A Revised Pascoletti–Serafini Scalarization Method for Multiobjective Optimization Problems

The presented study deals with the scalarization techniques for solving multiobjective optimization problems. The Pascoletti–Serafini scalarization technique is considered, and it is attempted to sidestep two weaknesses of this method, namely the inflexibility of the constraints and the difficulties of checking proper efficiency. To this end, two modifications for the Pascoletti–Serafini scalarization technique are proposed. First, by including surplus variables in the constraints and penalizing the violations in the objective function, the inflexibility of the constraints is resolved. Moreover, by including slack variables in the constraints, easy-to-check statements on proper efficiency are obtained. Thereafter, the two proposed modifications are combined to obtain the revised Pascoletti–Serafini scalarization method. Theorems are provided on the relation of (weakly, properly) efficient solutions of the multiobjective optimization problem and optimal solutions of the proposed scalarized problems. All the provided results are established with no convexity assumption. Moreover, the capability of the proposed approaches is demonstrated through numerical examples.     

Variational Iteration Method for Delay Differential Equations

Since1930'sand40's,theexamplesofdelaydifferentialequationsarisinginpracticalapplicationshavebeenescalatedrapidly,andhavebeenstudiedextensively(fordetails,see[1]).Inthispaperwewillproposeanovelmethodcalledvariationaliterationmethod[2]tosolvesuchproblems.Considerfollowingpopulationgrowthmodel[1]x′(t)+cθ(t-1)x(t)+cθ(t-1)=0(1a)x(0)=θ(0),-1≤t≤0(1b)　　Accordingtovariationaliterationmethod[2],thecorrectionfunctionalcanbeconstructedasfollowsxn+1(t)=xn(t)+∫t0λ[x′nτ+cθ(τ-1)xn(τ)+cθ(τ-1…     

Weak Convergence of an Iterative Method for Pseudomonotone Variational Inequalities and Fixed-Point Problems

We consider an iterative scheme for finding a common element of the set of solutions of a pseudomonotone, Lipschitz-continuous variational inequality problem and the set of common fixed points of N nonexpansive mappings. The proposed iterative method combines two well-known schemes: extragradient and approximate proximal methods. We derive a necessary and sufficient condition for weak convergence of the sequences generated by the proposed scheme.     

Homotopy Method for a General Multiobjective Programming Problem

In this paper, we study a class of vector minimization problems on a complete metric space X which is identified with the corresponding complete metric space of objective functions . We do not impose any compactness assumption on X. We show that, for most (in the sense of Baire category) functions , the corresponding vector optimization problem has a solution.     

A Generalized Newton-Like Method for Solving Variational Inequalities

In this paper,we present a generalized Newton-like method for solving variational inequalities,and prove Q-superlinear and Q-quadratic convergence without the strict complementarity slackness condition     

A Two-Level Method for Nonsymmetric Eigenvalue Problems

A two-level discretization method for eigenvalue problems is studied. Compared to the standard Galerkin finite element discretization technique performed on a fine grid this method discretizes the eigenvalue problem on a coarse grid and obtains an improved eigenvector (eigenvalue) approximation by solving only a linear problem on the fine grid (or two linear problems for the case of eigenvalue approximation of nonsymmetric problems). The improved solution has the asymptotic accuracy of the Galerkin discretization solution. The link between the method and the iterated Galerkin method is established. Error estimates for the general nonsymmetric case are derived.     

Parametric Duality Models for Semiinfinite Multiobjective Fractional Programming Problems Containing Generalized （α,η,ρ）-V-Invex Functions

In this paper, we present several parametric duality results under various generalized ( α,η,ρ )-V- invexity assumptions for a semiinfinite multiobjective fractional programming problem.     

The Aggregate Homotopy Method for Constrained Sequential Max-min Problems

Consider the Constrained Sequentialm二一min problem(CSMMP):梁bnn{蚕(x)S·t.垂(x)=二二1〕laX】刀In 1三坛三ml三J三认{人j(x)}}(l)1lla火11〕In1三坛三天1三J三p‘{夕‘,(x)}三。,which has many applieations in engineering design and deeision making(see【11 and thereferenees therein).This 15 a typieal elass of nondifferential optimization problems and,ingeneral,it 15 noneonvex even if all九5 and夕坛j are eonvex funetions even linear funetions,andhenee,it 15 diffieult in theoretieal researeh as well…     

A Homotopy Interior point Method for Nonlinear Complementarity Problems

ＡＨｏｍｏｔｏｐｙＩｎｔｅｒｉｏｒＰｏｉｎｔＭｅｔｈｏｄｆｏｒＮｏｎｌｉｎｅａｒＣｏｍｐｌｅｍｅｎｔａｒｉｔｙＰｒｏｂｌｅｍｓＷａｎｇＹｕ（Ｉｎｓｔ，ｏｆＥｎｇＭｅｃｈ，ＤａｌｉａｎＵｎｉｖ，ｏｆＴｅｃｈ，Ｄａｌｉａｎ１１６０２４）ＡＨｏｍｏｔｏｐｙ...     

