基于关系代数的软集合理论研究与应用

基于关系代数理论中的部分思想,定义了软集合理论中的差运算、选择运算和投影运算.探讨了关系代数和软集合的关系,运用关系代数的选择、投影、并、差等运算实现了软集合参数约简算法,并用SQL语言实现了算法.最后将算法运用到房屋置业选择问题中进行验证.结果表明,软集合方法能以一种更简单直接的形式为决策问题提供有效的参考依据.     

Accelerated reflection projection algorithm and its application to the LMI problem

We discuss accelerated version of the alternating projection method which can be applied to solve the linear matrix inequality (LMI) problem. The alternating projection method is a well-known algorithm for the convex feasibility problem, and has many generalizations and extensions. Bauschke and Kruk proposed a reflection projection algorithm for computing a point in the intersection of an obtuse cone and a closed convex set. We carry on this research in two directions. First, we present an accelerated version of the reflection projection algorithm, and prove its weak convergence in a Hilbert space; second, we prove the finite termination of an algorithm which is based on the proposed algorithm and provide an explicit upper bound for the required number of iterations under certain assumptions. Numerical experiments for the LMI problem are provided to demonstrate the effectiveness and merits of the proposed algorithms.     

The variable projection algorithm in time-resolved spectroscopy,microscopy and mass spectrometry applications

Nonlinear least squares optimization problems in which the parameters can be partitioned into two sets such that optimal estimates of parameters in one set are easy to solve for given fixed values of the parameters in the other set are common in practice. Particularly ubiquitous are data fitting problems in which the model function is a linear combination of nonlinear functions, which may be addressed with the variable projection algorithm due to Golub and Pereyra. In this paper we review variable projection, with special emphasis on its application to matrix data. The generalization of the algorithm to separable problems in which the linear coefficients of the nonlinear functions are subject to constraints is also discussed. Variable projection has been instrumental for model-based data analysis in multi-way spectroscopy, time-resolved microscopy and gas or liquid chromatography mass spectrometry, and we give an overview of applications in these domains, illustrated by brief case studies.     

Remark on the Successive Projection Algorithm for the Multiple-Sets Split Feasibility Problem

Recently, assuming that the metric projection onto a closed convex set is easily calculated, Liu et al. (Numer. Func. Anal. Opt. 35:1459–1466, 2014) presented a successive projection algorithm for solving the multiple-sets split feasibility problem (MSFP). However, in some cases it is impossible or needs too much work to exactly compute the metric projection. The aim of this remark is to give a modification to the successive projection algorithm. That is, we propose a relaxed successive projection algorithm, in which the metric projections onto closed convex sets are replaced by the metric projections onto halfspaces. Clearly, the metric projection onto a halfspace may be directly calculated. So, the relaxed successive projection algorithm is easy to implement. Its theoretical convergence results are also given.     

Adaptive Projected Subgradient Method for Asymptotic Minimization of Sequence of Nonnegative Convex Functions

Abstract

This paper presents an algorithm, named adaptive projected subgradient method that can minimize asymptotically a certain sequence of nonnegative convex functions over a closed convex set in a real Hilbert space. The proposed algorithm is a natural extension of the Polyak's subgradient algorithm, for nonsmooth convex optimization problem with a fixed target value, to the case where the convex objective itself keeps changing in the whole process. The main theorem, showing the strong convergence of the algorithm as well as the asymptotic optimality of the sequence generated by the algorithm, can serve as a unified guiding principle of a wide range of set theoretic adaptive filtering schemes for nonstationary random processes. These include not only the existing adaptive filtering techniques; e.g., NLMS, Projected NLMS, Constrained NLMS, APA, and Adaptive parallel outer projection algorithm etc., but also new techniques; e.g., Adaptive parallel min-max projection algorithm, and their embedded constraint versions. Numerical examples show that the proposed techniques are well-suited for robust adaptive signal processing problems.     

The relaxed inexact projection methods for the split feasibility problem

In this paper we present several relaxed inexact projection methods for the split feasibility problem (SFP). Each iteration of the first proposed algorithm consists of a projection onto a halfspace containing the given closed convex set. The algorithm can be implemented easily and its global convergence to the solution can be established under suitable conditions. Moreover,we present some modifications of the relaxed inexact projection method with constant stepsize by adopting Armijo-like search. We furthermore present a variable-step relaxed inexact projection method which does not require the computation of the matrix inverses and the largest eigenvalue of the matrix A^TA, and the objective function can decrease sufficiently at each iteration. We show convergence of these modified algorithms under mild conditions. Finally, we perform some numerical experiments, which show the behavior of the algorithms proposed.     

