Xinmin Yang

• Doctor of Philosophy
• Vice President at Chongqing Normal University

In this paper, we propose an efficient strategy for improving the multi-objective steepest descent method proposed by Fliege and Svaiter (Math Methods Oper Res, 51, 479–494, 2000). The core idea behind this strategy involves incorporating a positive modification parameter into the iterative formulation of the multi-objective steepest descent algori...

Over the past two decades, descent methods have received substantial attention within the multiobjective optimization field. Nonetheless, both theoretical analyses and empirical evidence reveal that existing first-order methods for multiobjective optimization converge slowly, even for well-conditioned problems, due to the objective imbalances. To a...

In this paper, we deal with multiobjective composite optimization problems, where each objective function is a combination of smooth and possibly non-smooth functions. We first propose a parameter-dependent conditional gradient method to solve this problem. The step size in this method requires prior knowledge of the parameters related to the H{\"o...

Stochastic multi-objective optimization (SMOO) has recently emerged as a powerful framework for addressing machine learning problems with multiple objectives. The bias introduced by the nonlinearity of the subproblem solution mapping complicates the convergence analysis of multi-gradient methods. In this paper, we propose a novel SMOO method called...

In this paper, we develop a unified framework and convergence analysis of descent methods for vector optimization problems (VOPs) from a majorization-minimization perspective. By choosing different surrogate functions, the generic method reduces to some existing descent methods with and without line search, respectively. The unified convergence ana...

In this paper, we propose an efficient strategy for improving the multi-objective steepest descent method proposed by Fliege and Svaiter (Math Methods Oper Res, 2000, 3: 479--494). The core idea behind this strategy involves incorporating a positive modification parameter into the iterative formulation of the multi-objective steepest descent algori...

We study the generalized sequential normal compactness in variational analysis and establish characterizations of the property of graphs of weakly differentiable mappings between Banach spaces, as well as calculus rules involving such functions.

Nonlinear conjugate gradient methods have recently garnered significant attention within the multiobjective optimization community. These methods aim to maintain consistency in conjugate parameters with their single-objective optimization counterparts. However, the preservation of the attractive conjugate property of search directions remains uncer...

Preconditioning is a powerful approach for solving ill-conditioned problems in optimization , where a preconditioning matrix is used to reduce the condition number and speed up the convergence of first-order method. Unfortunately, it is impossible to capture the curvature of all objective functions with a single preconditioning matrix in multiobjec...

In a recent study, Ansary (Optim Methods Softw 38(3):570-590,2023) proposed a Newton-type proximal gradient method for nonlinear multiobjective optimization problems (NPGMO). However, the favorable convergence properties typically associated with Newton-type methods were not established for NPGMO in Ansary's work. In response to this gap, we develo...

Over the past two decades, multiobejective gradient descent methods have received increasing attention due to the seminal work of Fliege and Svaiter. Recently, Chen et al. pointed out that imbalances among objective functions can lead to a small stepsize in Fliege and Svaiter's method, which significantly decelerates the convergence. To address the...

The filled function method is an efficient approach for finding a global minimizer of global optimization problems. The key of this kind of methods is the design of filled function. In this paper, we propose a new filled function with one parameter that is continuously differentiable and always contains local information of objective function. Then...

In this paper, we propose a conditional gradient method for solving constrained vector optimization problems with respect to a partial order induced by a closed, convex and pointed cone with nonempty interior. When the partial order under consideration is the one induced by the non-negative orthant, we regain the method for multiobjective optimizat...

In this paper, we consider the exact continuous relaxation model of ℓ0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\ell_{0}}$$\end{document} regularization problem,...

In this paper, we propose a variable metric method for unconstrained multiobjective optimization problems (MOPs). First, a sequence of points is generated using different positive definite matrices in the generic framework. It is proved that accumulation points of the sequence are Pareto critical points. Then, without convexity assumption, strong c...

In this paper, we consider the exact continuous relaxation model of $l_0$ regularization problem which was given by Bian and Chen (SIAM J. Numer. Anal 58(1): 858-883, 2020) and propose a smoothing proximal gradient algorithm with extrapolation (SPGE) for this kind of problem. We show that any accumulation point of the sequence generated by SPGE alg...