Variational Problems and Non-Diridhlet Problems of Qnasilinear Degenerate Elliptic Equations

In this paper,we discuss the variational problems ofstrongly quasiconvex functionals with boundary integrals and the non-Dirichlet problems of quasilinear degenemte elliptic equations.We haveestablished the solvability theorems of the variational problems and thenon-Dirichlet problems.     

Generalized Variational Relation Problems with Applications

In this paper, we first obtain an existence theorem of the solutions for a variational relation problem. An existence theorem for a variational inclusion problem, a KKM theorem and an extension of the well know Ky Fan inequality will be established, as particular cases. Some applications concerning a saddle point problem with constraints, existence of a common fixed point for two mappings and an optimization problem with constraints, will be given in the last section of the paper.     

Optimal Control Problems for Elliptic–Parabolic Variational Inequalities with Time-Dependent Constraints

In this paper, we study optimal control problems for quasi-linear elliptic–parabolic variational inequalities with time-dependent constraints. We prove the existence of an optimal control that minimizes the nonlinear cost functional. Moreover, we apply our general results to some model problems. In particular, we show the necessary condition of optimal pair for a problem of partial differential equation (PDE) with a non-homogeneous Dirichlet boundary condition.     

Multiobjective Variational Programming under Generalized Type I Univexity

In this paper, we extend the Mishra, Rueda and Giorgi generalized V-univexity type I, defined for multiobjective programming, to multiobjective variational programming problems and we derive various sufficient optimality conditions and mixed type duality results under generalized V-univexity type I conditions. The authors thank the referee for many valuable comments and helpful suggestions.     

A Smoothing Method for Zero–One Constrained Extremum Problems

In this paper, the zero–one constrained extremum problem is reformulated as an equivalent smooth mathematical program with complementarity constraints (MPCC), and then as a smooth ordinary nonlinear programming problem with the help of the Fischer–Burmeister function. The augmented Lagrangian method is adopted to solve the resulting problem, during which the non-smoothness may be introduced as a consequence of the possible inequality constraints. This paper incorporates the aggregate constraint method to construct a uniform smooth approximation to the original constraint set, with approximation controlled by only one parameter. Convergence results are established, showing that under reasonable conditions the limit point of the sequence of stationary points generated by the algorithm is a strongly stationary point of the original problem and satisfies the second order necessary conditions of the original problem. Unlike other penalty type methods for MPCC, the proposed algorithm can guarantee that the limit point of the sequence is feasible to the original problem.     

A Restarted Conjugate Gradient Method for Ill—posed Problems

This paper presents a restarted conjugate gradient iterative algorithm for solving ill-posed problems.The damped Morozov‘s discrepancy principle is used as a stopping rule,Numerical experiments are given to illustrate the efficiency of the method.     

New Parallel Descent-like Method for Solving a Class of Variational Inequalities

To solve a class of variational inequalities with separable structures, some classical methods such as the augmented Lagrangian method and the alternating direction methods require solving two subvariational inequalities at each iteration. The most recent work (B.S. He in Comput. Optim. Appl. 42(2):195–212, 2009) improved these classical methods by allowing the subvariational inequalities arising at each iteration to be solved in parallel, at the price of executing an additional descent step. This paper aims at developing this strategy further by refining the descent directions in the descent steps, while preserving the practical characteristics suitable for parallel computing. Convergence of the new parallel descent-like method is proved under the same mild assumptions on the problem data.     

Analysis on a Finite Volume Element Method for Stokes Problems

Finite volume element method for the Stokes problem is considered. We use a conforming piecewise linear function on a fine grid for velocity and piecewise constant element on a coarse grid for pressure. For general triangulation we prove the equivalence of the finite volume element method and a saddle-point problem, the inf-sup condition and the uniqueness of the approximation solution. We also give the optimal order H^1 norm error estimate. For two widely used dual meshes we give the L^2 norm error estimates, which is optimal in one case and quasi-optimal in another ease. Finally we give a numerical example.