Inertial projection and contraction algorithms for variational inequalities

In this article, we introduce an inertial projection and contraction algorithm by combining inertial type algorithms with the projection and contraction algorithm for solving a variational inequality in a Hilbert space H. In addition, we propose a modified version of our algorithm to find a common element of the set of solutions of a variational inequality and the set of fixed points of a nonexpansive mapping in H. We establish weak convergence theorems for both proposed algorithms. Finally, we give the numerical experiments to show the efficiency and advantage of the inertial projection and contraction algorithm.     

Algorithms for bound constrained quadratic programming problems

Summary We present an algorithm which combines standard active set strategies with the gradient projection method for the solution of quadratic programming problems subject to bounds. We show, in particular, that if the quadratic is bounded below on the feasible set then termination occurs at a stationary point in a finite number of iterations. Moreover, if all stationary points are nondegenerate, termination occurs at a local minimizer. A numerical comparison of the algorithm based on the gradient projection algorithm with a standard active set strategy shows that on mildly degenerate problems the gradient projection algorithm requires considerable less iterations and time than the active set strategy. On nondegenerate problems the number of iterations typically decreases by at least a factor of 10. For strongly degenerate problems, the performance of the gradient projection algorithm deteriorates, but it still performs better than the active set method.Work supported in part by the Applied Mathematical Sciences subprogram of the Office of Energy Research of the U.S. Department of Energy under Contract W-31-109-Eng-38     

Globally optimal solutions of max–min systems

A variety of problems in computer science, operations research, control theory, etc., can be modeled as non-linear and non-differentiable max–min systems. This paper introduces the global optimization into such systems. The criteria for the existence and uniqueness of the globally optimal solutions are established using the high matrix, optimal max-only projection set and k ^s -control vector of max–min functions. It is also shown that the global optimization can be accomplished through the partial max-only projection representation with algebraic and combinatorial features. The methods are constructive and lead to an algorithm of finding all globally optimal solutions.     

A superlinearly convergent projection method for constrained systems of nonlinear equations

In this paper, a new projection method for solving a system of nonlinear equations with convex constraints is presented. Compared with the existing projection method for solving the problem, the projection region in this new algorithm is modified which makes an optimal stepsize available at each iteration and hence guarantees that the next iterate is more closer to the solution set. Under mild conditions, we show that the method is globally convergent, and if an error bound assumption holds in addition, it is shown to be superlinearly convergent. Preliminary numerical experiments also show that this method is more efficient and promising than the existing projection method. This work was done when Yiju Wang visited Chongqing Normal University.     

About the Lipschitz property of the metric projection in the Hilbert space

We consider the metric projection operator from the real Hilbert space onto a strongly convex set. We prove that the restriction of this operator on the complement of some neighborhood of the strongly convex set is Lipschitz continuous with the Lipschitz constant strictly less than 1. This property characterizes the class of strongly convex sets and (to a certain degree) the Hilbert space. We apply the results obtained to the question concerning the rate of convergence for the gradient projection algorithm with differentiable convex function and strongly convex set.     

On fuzzy linear projection equation and applications

This paper introduces the concept of fuzzy projection of a fuzzy number on a set of fuzzy numbers based on r-cut approach. It is proved that the projection of a fuzzy number on the set of all fuzzy numbers is itself and under a special metric, the proposed fuzzy projection is a non-expansive mapping. By using this definition, the concept of fuzzy linear projection equation is defined and to solve it, a numerical method is applied. Based on the proposed algorithm and as an important application, two different types of system of fuzzy linear equations with fuzzy variables are solved. Numerical results illustrate the applicabilities of proposed approach.     

Arbitrary-norm separating plane

A plane separating two point sets in n-dimensional real space is constructed such that it minimizes the sum of arbitrary-norm distances of misclassified points to the plane. In contrast to previous approaches that used surrogates for distance-minimization, the present work is based on a precise norm-dependent explicit closed form for the projection of a point on a plane. This projection is used to formulate the separating-plane problem as a minimization of a convex function on a unit sphere in a norm dual to that of the arbitrary norm used. For the 1-norm, the problem can be solved in polynomial time by solving 2n linear programs or by solving a bilinear program. For a general p-norm, the minimization problem can be transformed via an exact penalty formulation to minimizing the sum ofa convex function and a bilinear function on a convex set. For the one and infinity norms, a finite successive linearization algorithm can be used for solving the exact penalty formulation.     

Implicitly and densely discrete black-box optimization problems

This paper addresses derivative-free optimization problems where the variables lie implicitly in an unknown discrete closed set. The evaluation of the objective function follows a projection onto the discrete set, which is assumed dense (and not sparse as in integer programming). Such a mathematical setting is a rough representation of what is common in many real-life applications where, despite the continuous nature of the underlying models, a number of practical issues dictate rounding of values or projection to nearby feasible figures. We discuss a definition of minimization for these implicitly discrete problems and outline a direct-search algorithm framework for its solution. The main asymptotic properties of the algorithm are analyzed and numerically illustrated.     

