On vector variational-like inequalities and vector optimization problems with (G, α)-invexity

The aim of this paper is to study the relationship among Minty vector variational-like inequality problem, Stampacchia vector variational-like inequality problem and vector optimization problem involving (G, α)-invex functions. Furthermore, we establish equivalence among the solutions of weak formulations of Minty vector variational-like inequality problem, Stampacchia vector variational-like inequality problem and weak efficient solution of vector optimization problem under the assumption of (G, α)-invex functions. Examples are provided to elucidate our results.

Robust Source Localization in Shallow Water Based on Vector Optimization

Owing to the multipath effect, the source localization in shallow water has been an area of active interest. However, most methods for source localization in shallow water are sensitive to the assumed model of the underwater environment and have poor robustness against the underwater channel uncertainty, which limit their further application in practical engineering. In this paper, a new method of source localization in shallow water, based on vector optimization concept, is described, which is highly robust against environmental factors affecting the localization, such as the channel depth, the bottom reflection coefficients, and so on. Through constructing the uncertainty set of the source vector errors and extracting the multi-path sound rays from the sea surface and bottom, the proposed method can accurately localize one or more sources in shallow water dominated by multipath propagation. It turns out that the natural formulation of our approach involves minimization of two quadratic functions subject to infinitely many nonconvex quadratic constraints. It shows that this problem （originally intractable） can be reformulated in a convex form as the so-called second-order cone program （SOCP） and solved efficiently by using the well-established interior point method, such as the sottware tool, SeDuMi. Computer simulations show better performance of the proposed method as compared with existing algorithms and establish a theoretical foundation for the practical engineering application.

NON-CONFLICTING ORDERING CONES AND VECTOR OPTIMIZATION IN INDUCTIVE LIMITS

Let （E,ξ）=ind（En,ξn） be an inductive limit of a sequence （En,ξn）n∈N of locally convex spaces and let every step （En,ξn） be endowed with a partial order by a pointed convex （solid） cone Sn. In the framework of inductive limits of partially ordered locally convex spaces, the notions of lastingly efficient points, lastingly weakly efficient points and lastingly globally properly efficient points are introduced. For several ordering cones, the notion of non-conflict is introduced. Under the requirement that the sequence （Sn）n∈N of ordering cones is non-conflicting, an existence theorem on lastingly weakly efficient points is presented. From this, an existence theorem on lastingly globally properly efficient points is deduced.

Mathematical Apparatus for Selection of Optimal Parameters of Technical, Technological Systems and Materials Based on Vector Optimization

We presented Mathematical apparatus of the choice of optimum parameters of technical, technological systems and materials on the basis of vector optimization. We have considered the formulation and solution of three types of tasks presented below. First, the problem of selecting the optimal parameters of technical systems depending on the functional characteristics of the system. Secondly, the problem of selecting the optimal parameters of the process depending on the technological characteristics of the process. Third, the problem of choosing the optimal structure of the material depending on the functional characteristics of this material. The statement of all problems is made in the form of vector problems of mathematical (nonlinear) programming. The theory and the principle of optimality of the solution of vector tasks it is explained in work of https://rdcu.be/bhZ8i. The implementation of the methodology is shown on a numerical example of the choice of optimum parameters of the technical, technological systems and materials. On the basis of mathematical methods of solution of vector problems we developed the software in the MATLAB system. The numerical example includes: input data (requirement specification) for modeling;transformation of mathematical models with uncertainty to the model under certainty;acceptance of an optimal solution with equivalent criteria (the solution of numerical model);acceptance of an optimal solution with the given priority of criterion.

Henig Efficiency in Vector Optimization with Nearly Cone-subconvexlike Set-valued Functions

In this paper, we study Henig efficiency in vector optimization with nearly cone-subconvexlike set-valued function. The existence of Henig efficient point is proved and characterization of Henig efficiency is established using the method of Lagrangian multiplier. As an interesting application of the results in this paper, we establish a Lagrange multiplier theorem for super efficiency in vector optimization with nearly conesubconvexlike set-valued function.

On First Order Optimality Conditions for Vector Optimization

We develop first order optimality conditions for constrained vector optimization. The partial orders for the objective and the constraints are induced by closed and convex cones with nonempty interior. After presenting some well known existence results for these problems, based on a scalarization approach, we establish necessity of the optimality conditions under a Slater-like constraint qualification, and then sufficiency for the K-convex case. We present two alternative sets of optimality conditions, with the same properties in connection with necessity and sufficiency, but which are different with respect to the dimension of the spaces to which the dual multipliers belong. We introduce a duality scheme, with a point-to-set dual objective, for which strong duality holds. Some examples and open problems for future research are also presented.

