核范数和谱范数下广义Sylvester方程最小二乘问题的有效算法

本文从数值角度讨论Schatten q-范数下的广义Sylvester方程约束最小二乘问题min x∈s‖N∑i=1A_iXB_i—C‖_q,其中S为闭凸约束集合,Schatten q-范数定义为‖M‖_q~q=∑_(i=1)~nσ_i~q(M),其中σ_i(M)为M∈R~(n×n)的奇异值.该问题的几类特殊情形在图像处理、控制论等领域有广泛的应用.q=2即Frobenius范数下该问题已被充分研究,故本文着重讨论q=1,+∞,即核范数和谱范数下该问题的数值求解.采用的数值方法是非精确标准容易执行的部分非精确交替方向法,并结合奇异值阈值算法,Moreau-Yosida正则化算法,谱投影算法和LSQR算法等求解相应子问题.给出算法的收敛性证明,并用数值算例验证其高效可行性.     

核范数和谱范数下广义Sylvester方程最小二乘问题的一类改进算法

在文献[10]中,作者从数值角度讨论核范数和谱范数下的广义Sylvester方程约束最小二乘问题min X∈ S|NΣI=1A_iXB_i-C|的算法,其中s为闭凸集合.采用的数值算法是非精确交替方向法,并结合阈值算法、 MoreauYosida正则化算法、谱投影算法、LSQR, SPG等算法求解相应子问题.本文在文献[10]的基础上,通过引入新变量,应用交替方向法简化子问题的求解,其中每个子问题都可以精确求解,更重要的是每个变量都具有显式的表达式.在理论方面我们证明了算法的收敛性,数值试验表明改进后的算法不管是在时间上还是在迭代步上,运行的结果得到很大的改善.     

用最小范数法解三峡梯级水电站短期经济调度问题

本文主要研究三峡梯级水电站与华中、华东和川东电网联网的短期经济调度问题,利用泛函分析和运筹学相结合的方法建立了三峡梯级水电站日负荷最优分配的数学模型。本文扩充和推广了Hawary和Christensen的最小范数法用来求解这个具有等式和不等式约束的高维非线性含时滞的动态最优化问题,最优策略由一组动态的非线性代数、微分方程确定。引入适当的变量并进行适当化简,最终可将三峡梯级水电系统的经济调度问题转化为一个最小范数问题,并给出了最优解的具体表达式.用Lagrange乘子和Kuhn-Tucker乘子将约束条件并入目标函数中形成一个增广价格函数。通过变换可将该无约束优化问题转化为求解非线性代数方程组的问题。本文选用Fletcher-Reeves共轭梯度法求解无约束极值问题.在IBM-PC型微机上进行了试算。试算结果表明用最小范数法求解三峡梯级水电站日负荷最优分配问题是完全可行的,梯级水耗率有明显下降,能获得一定的经济效益。     

解非线性规划问题的不精确线性搜索SQP滤子方法

本文用序列二次规划方法(SQP)结合Wolfe-Powell不精确线性搜索准则求解非线性规划问题.Wolfe-Powell准则是一种能够使目标函数获得充分下降而运行时间较省的确定步长方法.不精确线性搜索滤子方法比较其它结合精确线性搜索和信赖域方法求解问题的滤子方法更灵活更易实现.如果目标函数的预测下降量为负,我们的工作将主要利用可行恢复项改善可行性.一般条件下,本文提出的算法较易实现,且具有全局收敛性.数值试验显示了算法的有效性.     

基于增广Lagrange函数的约束优化问题的一个信赖域方法

本文提出一个求解不等式约束优化问题的基于指数型增广Lagrange函数的信赖域方法.基于指数型增广Lagrange函数,将传统的增广Lagrange方法的精确求解子问题转化为一个信赖域子问题,从而减少了计算量,并建立相应的信赖域算法.在一定的假设条件下,证明了算法的全局收敛性,并给出相应经典算例的数值实验结果.     

求解带箱式约束全局优化问题的滤子填充函数方法

提出一个基于滤子技术的填充函数算法, 用于求解带箱式约束的非凸全局优化问题. 填充函数算法是求解全局优化问题的有效方法之一, 而滤子技术以其良好的数值效果广泛应用于局部优化算法中. 为优化填充函数方法, 应用滤子来监控迭代过程. 首先给出一个新的填充函数并讨论了其特性, 在此基础上提出了理论算法及算法性质. 最后列出数值实验结果以说明算法的有效性.     

鲁棒稀疏重构问题的凝聚同伦算法

鲁棒稀疏重构问题是信号处理领域的重要问题,该问题的数学本质是一个NP难的数学优化问题.同伦算法是一类典型的路径跟踪算法,该算法是解非线性问题的一类成熟算法,具有全局收敛性,且易于并行实现.本文考虑同伦算法在鲁棒稀疏重构问题中的数值求解.基于l_∞范数及罚函数策略,我们首先将原始的基于l_0范数的最优化模型,转化为含参数的无约束极大极小值问题,进而构造凝聚函数光滑化模型中的极大值函数,并构造凝聚同伦算法数值求解.数值仿真实验验证了新方法的有效性,为大规模鲁棒重构问题的并行化数值求解奠定基础.     