A new numerical method is presented for bilevel programs with a nonconvex follower’s problem. The basic idea is to piecewise construct convex relaxations of the follower’s problems, replace the relaxed follower’s problems equivalently by their Karush–Kuhn–Tucker conditions and solve the resulting mathematical programs with equilibrium constraints....

In this paper, we investigate ε-strong efficiency of a set in ordered linear spaces. Firstly, a new conception of ε-strongly efficient point of a set is introduced. Secondly, some properties and the existence of ε-strongly efficient points of a set are studied. Finally, as the applications, the linear scalarization theorems of the set-valued optimi...

In this paper, the connectedness and path-connectedness of solution sets for weak generalized symmetric Ky Fan inequality problems with respect to addition-invariant set are studied. A class of weak generalized symmetric Ky Fan inequality problems via addition-invariant set is proposed. By using a nonconvex separation theorem, the equivalence betwe...

In this paper, we aim at applying improvement sets and image space analysis to investigate scalarizations and optimality conditions of the constrained set-valued optimization problem. Firstly, we use the improvement set to introduce a new class of generalized convex set-valued maps. Secondly, under suitable assumptions, some scalarization results o...

The existence of complementarity constraints causes the difficulties for studying mathematical programs with second-order cone complementarity constraints, since the standard constraint qualification, such as Robinson’s constraint qualification, is invalid. Therefore, various stationary conditions including strong, Mordukhovich and Clarke stationar...

This paper considers a mathematical problem with equilibrium constraints (MPEC) in which the objective is locally Lipschitz continuous but not continuously differentiable everywhere. Our focus is on constraint qualifications for the nonsmooth S-stationarity in the sense of the limiting subdifferentials. First, although the MPEC-LICQ is not a constr...

In this paper, we derive some optimality and stationarity conditions for a multiobjective problem with equilibrium constraints (MOPEC). In particular, under a generalized Guignard constraint qualification, we show that any locally Pareto optimal solution of MOPEC must satisfy the strong Pareto Kuhn-Tucker optimality conditions. We also prove that t...

In this paper, we consider a class of multiobjective problems with equilibrium constraints. Our first task is to extend the existing constraint qualifications for mathematical problems with equilibrium constraints from the single-objective case to the multiobjective case, and our second task is to derive some stationarity conditions under the prope...

This paper considers a class of vector variational inequalities. First, we present an equivalent formulation, which is a scalar variational inequality, for the deterministic vector variational inequality. Then we concentrate on the stochastic circumstance. By noting that the stochastic vector variational inequality may not have a solution feasible...

We study the directional Mordukhovich normal cones to nonsmooth sets, coderivatives of set-valued mappings in Asplund spaces and establish extensive calculus results on these constructions under various operations of sets and mappings. We also develop calculus of the directional sequential normal compactness both in general Banach spaces and in Asp...

This paper begins with a study on the dual representations of risk and regret measures and their impact on modeling multistage decision making under uncertainty. A relationship between risk envelopes and regret envelopes is established by using the Lagrangian duality theory. Such a relationship opens a door to a decomposition scheme, called progres...

This paper obtains some stability results for parametric generalized set-valued weak vector equilibrium problem. Under new assumptions, which do not contain any information about solution mappings, the authors establish the continuity of the solution mapping to a parametric generalized set-valued weak vector equilibrium problem without monotonicity...

By means of the Minkowski-type nonlinear scalarization functional, in this paper, we establish some nonlinear separation theorems via relative algebraic interior and vector closure in a general real linear space, via relative topological interior and topological closure in a real topological linear space and via quasi relative interior and topologi...

In this work we study the directional versions of Mordukhovich normal cones to nonsmooth sets, coderivatives of set-valued mappings, and subdifferentials of extended-real-valued functions in the framework of general Banach spaces. We establish some characterizations and basic properties of these constructions, and then develop calculus including su...

We focus on second order duality for a class of multiobjective programming problem subject to cone constraints. Four types of second order duality models are formulated. Weak and strong duality theorems are established in terms of the generalized convexity, respectively. Converse duality theorems, essential parts of duality theory, are presented un...

This paper mainly intends to present some semicontinuity and convergence results for perturbed vector optimization problems with approximate equilibrium constraints. We establish the lower semicontinuity of the efficient solution mapping for the vector optimization problem with perturbations of both the objective function and the constraint set. Th...

