Elliptic problems with discontinuities

In this paper we prove two existence theorems for elliptic problems with discontinuities. The first one is a noncoercive Dirichlet problem and the second one is a Neumann problem. We do not use the method of upper and lower solutions. For Neumann problems we assume that f is nondecreasing. We use the critical point theory for locally Lipschitz functionals.     

Augmented Lagrangian method for distributed optimal control problems with state constraints

We consider state-constrained optimal control problems governed by elliptic equations. Doing Slater-like assumptions, we know that Lagrange multipliers exist for such problems, and we propose a decoupled augmented Lagrangian method. We present the algorithm with a simple example of a distributed control problem.     

对带有盒约束的二次整数规划的一种线性化方法

本文主要讨论了二次整数规划问题的线性化方法.在目标函数为二次函数的情况下,我们讨论了带有二次约束的整数规划问题的线性化方法,并将文献中对二次0-1问题的研究拓展为对带有盒约束的二次整数规划问题的研究.最终将带有盒约束的二次整数规划问题转化为线性混合本文主要讨论了二次整数规划问题的线性化方法.在目标函数为二次函数的情况下,我们讨论了带有二次约束的整数规划问题的线性化方法,并将文献中对二次0-1问题的研究拓展为对带有盒约束的二次整数规划问题的研究.最终将带有盒约束的二次整数规划问题转化为线性混合0-1整数规划问题,然后利用Ilog-cplex或Excel软件中的规划求解工具进行求解,从而解决原二次整数规划.     

A regularized projection method for complementarity problems with non-Lipschitzian functions

We consider complementarity problems involving functions which are not Lipschitz continuous at the origin. Such problems arise from the numerical solution for differential equations with non-Lipschitzian continuity, e.g. reaction and diffusion problems. We propose a regularized projection method to find an approximate solution with an estimation of the error for the non-Lipschitzian complementarity problems. We prove that the projection method globally and linearly converges to a solution of a regularized problem with any regularization parameter. Moreover, we give error bounds for a computed solution of the non-Lipschitzian problem. Numerical examples are presented to demonstrate the efficiency of the method and error bounds.

     

A path following method for box-constrained multiobjective optimization with applications to goal programming problems

We propose a path following method to find the Pareto optimal solutions of a box-constrained multiobjective optimization problem. Under the assumption that the objective functions are Lipschitz continuously differentiable we prove some necessary conditions for Pareto optimal points and we give a necessary condition for the existence of a feasible point that minimizes all given objective functions at once. We develop a method that looks for the Pareto optimal points as limit points of the trajectories solutions of suitable initial value problems for a system of ordinary differential equations. These trajectories belong to the feasible region and their computation is well suited for a parallel implementation. Moreover the method does not use any scalarization of the multiobjective optimization problem and does not require any ordering information for the components of the vector objective function. We show a numerical experience on some test problems and we apply the method to solve a goal programming problem.     

Boundary-value problems for a hyperbolic equation with nonlocal conditions of the I and II kind

In this paper we consider two initial-boundary value problems with nonlocal conditions. The main goal is to propose a method for proving the solvability of nonlocal problems with integral conditions of the first kind. The proposed method is based on the equivalence of a nonlocal problem with an integral condition of the first kind and a nonlocal problem with an integral condition of the second kind in a special form. We prove the unique existence of generalized solutions to both problems.     

Convergence of a substructuring method with Lagrange multipliers

Summary. We analyze the convergence of a substructuring iterative method with Lagrange multipliers, proposed recently by Farhat and Roux. The method decomposes finite element discretization of an elliptic boundary value problem into Neumann problems on the subdomains plus a coarse problem for the subdomain nullspace components. For linear conforming elements and preconditioning by the Dirichlet problems on the subdomains, we prove the asymptotic bound on the condition number , or ,where is the characteristic element size and subdomain size. Received January 3, 1995     

Global Method for Monotone Variational Inequality Problems with Inequality Constraints

We consider optimization methods for monotone variational inequality problems with nonlinear inequality constraints. First, we study the mixed complementarity problem based on the original problem. Then, a merit function for the mixed complementarity problem is proposed, and some desirable properties of the merit function are obtained. Through the merit function, the original variational inequality problem is reformulated as simple bounded minimization. Under certain assumptions, we show that any stationary point of the optimization problem is a solution of the problem considered. Finally, we propose a descent method for the variational inequality problem and prove its global convergence.     

Stochastic Programming with Multivariate Second Order Stochastic Dominance Constraints with Applications in Portfolio Optimization

In this paper we study optimization problems with multivariate stochastic dominance constraints where the underlying functions are not necessarily linear. These problems are important in multicriterion decision making, since each component of vectors can be interpreted as the uncertain outcome of a given criterion. We propose a penalization scheme for the multivariate second order stochastic dominance constraints. We solve the penalized problem by the level function methods, and a modified cutting plane method and compare them to the cutting surface method proposed in the literature. The proposed numerical schemes are applied to a generic budget allocation problem and a real world portfolio optimization problem.     