高阶优化算法分析简介

高阶优化算法是利用目标函数的高阶导数信息进行优化的算法,是最优化领域中的一个新兴的研究方向.高阶算法具有更低的迭代复杂度,但是需要求解一个更难的子问题.主要介绍三种高阶算法,分别为求解凸问题的高阶加速张量算法和A-HPE框架下的最优张量算法,以及求解非凸问题的ARp算法.同时也介绍了怎样求解高阶算法的子问题.希望通过对高阶算法的介绍,引起更多学者的关注与重视.     

基于SCN函数共轭梯度方向的稀疏向量特征提取算法

稀疏向量特征提取是指在优化时利用各种范数对解进行约束,从而获得带有稀疏特征的最优解,其广泛应用于复杂系统中的机器学习、深度学习和大数据分析等领域的特征提取问题.大量的研究表明各种范数如L₀范数、L₁范数和L₂范数的方法都存在各自的缺点,主要表现在越容易求解的范数越不精准稀疏,越精准稀疏的范数越难求解.文章提出了一种基于SCN函数共轭梯度方向的稀疏向量特征发现算法(CGDL),稀疏向量特征发现可以用一个稀疏特征提取优化模型建立,其目标函数是一个SCN函数,对其中的L₀范数进行转换,形成一个具有特殊结构优化问题,这个问题等价于双层规划的凸-凹极小极大化问题,这类问题可以解决稀疏回归、图像特征和压缩感知等问题.文章给出了上述模型的稀疏特征提取算法的详细计算步骤和收敛性分析证明,并且对给定的实际数据集和高维模拟数据集对算法的有效性、复杂性和收敛速度进行了数值对比实验,表明了该算法在精准度和稀疏性上显著优于其他对比方法,并且具有较好的收敛速度.     

整数规划问题的滤子填充函数算法

全局优化是最优化的一个分支,非线性整数规划问题的全局优化在各个方面都有广泛的应用.填充函数是解决全局优化问题的方法之一,它可以帮助目标函数跳出当前的局部极小点找到下一个更好的极小点.滤子方法的引入可以使得目标函数和填充函数共同下降,省却了以往算法要设置两个循环的麻烦,提高了算法的效率.本文提出了一个求解无约束非线性整数规划问题的无参数填充函数,并分析了其性质.同时引进了滤子方法,在此基础上设计了整数规划的无参数滤子填充函数算法.数值实验证明该算法是有效的.     

一类自适应广义交替方向乘子法

广义交替方向乘子法是求解凸优化问题的有效算法.当实际问题中子问题难以求解时,可以采用在子问题中添加邻近项的方法处理,邻近矩阵正定时,算法收敛,然而这也会使迭代步长较小.最新研究表明,邻近矩阵可以有一定的不正定性.本文在基于不定邻近项的广义交替方向乘子法框架下,提出一种自适应的广义交替方向乘子法,动态地选择邻近矩阵,增大迭代步长.在一些较弱的假设下,证明了算法的全局收敛性.我们进行一些初等数值实验,验证了算法的有效性.     

交替方向法求解带线性约束的变分不等式

１引言变分不等式是一个有广泛应用的数学问题,它的一般形式是：确定一个向量,使其满足这里ｆ是一个从到自身的一个映射,Ｓ是Ｒ中的一个闭凸集．在许多实际问题中集合Ｓ往往具有如下结构其中ＡｂＫ是中的一个简单闭凸集．例如一个正卦限,一个框形约束结构,或者一个球简言之,Ｓ是Ｒ中的一个超平面与一个简单闭凸集的交．求解问题（１）－（２）,往往是通过对线性约束Ａ引人Ｌａｇｒａｎｇｅ乘子,将原问题化为如下的变分不等式：确定使得我们记问题（３）－（４）为ＶＩ（Ｆ）．熟知［３］,ＶＩ（,Ｆ）等价于投影方程其中凡（·）表…     

Properties and methods for finding the best rank-one approximation to higher-order tensors

The problem of finding the best rank-one approximation to higher-order tensors has extensive engineering and statistical applications. It is well-known that this problem is equivalent to a homogeneous polynomial optimization problem. In this paper, we study theoretical results and numerical methods of this problem, particularly focusing on the 4-th order symmetric tensor case. First, we reformulate the polynomial optimization problem to a matrix programming, and show the equivalence between these two problems. Then, we prove that there is no duality gap between the reformulation and its Lagrangian dual problem. Concerning the approaches to deal with the problem, we propose two relaxed models. The first one is a convex quadratic matrix optimization problem regularized by the nuclear norm, while the second one is a quadratic matrix programming regularized by a truncated nuclear norm, which is a D.C. function and therefore is nonconvex. To overcome the difficulty of solving this nonconvex problem, we approximate the nonconvex penalty by a convex term. We propose to use the proximal augmented Lagrangian method to solve these two relaxed models. In order to obtain a global solution, we propose an alternating least eigenvalue method after solving the relaxed models and prove its convergence. Numerical results presented in the last demonstrate, especially for nonpositive tensors, the effectiveness and efficiency of our proposed methods.     