In this paper, E-super efficiency of set-valued optimization problems is investigated. Firstly, based on the improvement set, a new notion of E-super efficient point is introduced in real locally convex spaces. Secondly, under the assumption of near E-subconvexlikeness of set-valued maps, scalarization theorems of set-valued optimization problems a...

In this paper we first review the theory of weak differentiability with some improvements and unifications of existing results; then we introduce an extended variant of this notion and establish its basic properties; finally we use the weak differentiability and its variant to develop new calculus results in variational analysis for the theory of g...

In this paper, we study approximate solutions of vector optimization problems. We introduce the concept of cone convex functions with respect to (in short w.r.t.) a mapping. Under this kind of cone convexity assumption, we obtain the Karush-Kuhn-Tucker type necessary and sufficient optimality conditions for quasiminimal solutions w.r.t. a mapping o...

In this paper, we obtain some stability results for parametric weak vector equilibrium problem with set-valued mappings. By using a scalarization method, we establish sufficient conditions for the semicontinuity of the approximate solution mappings to parametric set-valued weak vector equilibrium problem under weak assumptions. These results extend...

Many inverse problems can be formulated as split feasibility problems. To find feasible solutions, one has to minimize proximity functions. We show that the existence of minimizers to the proximity function for Censor-Elfving’s split feasibility problem is equivalent to the existence of projections on appropriate convex sets and provide conditions...

In this paper, we discuss the closedness of set of efficient solutions for generalized Ky Fan inequality problems in topological vector spaces. We introduce a concept of section mapping of a bifunction. By using the lower semicontinuity of the section mapping, we present sufficient conditions for the closedness of set of efficient solutions to the...

In this note, we obtain an important property from Condition C. Using the property, we can provide short proofs for some properties of (generalized) preinvex functions.

In this paper, we introduce two types of Levitin-Polyak well-posed-ness for a system of generalized vector variational inequality problems. By means of a gap function of the system of generalized vector variational inequal-ity problems, we establish equivalence between the two types of Levitin-Polyak well-posedness of the system of generalized vect...

The concept of \(\varphi \) -strongly preinvex functions is introduced, and some properties of \(\varphi \) -strongly preinvex functions are given. Several new and simple characterizations of the approximate solution sets for nonsmooth optimization problems with \(\varphi \) -strong preinvexity are obtained. We establish the relationships between t...

In the present paper, we are devoted to exploring conditions of well-posedness for hemivariational inequalities in reflexive Banach spaces. By using some equivalent formulations of the hemivariational inequality considered under different monotonicity assumptions, we establish two kinds of conditions under which the strong well-posedness and the we...

In this paper, we characterize approximate solutions of vector optimization problems with set-valued maps. We gives several characterizations of generalized subconvexlike set-valued functions(see [10]), which is a generalization of nearly subconvexlike functions introduced in [34]. We present alternative theorem and derived scalarization theorems f...

In this paper, by using a scalarization method, we establish sufficient conditions for Ho¨lder continuity of approximate solution mapping to a class of parametric weak generalized Ky Fan Inequality with set-valued mappings. These results extend and improve some known results in the literature. Furthermore, some examples are given to illustrate the...

In recent years, some different kinds of approximate proper efficiency have been proposed by means of different tools including co-radiant set and improvement set for a vector optimization problem. In this paper, we first summarize some known concepts of approximate proper efficiency, relations among them and some corresponding linear scalarization...

In this paper, under the nearly E-subconvexlikeness, some characterizations of the E-Benson proper efficiency are established in terms of scalarization, Lagrange multipliers, saddle point criteria and duality for a vector optimization problem with set-valued maps. Our main results generalize and unify some previously known results.

In this paper, a unified framework of a nonlinear augmented Lagrangian dual problem is investigated for the primal problem of minimizing an extended real-valued function by virtue of a nonlinear augmenting penalty function. Our framework is more general than the ones in the literature in the sense that our nonlinear augmenting penalty function is d...

For semi-infinite programming (SIP), we consider a class of smoothed penalty functions, which approximate the exact $l_\rho (0<\rho \le 1)$ penalty functions. On base of the smoothed penalty function, we present a feasible penalty algorithm for solving SIP. Without any boundedness condition or coercive condition, we establish the global convergen...

