A smoothing Newton method for mathematical programs constrained by parameterized quasi-variational inequalities

We consider a class of mathematical programs governed by parameterized quasi-variational inequalities(QVI).The necessary optimality conditions for the optimization problem with QVI constraints are reformulated as a system of nonsmooth equations under the linear independence constraint qualification and the strict slackness condition.A set of second order sufficient conditions for the mathematical program with parameterized QVI constraints are proposed,which are demonstrated to be sufficient for the second o...     

Mathematical Programs with Semidefinite Cone Complementarity Constraints: Constraint Qualifications and Optimality Conditions

The tangent cone of gph $N_{S^n_+}$ plays an important role in developing necessary conditions for mathematical programs with semidefinite cone complementarity constraints. We demonstrate an elegant formula for the tangent cone of gph $N_{S^n_+}$ , based on which the Bouligand stationary point is characterized explicitly. The relationships among different stationary points under certain constraint qualifications are discussed. Then we propose a second order sufficient condition which can be weakened under the strict complementarity condition. Importantly, for the sake of algorithm design, under the assumption of strict complementarity condition, we give a nonsmooth equation reformulation of the stationary point, whose smoothing system is verified to be nonsingular at the stationary point under the proposed second order sufficient condition.     

均衡约束为二阶锥约束广义方程的数学规划问题的二阶充分条件(英)

本文考虑一类均衡约束为二阶锥约束广义方程的数学规划问题. 我们通过一个非光滑映射的方向导数, 给出了临界锥的定义, 并建立它在可行点处的等价形式. 基于此临界锥, 我们提出了均衡约束为二阶锥约束广义方程的数学规划问题的二阶充分性条件, 并且验证了在适当的条件下, M-稳定点处的二阶充分性条件是二阶增长条件成立的充分条件.     

Higher-order optimality conditions for strict local minima

In this work, we study a nonsmooth optimization problem with generalized inequality constraints and an arbitrary set constraint. We present necessary conditions for a point to be a strict local minimizer of order k in terms of higher-order (upper and lower) Studniarski derivatives and the contingent cone to the constraint set. In the same line, when the initial space is finite dimensional, we develop sufficient optimality conditions. We also provide sufficient conditions for minimizers of order k using the lower Studniarski derivative of the Lagrangian function. Particular interest is put for minimizers of order two, using now a special second order derivative which leads to the Fréchet derivative in the differentiable case.     

Second-order conditions in c1, 1 optimization with applications

Second-order necessary conditions for inequality and equality constrained C^{1, 1} optimization problems are derived. A constraint qualification condition which uses the recent generalized second-order directional derivative is employed to obtain these conditions. Various second-order sufficient conditions are given under appropriate conditions on the generalized second-order directional derivative in a neighborhood of a given point. An application of the secondorder conditions to a new class of nonsmooth C^{1, 1} optimization problems with infinitely many constraints is presented.     

Nonsmooth minimax programming under locally Lipschitz (Φ, ρ)-invexity

Minimax programming problems involving locally Lipschitz (Φ, ρ)-invex functions are considered. The parametric and non-parametric necessary and sufficient optimality conditions for a class of nonsmooth minimax programming problems are obtained under nondifferentiable (Φ, ρ)-invexity assumption imposed on objective and constraint functions. When the sufficient conditions are utilized, parametric and non-parametric dual problems in the sense of Mond-Weir and Wolfe may be formulated and duality results are derived for the considered nonsmooth minimax programming problem. With the reference to the said functions we extend some results of optimality and duality for a larger class of nonsmooth minimax programming problems.     

On the Dirichlet Problem for a System of Degenerate at a Point Elliptic Equations in the Class of Bounded Functions

We consider a weakly connected (by the lowest terms) system of elliptic equations of second order with the main part in the form of the Laplace operator, the order of which becomes degenerate at an interior point of the domain. We investigate a Dirichlet-type problem in the class of bounded Hölder vector functions. We obtain sufficient conditions for the existence and uniqueness of a solution.     

Global Complexity Bound Analysis of the Levenberg–Marquardt Method for Nonsmooth Equations and Its Application to the Nonlinear Complementarity Problem

We investigate a global complexity bound of the Levenberg–Marquardt Method (LMM) for nonsmooth equations. The global complexity bound is an upper bound to the number of iterations required to get an approximate solution that satisfies a certain condition. We give sufficient conditions under which the bound of the LMM for nonsmooth equations is the same as smooth cases. We also show that it can be reduced under some regularity assumption. Furthermore, by applying these results to nonsmooth equations equivalent to the nonlinear complementarity problem (NCP), we get global complexity bounds for the NCP. In particular, we give a reasonable bound when the mapping involved in the NCP is a uniformly P-function.     

Boundary value problems of second order nonlinear functional difference equations

By using the critical point theory, some sufficient conditions for the existence and multiplicity of the boundary value problems to a class of second order functional difference equations are obtained.     