A Kind of Equivalence of Three Nonlinear Scalarization Functions in Vector Optimization

In this paper,by the notions of base functionals and augmented dual cones,the authors indicate firstly that the norms,Gerstewitz functionals and oriented distance functions have common characteristics with base functionals.After that,the equivalence of these three sublinear functions on the ordering cone is established by using the structures of augmented dual cones under the assumption that it has a bounded base.However,the authors show that two superlinear functions do not have similar relations with the norms ahead.More generally,the equivalence of three sublinear functions outside the negative cone has also been obtained in the end.

On Connectedness of Efficient Solution Sets for a Star Cone-Quasiconvex Vector Optimization Problem

In this note, we prove that the efficient solution set for a vector optimization problem with a continuous, star cone-quasiconvex objective mapping is connected under the assumption that the ordering cone is a D-cone. A D-cone includes any closed convex pointed cones in a normed space which admits strictly positive continuous linear functionals.

Existence of Weakly Efficient Solutions in Vector Optimization

In this paper, we present an existence result for weak efficient solution for the vector optimization problem. The result is stated for invex strongly compactly Lipschitz functions.

A Kind of Unified Strict Efficiency via Improvement Sets in Vector Optimization

In this paper,we propose a kind of unified strict efficiency named E-strict efficiency via improvement sets for vector optimization.This kind of efficiency is shown to be an extension of the classical strict efficiency andε-strict efficiency and has many desirable properties.We also discuss some relationships with other properly efficiency based on improvement sets and establish the corresponding scalarization theorems by a base-functional and a nonlinear functional.Moreover,some examples are given to illustrate the main conclusions.

Generalized Lagrangian Duality in Set-valued Vector Optimization via Abstract Subdifferential

In this paper,we investigate dual problems for nonconvex set-valued vector optimization via abstract subdifferential.We first introduce a generalized augmented Lagrangian function induced by a coupling vector-valued function for set-valued vector optimization problem and construct related set-valued dual map and dual optimization problem on the basic of weak efficiency,which used by the concepts of supremum and infimum of a set.We then establish the weak and strong duality results under this augmented Lagrangian and present sufficient conditions for exact penalization via an abstract subdifferential of the object map.Finally,we define the sub-optimal path related to the dual problem and show that every cluster point of this sub-optimal path is a primal optimal solution of the object optimization problem.In addition,we consider a generalized vector variational inequality as an application of abstract subdifferential.

Semicontinuity of the Minimal Solution Set Mappings for Parametric Set-Valued Vector Optimization Problems

With the help of a level mapping,this paper mainly investigates the semicontinuity of minimal solution set mappings for set-valued vector optimization problems.First,we introduce a kind of level mapping which generalizes one given in Han and Gong(Optimization 65:1337–1347,2016).Then,we give a sufficient condition for the upper semicontinuity and the lower semicontinuity of the level mapping.Finally,in terms of the semicontinuity of the level mapping,we establish the upper semicontinuity and the lower semicontinuity of the minimal solution set mapping to parametric setvalued vector optimization problems under the C-Hausdorff continuity instead of the continuity in the sense of Berge.

Systems of generalized vector quasi-variational inclusions and systems of generalized vector quasi-optimization problems in locally FC-uniform spaces

In this paper, we introduce some new systems of generalized vector quasi-variational inclusion problems and system of generalized vector ideal （resp., proper, Pareto, weak） quasi-optimization problems in locally FC-uniform spaces without convexity structure. By using the KKM type theorem and Himmelberg type fixed point theorem proposed by the author, some new existence theorems of solutions for the systems of generalized vector quasi-variational inclusion problems are proved. As to its applications, we obtain some existence results of solutions for systems of generalized vector quasi-optimization problems.

Fenchel-Lagrange Duality and Saddle-Points for Constrained Vector Optimization

The aim of this paper is to apply a perturbation approach to deal with Fenchel- Lagrange duality based on weak efficiency to a constrained vector optimization problem. Under the stability criterion, some relationships between the solutions of primal problem and the Fenchel-Lagrange duality are discussed. Moreover, under the same condition, two saddle-points theorems are proved.

Improved particle swarm optimization algorithm for fuzzy multi-class SVM

An improved particle swarm optimization（PSO） algorithm is proposed to train the fuzzy support vector machine（FSVM） for pattern multi-classification.In the improved algorithm,the particles studies not only from itself and the best one but also from the mean value of some other particles.In addition,adaptive mutation was introduced to reduce the rate of premature convergence.The experimental results on the synthetic aperture radar（SAR） target recognition of moving and stationary target acquisition and recognition（MSTAR） dataset and character recognition of MNIST database show that the improved algorithm is feasible and effective for fuzzy multi-class SVM training.