In this paper, using a scalarization technique, we provide sufficient conditions for the upper/lower semicontinuity of the solution mappings to parametric set-valued weak vector equilibrium problems without monotonicity and C-concavity in linear metric spaces. These results improve the recent ones in the literature. Some examples are given for illu...

In this paper, some scalar characterizations of approximate weakly efficient solutions and approximate Henig efficient solutions for vector equilibrium problems are derived without imposing any convexity assumption on objective functions and feasible set. Meanwhile, the linear scalar characterization of approximate weakly efficient solutions is als...

We obtain two new characterizations of preinvex functions. More specifically, a real valued function is preinvex function if and only if it is intermediate-point preinvex and semi-strictly quasi-preinvex; and a real valued function is preinvex function if and only if it is intermediate-point preinvex and semi-locally semi-strictly quasi-preinvex.

In this paper, cone preinvex and related functions are studied. The concept of cone subpreinvex functions is introduced. Some properties of cone subpreinvex functions are established and their relationships with cone convex, cone subconvex, cone preinvex functions are explored. Under the condition of cone subpreinvex functions, optimality condition...

A new concept of nondifferentiable pseudoinvex functions is introduced. Based on the basic properties of this class of pseudoinvex functions, several new and simple characterizations of the solution sets for nondifferentiable pseudoinvex programs are given. Our results are extension and improvement of some results obtained by Mangasarian (Oper. Res...

We point out some errors in a recent paper of M. A. E.-H. Kassem [“Symmetric and self duality in vector optimization problem”, Appl. Math. Comput. 183, 1121–1126 (2006)]. And a pair of the first-order symmetric dual model for vector optimization problem is proposed in this paper. Then, we prove the weak, strong and converse duality theorems for the...

A class of multiobjective optimization problems in which inequality constraints are involved is considered. The necessary and sufficient conditions are established for an efficient solution. Moreover, the equivalence is proved between efficiency and properly efficiency by linear scalarization methods under some suitable conditions. Our results impr...

We establish a mixed type converse duality for a class of multiobjective programming programs. This clarifies several omissions in an earlier work by X. M. Yang, X. Q. Yang and K. L. Teo [J. Math. Anal. Appl. 304, No. 1, 394–398 (2005; Zbl 1075.90069)].

In this paper, we study the approximate solutions for vector optimization problem with set-valued functions. The scalar characterization is derived without imposing any convexity assumption on the objective functions. The relationships between approximate solutions and weak efficient solutions are discussed. In particular, we prove the connectednes...

In this paper, a class of nonsmooth multiobjective optimization problems with equality constraints and inequality constraints is considered. The generalized Karush-Kuhn-Tucker necessary and sufficient optimality conditions are given. Furthermore, the mixed type dual model is discussed, and theorems of weak duality, strong duality, converse duality,...

We study first- and second-order necessary and sufficient optimality conditions for approximate (weakly, properly) efficient solutions of multiobjective optimization problems. Here, tangent cone, is an element of-normal cone, cones of feasible directions, second-order tangent set, asymptotic second-order cone, and Hadamard upper (lower) directional...

In this paper, we point out an inconsistency between assumptions and results on the second order strong and converse duality in a recent paper of I. Ahmad (Information Sciences 173 (2005) 23-34). We then provide appropriate modifications to rectify this deficiency.

New concepts of generalized (ρ,θ)-η invex functions for non-differentiable functions and generalized (ρ,θ)-η invariant monotone operators for set-valued mappings are introduced. The relationships between generalized (ρ,θ)-η invexity of functions and generalized (ρ,θ)-η invariant monotonicity of the corresponding Clarke’s subdifferentials are studie...

In this article, we introduce a unified class of augmented Lagrangian functions for constrained non-convex optimization problems which include many types of the augmented Lagrangians. We first get the zero duality gap property between the primal problem and the augmented Lagrangian dual problem. Then, under second-order sufficiency conditions, we p...

In this paper, a pair of higher order symmetric dual models for multiobjective nonlinear programming is introduced. The weak, strong and converse duality theorems are proven for the formulated higher order symmetric dual programs under invexity conditions.