Multiple and sign‐changing solutions for the multivalued p ‐Laplacian equation

We consider a class of elliptic inclusions under Dirichlet boundary conditions involving multifunctions of Clarke's generalized gradient. Under conditions given in terms of the first eigenvalue as well as the Fu?ik spectrum of the p ‐Laplacian we prove the existence of a positive, a negative and a sign‐changing solution. Our approach is based on variational methods for nonsmooth functionals (nonsmooth critical point theory, second deformation lemma), and comparison principles for multivalued elliptic problems. In particular, the existence of extremal constant‐sign solutions plays a key role in the proof of sign‐changing solutions (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Fourth order nonselfadjoint differential equations with clamped-free boundary conditions

Summary A class of nonselfadjoint fourth order differential equations is investigated in this paper by a pair of equations of the second order. Special attention is given to establishing conditions for the existence of solutions subject to two point boundary conditions, and is achieved basically through various characterizations, comparison theorems and related eigenvalue problems. Entrata in Redazione il 1^o giugno 1977. The first author is supported by the NRC of Canada under Grant number A3105.     

Boundary value problems of second order nonlinear difference equations with Jacobi operators

The paper is devoted to a question of existence and multiplicity of solutions of boundary value problems for a class of second order nonlinear difference equations with Jacobi operators. By using the critical point theory, some sufficient conditions are obtained.     

On Boundary Value Problems for Second Order Nonlinear Functional Difference Equations

In this paper, some sufficient conditions for the existence and multiplicity of the boundary value problems to a class of second order functional difference equations are obtained by using the critical point theory.     

Nonsmooth Equation Based BFGS Method for Solving KKT Systems in Mathematical Programming

In this paper, we present a BFGS method for solving a KKT system in mathematical programming, based on a nonsmooth equation reformulation of the KKT system. We split successively the nonsmooth equation into equivalent equations with a particular structure. Based on the splitting, we develop a BFGS method in which the subproblems are systems of linear equations with symmetric and positive-definite coefficient matrices. A suitable line search is introduced under which the generated iterates exhibit an approximate norm descent property. The method is well defined and, under suitable conditions, converges to a KKT point globally and superlinearly without any convexity assumption on the problem.     

具高阶Laplace算子的非线性脉冲时滞双曲型方程的振动性

研究一类具高阶Laplace算子的非线性脉冲时滞双曲型偏泛函微分方程,利用二阶脉冲时滞微分不等式,得到了该类方程在两类不同边值条件下所有有界解振动的若干充分判据.     

Simple layer potential and the first boundary value problem for a parabolic system on the plane

By the method of boundary integral equations, we construct a classical solution of the first initial–boundary value problem for a one-dimensional (with respect to x) parabolic system in a domain with nonsmooth lateral boundary for the case in which the right-hand sides of the boundary conditions only have continuous derivatives of order 1/2. We study the smoothness of the solution.     

一类具连续分布滞量的非线性中立型偏微分方程解的振动性

In this paper,some sufficient conditions are obtained for the oscillation of solutions for a class of second order nonlinear neutral partial differential equations with continuous distribution delay under Robin and Dirichlet's boundary value conditions.     

Multiderivative Runge-Kutta processes for two-point boundary value problems

A class of finite difference schemes for the solution of a nonlinear system of first order differential equations with two point boundary conditions which shares properties with Runge-Kutta processes and gap schemes is discussed. The order conditions for the coefficients of these processes, techniques for reducing these order conditions in number and the symmetry conditions are given. A symmetricA-stable eight order process which has second, fourth and sixth orderA-stable processes embedded in it is given as an example.Research supported in part by the United States Air Force under contract AFOSR-89-0383.     

Smoothing Newton method for nonsmooth second-order cone complementarity problems with application to electric power markets

This paper is dedicated to solving a nonsmooth second-order cone complementarity problem, in which the mapping is assumed to be locally Lipschitz continuous, but not necessarily to be continuously differentiable everywhere. With the help of the vector-valued Fischer-Burmeister function associated with second-order cones, the nonsmooth second-order cone complementarity problem can be equivalently transformed into a system of nonsmooth equations. To deal with this reformulated nonsmooth system, we present an approximation function by smoothing the inner mapping and the outer Fischer-Burmeister function simultaneously. Different from traditional smoothing methods, the smoothing parameter introduced is treated as an independent variable. We give some conditions under which the Jacobian of the smoothing approximation function is guaranteed to be nonsingular. Based on these results, we propose a smoothing Newton method for solving the nonsmooth second-order cone complementarity problem and show that the proposed method achieves globally superlinear or quadratic convergence under suitable assumptions. Finally, we apply the smoothing Newton method to a network Nash-Cournot game in oligopolistic electric power markets and report some numerical results to demonstrate its effectiveness.

     

具连续变量和最大值项的二阶差分方程的振动性

研究了一类具有最大值项和连续变量的非线性二阶中立型时滞差分方程的振动性,利用Banach空间的不动点原理和一些不等式技巧,得到了这类方程存在最终正解的充分条件,并得到了该方程振动的一些判别准则.