Springback Prediction and Optimization of Variable Stretch Force Trajectory in Three-dimensional Stretch Bending Process

Most of the existing studies use constant force to reduce springback while researching stretch force. However, variable stretch force can reduce springback more efficiently. The current research on springback prediction in stretch bending forming mainly focuses on artificial neural networks combined with the finite element simulation. There is a lack of springback prediction by support vector regression（SVR）. In this paper, SVR is applied to predict springback in the three-dimensional stretch bending forming process, and variable stretch force trajectory is optimized. Six parameters of variable stretch force trajectory are chosen as the input parameters of the SVR model. Sixty experiments generated by design of experiments（DOE） are carried out to train and test the SVR model. The experimental results confirm that the accuracy of the SVR model is higher than that of artificial neural networks. Based on this model, an optimization algorithm of variable stretch force trajectory using particle swarm optimization（PSO） is proposed. The springback amount is used as the objective function. Changes of local thickness are applied as the criterion of forming constraints. The objection and constraints are formulated by response surface models. The precision of response surface models is examined. Six different stretch force trajectories are employed to certify springback reduction in the optimum stretch force trajectory, which can efficiently reduce springback. This research proposes a new method of springback prediction using SVR and optimizes variable stretch force trajectory to reduce springback.

Density-wave tendency from a topological nodal-line perspective

The understanding of density waves is a vital component of our insight into electronic quantum matters. Here, we propose an additional mosaic to the existing mechanisms such as Fermi-surface nesting, electron–phonon coupling, and exciton condensation. In particular, we find that certain two-dimensional(2D) spin density-wave systems are equivalent to three-dimensional(3D) Dirac nodal-line systems in the presence of a magnetic field, whose electronic structure takes the form of Dirac-fermion Landau levels and allows a straightforward analysis of its optimal filling. The subsequent minimumenergy wave vector varies over a continuous range and shows no direct connection to the original Fermi surfaces in 2D.Also, we carry out numerical calculations where the results on model examples support our theory. Our study points out that we have yet to attain a complete understanding of the emergent density wave formalism.

Multiobjective Decision Making: the Combined Approach

In this paper, a new approach for generating all or partly efficient solutions called the Combined Approach is developed. The property of efficient solutions generated by the combined approach and its relationships with other four approaches: weighting approach, sequential approach, ε-constraint approach and hybrid approach, are discussed. Based on this combined approach, a decision-making support method called the Combined Decision-Making Method (CDMM) for multiobjective problems is developed, which is an interactive process with the decision maker. Only the aspiration levels, which reflect the decision maker's satisfying degrees for corresponding objectives, are needed to be supplied by the decision maker step by step as he will. This interactive way for objectives can easily be accepted. Finally, the application of the proposed decision making method in the resource allocation problem is discussed, and an example for the production decision analysis of the solar energy cells given.

Synchronous self-learning Pareto strategy An ensemble framework for vector and multi-criterion optimization

Purpose–In recent decades,development of effective methods for optimizing a set of conflicted objective functions has been absorbing an increasing interest from researchers.This refers to the essence of real-life engineering systems and complex natural mechanisms which are generally multi-modal,non-convex and multi-criterion.Until now,several deterministic and stochastic methods have been proposed to cope with such complex systems.Advanced soft computational methods such as evolutionary games(cooperative and non-cooperative),Pareto-based techniques,fuzzy evolutionary methods,cooperative bio-inspired algorithms and neuro-evolutionary systems have effectively come to the aid of researchers to build up efficient paradigms with application to vector optimization.The paper aims to discuss this issue.Design/methodology/approach–A novel hybrid algorithm called synchronous self-learning Pareto strategy(SSLPS)is presented for the sake of vector optimization.The method is the ensemble of evolutionary algorithms(EA),swarm intelligence(SI),adaptive version of self-organizing map(CSOM)and a data shuffling mechanism.EA are powerful numerical optimization algorithms capable of finding a global extreme point over a wide exploration domain.SI techniques(the swarm of bees in our case)can improve both intensification and robustness of exploration.CSOM network is an unsupervised learning methodology which learns the characteristics of non-dominated solutions and,thus,enhances the quality of the Pareto front.Findings–To prove the effectiveness of the proposed method,the authors engage a set of well-known benchmark functions and some well-known rival optimization methods.Additionally,SSLPS is employed for optimal design of shape memory alloy actuator as a nonlinear multi-modal real-world engineering problem.The experiments show the acceptable potential of SSLPS for handling both numerical and engineering multi-objective problems.Originality/value–To the author’s best knowledge,the proposed algorithm is among the rare multiobjective methods which fosters the use of automated unsupervised learning for increasing the intensity of Pareto front(while preserving the diversity).Also,the research evaluates the power of hybridization of SI and EA for efficient search.

Stability of Generalized Minimax Regret Equilibria with Scalar Set Payoff

In this paper, we first introduce the notion and model of generalized minimax regret equilibria with scalar set payoffs. After that, we study its general stability theorem under the conditions that the existence theorem of generalized minimax regret equilibrium point with scalar set payoffs holds. In other words, when the scalar set payoffs functions and feasible constraint mappings are slightly disturbed, by using Fort theorem and continuity results of set-valued mapping optimal value functions, we obtain a general stability theorem for generalized minimax regret equilibria with scalar set payoffs. At the same time, an example is given to illustrate our result.