The augmented Lagrangian method for a type of inverse quadratic programming problems over second-order cones

The focus of this paper is on studying an inverse second-order cone quadratic programming problem, in which the parameters in the objective function need to be adjusted as little as possible so that a known feasible solution becomes the optimal one. We formulate this problem as a minimization problem with cone constraints, and its dual, which has fewer variables than the original one, is a semismoothly differentiable (SC ¹) convex programming problem with both a linear inequality constraint and a linear second-order cone constraint. We demonstrate the global convergence of the augmented Lagrangian method with an exact solution to the subproblem and prove that the convergence rate of primal iterates, generated by the augmented Lagrangian method, is proportional to 1/r, and the rate of multiplier iterates is proportional to $1/\sqrt{r}$ , where r is the penalty parameter in the augmented Lagrangian. Furthermore, a semismooth Newton method with Armijo line search is constructed to solve the subproblems in the augmented Lagrangian approach. Finally, numerical results are reported to show the effectiveness of the augmented Lagrangian method with both an exact solution and an inexact solution to the subproblem for solving the inverse second-order cone quadratic programming problem.     

Inverse optimization for linearly constrained convex separable programming problems

In this paper, we study inverse optimization for linearly constrained convex separable programming problems that have wide applications in industrial and managerial areas. For a given feasible point of a convex separable program, the inverse optimization is to determine whether the feasible point can be made optimal by adjusting the parameter values in the problem, and when the answer is positive, find the parameter values that have the smallest adjustments. A sufficient and necessary condition is given for a feasible point to be able to become optimal by adjusting parameter values. Inverse optimization formulations are presented with ℓ₁ and ℓ₂ norms. These inverse optimization problems are either linear programming when ℓ₁ norm is used in the formulation, or convex quadratic separable programming when ℓ₂ norm is used.     

Minimum norm solution to the positive semidefinite linear complementarity problem

In this article, we present an algorithm to compute the minimum norm solution of the positive semidefinite linear complementarity problem. We show that its solution can be obtained using the alternative theorems and a convenient characterization of the solution set of a convex quadratic programming problem. This problem reduces to an unconstrained minimization problem with once differentiable convex objective function. We propose an extension of Newton's method for solving the unconstrained optimization problem. Computational results show that convergence to high accuracy often occurs in just a few iterations.     

Simple solution methods for separable mixed linear and quadratic knapsack problem

Quadratic knapsack problem has a central role in integer and nonlinear optimization, which has been intensively studied due to its immediate applications in many fields and theoretical reasons. Although quadratic knapsack problem can be solved using traditional nonlinear optimization methods, specialized algorithms are much faster and more reliable than the nonlinear programming solvers. In this paper, we study a mixed linear and quadratic knapsack with a convex separable objective function subject to a single linear constraint and box constraints. We investigate the structural properties of the studied problem, and develop a simple method for solving the continuous version of the problem based on bi-section search, and then we present heuristics for solving the integer version of the problem. Numerical experiments are conducted to show the effectiveness of the proposed solution methods by comparing our methods with some state of the art linear and quadratic convex solvers.     

Iteration-complexity of first-order penalty methods for convex programming

This paper considers a special but broad class of convex programming problems whose feasible region is a simple compact convex set intersected with the inverse image of a closed convex cone under an affine transformation. It studies the computational complexity of quadratic penalty based methods for solving the above class of problems. An iteration of these methods, which is simply an iteration of Nesterov’s optimal method (or one of its variants) for approximately solving a smooth penalization subproblem, consists of one or two projections onto the simple convex set. Iteration-complexity bounds expressed in terms of the latter type of iterations are derived for two quadratic penalty based variants, namely: one which applies the quadratic penalty method directly to the original problem and another one which applies the latter method to a perturbation of the original problem obtained by adding a small quadratic term to its objective function.     

Nonnegative tensor factorizations using an alternating direction method

The nonnegative tensor (matrix) factorization finds more and more applications in various disciplines including machine learning, data mining, and blind source separation, etc. In computation, the optimization problem involved is solved by alternatively minimizing one factor while the others are fixed. To solve the subproblem efficiently, we first exploit a variable regularization term which makes the subproblem far from ill-condition. Second, an augmented Lagrangian alternating direction method is employed to solve this convex and well-conditioned regularized subproblem, and two accelerating skills are also implemented. Some preliminary numerical experiments are performed to show the improvements of the new method.     

On an inverse linear programming problem

A method for solving the following inverse linear programming (LP) problem is proposed. For a given LP problem and one of its feasible vectors, it is required to adjust the objective function vector as little as possible so that the given vector becomes optimal. The closeness of vectors is estimated by means of the Euclidean vector norm. The inverse LP problem is reduced to a problem of unconstrained minimization for a convex piecewise quadratic function. This minimization problem is solved by means of the generalized Newton method.