On variable-step relaxed projection algorithm for variational inequalities

Projection algorithms are practically useful for solving variational inequalities (VI). However some among them require the knowledge related to VI in advance, such as Lipschitz constant. Usually it is impossible in practice. This paper studies the variable-step basic projection algorithm and its relaxed version under weakly co-coercive condition. The algorithms discussed need not know constant/function associated with the co-coercivity or weak co-coercivity and the step-size is varied from one iteration to the next. Under certain conditions the convergence of the variable-step basic projection algorithm is established. For the practical consideration, we also give the relaxed version of this algorithm, in which the projection onto a closed convex set is replaced by another projection at each iteration and latter is easy to calculate. The convergence of relaxed scheme is also obtained under certain assumptions. Finally we apply these two algorithms to the Split Feasibility Problem (SFP).     

基于迭代投影的梯度硬阈值追踪算法

梯度硬阈值追踪算法是求解稀疏优化问题的有效算法之一.考虑到算法中投影对最优解的影响,提出一种比贪婪策略更好的投影算法是很有必要的.针对一般的稀疏约束优化问题,利用整数规划提出一种迭代投影策略,将梯度投影算法中的投影作为一个子问题求解.通过迭代求解该子问题得到投影的指标集,并以此继续求解原问题,以提高梯度硬阈值追踪算法的计算效果.证明了算法的收敛性,并通过数值实例验证了算法的有效性.     

A superlinearly convergent strongly sub-feasible SSLE-type algorithm with working set for nonlinearly constrained optimization

In this paper, by means of a new efficient identification technique of active constraints and the method of strongly sub-feasible direction, we propose a new sequential system of linear equations (SSLE) algorithm for solving inequality constrained optimization problems, in which the initial point is arbitrary. At each iteration, we first yield the working set by a pivoting operation and a generalized projection; then, three or four reduced linear equations with a same coefficient are solved to obtain the search direction. After a finite number of iterations, the algorithm can produced a feasible iteration point, and it becomes the method of feasible directions. Moreover, after finitely many iterations, the working set becomes independent of the iterates and is essentially the same as the active set of the KKT point. Under some mild conditions, the proposed algorithm is proved to be globally, strongly and superlinearly convergent. Finally, some preliminary numerical experiments are reported to show that the algorithm is practicable and effective.     

A derivative-free three-term projection algorithm involving spectral quotient for solving nonlinear monotone equations

ABSTRACT

In this paper, we develop a three-term conjugate gradient method involving spectral quotient, which always satisfies the famous Dai-Liao conjugacy condition and quasi-Newton secant equation, independently of any line search. This new three-term conjugate gradient method can be regarded as a variant of the memoryless Broyden-Fletcher-Goldfarb-Shanno quasi-Newton method with regard to spectral quotient. By combining this method with the projection technique proposed by Solodov and Svaiter in 1998, we establish a derivative-free three-term projection algorithm for dealing with large-scale nonlinear monotone system of equations. We prove the global convergence of the algorithm and obtain the R-linear convergence rate under some mild conditions. Numerical results show that our projection algorithm is effective and robust, and is more competitive with the TTDFP algorithm proposed Liu and Li [A three-term derivative-free projection method for nonlinear monotone system of equations. Calcolo. 2016;53:427–450].     

一般线性或非线性约束下的共轭投影变尺度方法

梯度投影法是一类有效的约束最优化算法，在最优化领域中占有重要的地位．但是，梯度投影法所采用的投影是正交投影，不包含目标函数和约束函数的二阶导数信息·因而；收敛速度不太令人满意．本文介绍一种共轭投影概念，利用共轭投影构造了一般线性或非线性约束下的共轭投影变尺度算法，并证明了算法在一定条件下具有全局收敛性．由于算法中的共轭投影恰当地包含了目标函数和约束函数的二阶导数信息，因而收敛速度有希望加快．数值试验的结果表明算法是有效的．     

Complexity Estimation for an Algorithm of Searching for Zero of a Piecewise Linear Convex Function

It is known that the problem of the orthogonal projection of a point to the standard simplex can be reduced to solution of a scalar equation. In this article, the complexity is analyzed of an algorithm of searching for zero of a piecewise linear convex function which is proposed in [30]. The analysis is carried out of the best and worst cases of the input data for the algorithm. To this end, the largest and smallest numbers of iterations of the algorithm are studied as functions of the size of the input data. It is shown that, in the case of equality of elements of the input set, the algorithm performs the smallest number of iterations. In the case of different elements of the input set, the number of iterations is maximal and depends rather weakly on the particular values of the elements of the set. The results of numerical experiments with random input data of large dimension are presented.