Fixed point optimization algorithm and its application to network bandwidth allocation

A convex optimization problem for a strictly convex objective function over the fixed point set of a nonexpansive mapping includes a network bandwidth allocation problem, which is one of the central issues in modern communication networks. We devised an iterative algorithm, called a fixed point optimization algorithm, for solving the convex optimization problem and conducted a convergence analysis on the algorithm. The analysis guarantees that the algorithm, with slowly diminishing step-size sequences, weakly converges to a unique solution to the problem. Moreover, we apply the proposed algorithm to a network bandwidth allocation problem and show its effectiveness.     

Deepest point of a polyhedron and linear programming

Abstract

The problem of finding a deepest point (a ball centre) of a polyhedron is studied. A finite combinatorial interior point method is presented for this problem which yields an algorithm for linear programming. We conjecture that this is a strongly polynomial algorithm. Meanwhile developing the algorithm, several auxiliary results were found; among others, Gorokh and Werner’s algorithm for linear inequalities is slightly extended. Our numerical experiments with the problem detected bugs in a linear interior point solver used in MATLAB 6 Optimization Toolbox.     

Minimax feature selection problem for constructing a classifier using support vector machines

A minimax feature selection problem for constructing a classifier using support vector machines is considered. Properties of the solutions of this problem are analyzed. An improvement of the saddle point search algorithm based on extending the bound for the step parameter is proposed. A new nondifferential optimization algorithm is developed that, together with the saddle point search algorithm, forms a hybrid feature selection algorithm. The efficiency of the algorithm for computing Dykstra’s projections as applied for the feature selection problem is experimentally estimated.     

A new iterative algorithm for the variational inequality problem over the fixed point set of a firmly nonexpansive mapping

Many problems to appear in signal processing have been formulated as the variational inequality problem over the fixed point set of a nonexpansive mapping. In particular, convex optimization problems over the fixed point set are discussed, and operators which are considered to the problems satisfy the monotonicity. Hence, the uniqueness of the solution of the problem is not always guaranteed. In this article, we present the variational inequality problem for a monotone, hemicontinuous operator over the fixed point set of a firmly nonexpansive mapping. The main aim of the article is to solve the proposed problem by using an iterative algorithm. To this goal, we present a new iterative algorithm for the proposed problem and its convergence analysis. Numerical examples for the proposed algorithm for convex optimization problems over the fixed point set are provided in the final section.     

Computing the nadir point for multiobjective discrete optimization problems

We investigate the problem of finding the nadir point for multiobjective discrete optimization problems (MODO). The nadir point is constructed from the worst objective values over the efficient set of a multiobjective optimization problem. We present a new algorithm to compute nadir values for MODO with \(p\) objective functions. The proposed algorithm is based on an exhaustive search of the \((p-2)\)-dimensional space for each component of the nadir point. We compare our algorithm with two earlier studies from the literature. We give numerical results for all algorithms on multiobjective knapsack, assignment and integer linear programming problems. Our algorithm is able to obtain the nadir point for relatively large problem instances with up to five-objectives.     

A splitting method for quadratic programming problem

1. IntroductionThe quadratic programming (QP) problem is the most simple one in nonlinear pro-gramming and plays a very important role in optimization theory and applications.It is well known that matriX splitting teChniques are widely used for solving large-scalelinear system of equations very successfully. These algorithms generate an infinite sequence,in contrast to the direct algorithms which terminate in a finite number of steps. However,iterative algorithms are considerable simpler tha…     

Strong Convergence of a Split Common Fixed Point Problem

In this article, we present a new general algorithm for solving the split common fixed point problem in an infinite dimensional Hilbert space, which is to find a point which belongs to the common fixed point of a family of quasi-nonexpansive mappings such that its image under a linear transformation belongs to the common fixed point of another family of quasi-nonexpansive mappings in the image space. We establish the strong convergence for the algorithm to find a unique solution of the variational inequality, which is the optimality condition for the minimization problem. The algorithm and its convergence results improve and develop previous results in this field.     

On the convergence of a modified algorithm for the spherical facility location problem

We study the spherical facility location problem which is a more realistic model than the Euclidean facilities location. We present a modified algorithm for this problem, which has the following good properties: (a) It is very easy to initialize the algorithm with an arbitrary point as its starting point; (b) Under suitable assumptions, it is proved that the algorithm globally converges to a global minimizer of the problem.     

一个关于二次规划问题的分段线性同伦算法

本文发展了一个关于二次规划问题的分段线性同伦算法。该算法可看作是外点罚函数法的一个变体。凡是符合外点罚函数法收敛条件的二次规划问题用该算法均可经有限次轮回运算得到稳定解。大量的关于随机的凸二次规划问题的数值实验结果表明它的计算效率是高的，在某些条件下可能是多项式时间算法。     

一类线性与框式约束凸规划问题的原始-对偶内点算法

本文对一类具有线性和框式约束的凸规划问题给出了一个原始-对偶内点算法, 该算法可在任一原始-对偶可行内点启动, 并且全局收敛,当初始点靠近中心路径时, 算法成为中心路径跟踪算法。 数值实验表明, 算法对求解大型的这类问题是有效的。     

A new iterative method for the split common fixed point problem in Hilbert spaces

The split common fixed point problem is an inverse problem that consists in finding an element in a fixed point set such that its image under a bounded linear operator belongs to another fixed-point set. In this paper, we propose a new algorithm for this problem that is completely different from the existing algorithms. Moreover, our algorithm does not need any prior information of the operator norm. Under standard assumptions, we establish a weak convergence theorem of the proposed algorithm and a strong convergence theorem of its variant.     