The relationships between strongly pseudo-invexity of non-differentiable functions and strongly invariant pseudo-monotonicity of set-valued mapping are discussed. Pseudo-invexity and semi-strictly prequasi-invexity of non-differentiable functions are studied also. In addition, an equivalent definition of invariant pseudo-monotonicity of set-valued...

First order and second order dual models for a class of nondifferentiable programming problems in which the objective function contains a support function of a compact convex set are formulated. Weak and converse duality theorems for the two dual models are established by using Fritz John necessary optimality conditions and some suitable conditions...

For constrained nonconvex optimization, we first show that under second-order sufficient conditions, a class of augmented Lagrangian functions possess local saddle points, and then prove that global saddle points of these augmented Lagrangian functions exist under certain mild additional conditions.

We study efficient point sets in terms of extreme points, positive support points and strongly positive exposed points. In the case when the ordering cone has a bounded base, we prove that the efficient point set of a weakly compact convex set is contained in the closed convex hull of its strongly positive exposed points, thereby extending the Phel...

T.R. Gulati and Divya Agarwal pointed out that the statement: α>0α>0 and θ=αr∗ imply θ>0θ>0, on line 4 of page 208 of [X.M. Yang, X.Q. Yang and K.L. Teo, Appl. Math. Lett. 18 (2005) 205–208], is erroneous since θ=αr∗ is obtained letting α=0α=0. Here is a simple proof that θ>0θ>0.

For unconstrained optimization, a new hybrid projection algorithm is presented in the paper. This algorithm has some attractive convergence properties. Convergence theory can be obtained under the condition that ∇ f(x) is uniformly continuous. If ∇ f(x) is continuously differentiable pseudo-convex, the whole sequence of iterates converges to a solu...

In this paper, such new definitions as D-B-preinvexity, strictly D-B-preinvexity and explicitly D-B-preinvexity for vector-valued mappings are firstly introduced. Then, a sufficient condition of D-B-preinvex mappings is shown. And then the relationship between the explicitly D-B-preinvexity and the strictly D-B-preinvexity is discussed. Finally, so...

In this article, some properties of pseudoinvex functions are given and relations between vector variational-like inequalities and vector optimization problems are discussed under pseudoinvexity and invariant pseudomonotonicity conditions, respectively.

In a recent paper, a nonmonotone spectral projected gradient (SPG) method was introduced by Birgin et al. for the minimization of differentiable functions on closed convex sets and extensive presented results showed that this method was very efficient. In this paper, we give a more comprehensive theoretical analysis of the SPG method. In doing so,...

We introduce a new model of the system of generalized vector quasi-equilibrium problems with upper semicontinuous set-valued maps and present several existence results of a solution for this system of generalized vector quasi-equilibrium problems and its special cases. The results in this paper extend and improve some results in the literature.

In this paper, we prove a new existence result for multivalued complementarity problems in Banach spaces. Our result represents a refinement and improvement of the previously known results.

In this paper, the notion of affinelike set-valued maps is introduced and some properties of these maps are presented. Then a new Hahn–Banach extension theorem with a K-convex set-valued map dominated by an affinelike set-valued map is obtained.

In this paper, we study optimal value functions of generalized semi-infinite minmax programming problems on a noncompact set. Directional derivatives and subdifferential characterizations of optimal value functions are given. Using these properties, we establish first order optimality conditions for unconstrained generalized semi-infinite programmi...

If a nonconvex minimization problem can be converted into an equivalent convex minimization problem, the primal nonconvex minimization problem is called a hidden convex minimization problem. Sufficient conditions are developed in this paper to identify such hidden convex minimization problems. Hidden convex minimization problems possess the same de...

Consider the class of prequasiinvex functions. In this paper, some characterizations of prequasiinvex function are provided. Under prequasiinvexity conditions, determination of the satisfaction of semistrictly prequasiinvexity for a function can be achieved via an intermediate-point semistrictly prequasiinvexity check.

In this paper, we first discuss some basic properties of semipreinvex functions. We then show that the ratio of semipreinvex functions is semipreinvex, which extends earlier results by Khan and Hanson [6] and Craven and Mond [3]. Finally, saddle point optimality criteria are developed for a multiobjective fractional programming problem under semipr...