Stationarity Conditions and Their Reformulations for Mathematical Programs with Vertical Complementarity Constraints

We consider the mathematical program with vertical complementarity constraints. We show that the min-max-min problems and the problems with max-min constraints can be reformulated as the above problem. As a complement of the work of Scheel and Scholtes in 2000, we derive the Mordukhovich-type stationarity conditions for the considered problem. We further reformulate various popular stationarity systems as nonlinear equations with simple constraints. A modified Levenberg–Marquardt method is employed to solve these constrained equations.     

On the solvability of a nonlocal boundary value problem with opposite fluxes at a part of the boundary and of the adjoint problem

We consider a nonlocal boundary value problem for the Laplace operator in a circular sector with opposite fluxes on the radii and with zero value of the solution on one of the radii; we also consider the adjoint problem. We prove the unique solvability of these problems and obtain the solution in an explicit form by the spectral method. As a by-product, we study the completeness and the basis property of systems of roots functions for problems of Samarskii-Ionkin type, which may be of interest in itself.     

On the number of semiregular solutions in problems with spectral parameter for higher-order equations of elliptic type with discontinuous nonlinearities

We consider the problem of the existence of solutions of the spectral problem for higher-order equations of elliptic type with discontinuous nonlinearities. By using the variational method, we prove theorems on the number of semiregular solutions for the problems in question.     

Well-posedness for a cauchy problem associated to time dependent free boundaries with nonlocal leading terms

We study a class of free boundary problems, where the normal velocity of the interface is proportional to the derivative of the solution of an elliptic PDE; we give a simple, explicit criterion for the well-posedness of the linearized Cauchy problem. The method is then applied to two classical problems; the stefan problem and the Muskat problem.     

Solving Fractional Multicriteria Optimization Problems with Sum of Squares Convex Polynomial Data

This paper focuses on the study of finding efficient solutions in fractional multicriteria optimization problems with sum of squares convex polynomial data. We first relax the fractional multicriteria optimization problems to fractional scalar ones. Then, using the parametric approach, we transform the fractional scalar problems into non-fractional problems. Consequently, we prove that, under a suitable regularity condition, the optimal solution of each non-fractional scalar problem can be found by solving its associated single semidefinite programming problem. Finally, we show that finding efficient solutions in the fractional multicriteria optimization problems is tractable by employing the epsilon constraint method. In particular, if the denominators of each component of the objective functions are same, then we observe that efficient solutions in such a problem can be effectively found by using the hybrid method. Some numerical examples are given to illustrate our results.     

On the solvability of a nonlocal boundary value problem with the equality of fluxes at a part of the boundary and of the adjoint problem

In the present paper, we consider a nonlocal boundary value problem for the Laplace operator in a circular sector with the equality of fluxes on the radii and with zero value of the solution on one of the radii. We also consider the adjoint problem. We prove the uniqueness of the solution of these problems and obtain an explicit form for the solution by the spectral method. When proving the solvability of the problems, we study the completeness and the basis property of systems of root functions for problems of the type of the Samarskii-Ionkin problem in L _p , which can be of interest in itself.     

Averaging of Boundary-Value Problems with Parameters for Multifrequency Impulsive Systems

By using the averaging method, we prove the solvability of boundary-value problems with parameters for nonlinear oscillating systems with pulse influence at fixed times. We also obtain estimates for the deviation of solutions of the averaged problem from solutions of the original problem.     

A proximal method with logarithmic barrier for nonlinear complementarity problems

We study the proximal method with the regularized logarithmic barrier, originally stated by Attouch and Teboulle for positively constrained optimization problems, in the more general context of nonlinear complementarity problems with monotone operators. We consider two sequences generated by the method. We prove that one of them, called the ergodic sequence, is globally convergent to the solution set of the problem, assuming just monotonicity of the operator and existence of solutions; for convergence of the other one, called the proximal sequence, we demand some stronger property, like paramonotonicity of the operator or the so called “cut property” of the problem.     

Approximation of optimal control problems with state constraints

We study the approximation of control problems governed by elliptic partial differential equations with pointwise state constraints. For a finite dimensional approximation of the control set and for suitable perturbations of the state constraints, we prove that the corresponding sequence of discrete control problems converges to a relaxed problem. A similar analysis is carried out for problems in which the state equation is discretized by a finite element method.     

Averaging of a multipoint problem with parameters for an impulsive oscillation system

By using the averaging method, we prove the solvability of a multipoint problem with parameters for a nonlinear oscillation system with pulse influence at fixed times. We establish estimates for the deviation of solutions of the original and averaged problems.