Strong Convergence of an Inexact Proximal Point Algorithm for Equilibrium Problems in Banach Spaces

We develop an inexact proximal point algorithm for solving equilibrium problems in Banach spaces which consists of two principal steps and admits an interesting geometric interpretation. At a certain iterate, first we solve an inexact regularized equilibrium problem with a flexible error criterion to obtain an axillary point. Using this axillary point and the inexact solution of the previous iterate, we construct two appropriate hyperplanes which separate the current iterate from the solution set of the given problem. Then the next iterate is defined as the Bregman projection of the initial point onto the intersection of two halfspaces obtained from the two constructed hyperplanes containing the solution set of the original problem. Assuming standard hypotheses, we present a convergence analysis for our algorithm, establishing that the generated sequence strongly and globally converges to a solution of the problem which is the closest one to the starting point of the algorithm.     

A new approach for solving linear bilevel problems using genetic algorithms

Bilevel programming involves two optimization problems where the constraint region of the first level problem is implicitly determined by another optimization problem. This paper develops a genetic algorithm for the linear bilevel problem in which both objective functions are linear and the common constraint region is a polyhedron. Taking into account the existence of an extreme point of the polyhedron which solves the problem, the algorithm aims to combine classical extreme point enumeration techniques with genetic search methods by associating chromosomes with extreme points of the polyhedron. The numerical results show the efficiency of the proposed algorithm. In addition, this genetic algorithm can also be used for solving quasiconcave bilevel problems provided that the second level objective function is linear.     

一种基于邻近点算法的变步长原始-对偶算法

本文考虑求解一种源于信号及图像处理问题的鞍点问题.基于邻近点算法的思想,我们对原始-对偶算法进行改进,构造一种对称正定且可变的邻近项矩阵,得到一种新的原始-对偶算法.新算法可以看成一种邻近点算法,因此它的收敛性易于分析,且无需较强的假设条件.初步实验结果表明,当新算法被应用于求解图像去模糊问题时,和其他几种主流的高效算法相比,新算法能得到较高质量的结果,且计算时间也是有竞争力的.     

Numerical solution of an equilibrium problem of based on the generalized level method

An equilibrium problem is studied whose special case is finding a Nash point in a noncooperative multiperson game. A numerical algorithm for solving this problem is described. Conditions on the problem are stated under which an estimate is obtained for the convergence rate of the algorithm to a unique solution of the problem. The results are used for a numerical analysis of noncooperative games.     

On a decoupled algorithm for solving a finite element problem for the approximation of viscoelastic fluid flow

Summary. The aim of this work is to study a decoupled algorithm of a fixed point for solving a finite element (FE) problem for the approximation of viscoelastic fluid flow obeying an Oldroyd B differential model. The interest for this algorithm lies in its applications to numerical simulation and in the cost of computing. Furthermore it is easy to bring this algorithm into play. The unknowns are the viscoelastic part of the extra stress tensor, the velocity and the pressure. We suppose that the solution is sufficiently smooth and small. The approximation of stress, velocity and pressure are resp. discontinuous, continuous, continuous FE. Upwinding needed for convection of , is made by discontinuous FE. The method consists to solve alternatively a transport equation for the stress, and a Stokes like problem for velocity and pressure. Previously, results of existence of the solution for the approximate problem and error bounds have been obtained using fixed point techniques with coupled algorithm. In this paper we show that the mapping of the decoupled fixed point algorithm is locally (in a neighbourhood of ) contracting and we obtain existence, unicity (locally) of the solution of the approximate problem and error bounds. Received July 29, 1994 / Revised version received March 13, 1995     

有向网络中具有一个枢纽点的最小支撑树的计算方法

对有向网络中具有一个枢纽点的支撑树的问题和性质进行了研究,给出了在有向网络图中寻找以某一定点为枢纽点的最小支撑树的计算方法,并对算法的复杂性进行了讨论,最后将该算法应用于实际算例的计算．     

A trust region algorithm for solving bilevel programming problems

In this paper, we present a new trust region algorithm for a nonlinear bilevel programming problem by solving a series of its linear or quadratic approximation subproblems. For the nonlinear bilevel programming problem in which the lower level programming problem is a strongly convex programming problem with linear constraints, we show that each accumulation point of the iterative sequence produced by this algorithm is a stationary point of the bilevel programming problem.     

A hybrid algorithm for the minimum bounding sphere problem

Two primal approaches, the shrinking approach and the dual approach, have been studied for the exact Minimum Bounding Sphere (MBS) problem. In this paper, we present a dual algorithm that uses the shrinking approach to solve subproblems. The experiments show our hybrid algorithm is faster than the dedicated shrinking algorithm and dual algorithm for solving the exact MBS problem in large point sets.     

An infeasible interior-point algorithm based on modified Nesterov and Todd directions for symmetric linear complementarity problem

We present an infeasible interior-point algorithm for symmetric linear complementarity problem based on modified Nesterov–Todd directions by using Euclidean Jordan algebras. The algorithm decreases the duality gap and the feasibility residual at the same rate. In this algorithm, we construct strictly feasible iterates for a sequence of perturbations of the given problem. Each main iteration of the algorithm consists of a feasibility step and a number of centring steps. The starting point in the first iteration is strictly feasible for a perturbed problem. The feasibility steps lead to a strictly feasible iterate for the next perturbed problem. By using centring steps for the new perturbed problem, a strictly feasible iterate is obtained to be close to the central path of the new perturbed problem. Furthermore, giving a complexity analysis of the algorithm, we derive the currently best-known iteration bound for infeasible interior-point methods.