by analyzing the characteristics of KMTOA cluster behavior and combining the opposition-based learning strategy with varying accelerated motion in physics. The algorithm first applies different opposition-based learning strategies to the population initialization and iterative process of the algorithm. The two-stage strategy is beneficial to improving the quality of the solution set and accelerating the convergence of the algorithm. Then, based on the concept of varying accelerated motion, the acceleration formula is improved to increase the ability to escape local optimum. The experimental results show that the algorithm has good performance in solution precision, convergence speed and can be well applied to the functions with different shift values.

Choo (Oper Res 32:216–220, 1984) has proved that any efficient solution of a linear fractional vector optimization problem with a bounded constraint set is properly efficient in the sense of Geoffrion. This paper studies Geoffrion’s properness of the efficient solutions of linear fractional vector optimization problems with unbounded constraint sets. By examples, we show that there exist linear fractional vector optimization problems with the efficient solution set being a proper subset of the unbounded constraint set, which have improperly efficient solutions. Then, we establish verifiable sufficient conditions for an efficient solution of a linear fractional vector optimization to be a Geoffrion properly efficient solution by using the recession cone of the constraint set. For bicriteria problems, it is enough to employ a system of two regularity conditions. If the number of criteria exceeds two, a third regularity condition must be added to the system. The obtained results complement the above-mentioned remarkable theorem of Choo and are analyzed through several interesting examples, including those given by Hoa et al. (J Ind Manag Optim 1:477–486, 2005).

This article deals with considering stability properties of Pareto minimal solutions to set optimization problems with the set less order relation in real topological Hausdorff vector spaces. We focus on studying the Painleve–Kuratowski convergence of Pareto minimal elements in the image space. Employing convexity properties, we study the external stability of Pareto minimal solutions via weak ones. Then, we use converse properties to investigate external stability conditions to such problems where Pareto minimal solution sets and weak/ideal ones are distinct. For the internal stability, we propose a concept of compact convergence in the sense of Painleve–Kuratowski and use it together with a domination property to analyze stability conditions for the reference problems.