一类大规模二次规划问题的可行下降分解算法

本文对一类大规模二次规划问题，提出了矩阵剖分的概念和方法，并将问题转化为求解一系列容易求解的小规模二次规划子问题．另外，通过施加某些约束机制，使子问题所产生的迭代点均为可行下降点．在通常的假定下，证明算法具有全局收敛性，大量数值实验表明，本文所提出的新算法是有效的。     

两分块非凸优化Peaceman-Rachford分裂序列二次规划双步长算法

本文研究大规模两分块非凸约束优化的分解降维算法,提出Peaceman-Rachford (PR)分裂序列二次规划双步长求解方法.本文主要工作和贡献如下:(1)借助PR分裂算法思想将传统二次规划(quadratic programming, QP)子问题的增广Lagrange问题分解为两个小规模QP子问题;(2)通过求解小规模QP产生搜索方向;(3)以增广Lagrange函数为效益函数,沿搜索方向先后进行Armijo线搜索产生双迭代步长,在较弱的条件下保证了算法的全局收敛性、强收敛性和合理的迭代复杂性,克服了Maratos效应;(4)提出乘子新的对称型修正技术;(5)基于一类数学模型和电力系统经济调度模型以及?₂正则二分类问题,对算法进行大量中等规模的比较数值实验,验证了算法的有效性.     

极大熵方法与二次规划子问题的显式解

本文对混合约束极大极小问题的目标函数与约束分别用熵函数来逼近,讨论了逼近问题的二次规划子问题的搜索方向的显式形式,并给出了极大极小问题和多目标规划的二次规划予问题的显式解。将所得结果用于相应的算法中,可提高算法的有效性。     

基于松弛策略解半无限规划模型的显式修正算法

讨论了一类线性半无限最优规划模型的求解算法.采用松弛方法解其系列子问题LP(T_k)及DLP(T_k),基于松弛策略和在适当的假设条件下,提出了一个我们称之为显式算法的新型算法.新算法的主要改进之处是算法在每一步迭代计算时,允许丢弃一些不必要的约束.在这种方式下,算法避免了求解系列太大规模的子问题.最后,基于提出的显式修正算法,并与传统割平面方法和已有文献中的松弛修正算法、对同一问题作了初步的数值比较实验.     

一种改进的求解支持向量机模型的坐标梯度下降算法

针对支持向量机模型问题,给出了一种新的坐标梯度下降算法.算法首先求解一个特殊的二次规划问题,将所得的结果进行分解后,得到每次迭代所需要的工作集,然后,求解一个降维的二次规划子问题得到下降方向.新算法无需进行线搜索,避免了线搜索带来的时间和空间上的开销,使得计算量大大减少.最后,在较弱的条件下证明了算法的全局收敛性,并利用数值实验证明了算法的可行性和有效性.     

非线性无约束优化问题的滤子填充函数算法

提出了一个求解无约束非线性规划问题的无参数填充函数,并分析了其性质.同时引进了滤子技术,在此基础上设计了无参数滤子填充函数算法,数值实验证明该算法是有效的.     

一类带线搜索的自适应信赖域算法

本文对无约束优化问题提出了一类带线搜索的自适应信赖域算法,新算法在试验步失败时不重解子问题,而是采用线搜索,从而减少了计算量,不同于一般的带线搜索的信赖域算法,新算法根据实际下降量与预估下降量的比值按照变化的速率对信赖域半径进行调整.文中在一定的条件下证明了算法的收敛性,并且给出了相应的数值实验结果.     

基于分支定价算法的电动汽车车辆路径问题

目前,随着电动汽车的普及,物流企业逐渐重视电动汽车的应用。本文考虑到电动汽车在实际应用中的行驶里程、充电耗时以及配送时间等因素,研究含时间窗的电动汽车车辆路径问题,建立了相应的混合整数规划模型,然后改进分支定价算法以求得其最优解。改进的分支定价算法首先根据Dantzig-Wolfe分解原理将原问题分解为基于路径的主问题(MP)和求最短路径的子问题,然后用列生成和动态规划算法在主问题和子问题之间进行迭代以求得主问题线性松弛后的最优解,最后采用基于弧的分支策略求得其整数解。通过用改进的Solomon算例的实验数据,与CPLEX比较验证了模型和算法结果的准确性,并对该问题进行了灵敏度分析,证明了本文提出的算法具有一定的应用价值。     

解新锥模型信赖域子问题的折线法

本文以新锥模型信赖域子问题的最优性条件为理论基础,认真讨论了新子问题的锥函数性质,分析了此函数在梯度方向及与牛顿方向连线上的单调性.在此基础上本文提出了一个求解新锥模型信赖域子问题折线法,并证明了这一子算法保证解无约束优化问题信赖域法全局收敛性要满足的下降条件.本文获得的数值实验表明该算法是有效的.     

具有相容子问题的序列二次规划新算法

借助ε-约束集与一种特殊的罚函数,给出了一个具有相容子问题的序列二次规划新算法,较圆满地解决了SQP算法中的相容性问题,并证明了该算法仍保持全局收敛和超线性收敛的性质.     

A new long-step interior point algorithm for linear programming based on the algebraic equivalent transformation

In this paper, we investigate a new primal-dual long-step interior point algorithm for linear optimization. Based on the step size, interior point algorithms can be divided into two main groups, short-step, and long-step methods. In practice, long-step variants perform better, but usually, a better theoretical complexity can be achieved for the short-step methods. One of the exceptions is the large-update algorithm of Ai and Zhang. The new wide neighborhood and the main characteristics of the presented algorithm are based on their approach. In addition, we use the algebraic equivalent transformation technique of Darvay to determine new modified search directions for our method. We show that the new long-step algorithm is convergent and has the best known iteration complexity of short-step variants. We present our numerical results and compare the performance of our algorithm with two previously introduced Ai-Zhang type interior point algorithms on a set of linear programming test problems from the Netlib library.

     

一类非单调线性互补问题的高阶仿射尺度算法

In this paper, a new interior point algorithm-high-order atone scaling for a class of nonmonotonic linear complementary problems is developed. On the basis of idea of primal-dual affine scaling method for linear programming , the search direction of our algorithm is obtained by a linear system of equation at each step . We show that, by appropriately choosing the step size, the algorithm has polynomial time complexity. We also give the numberical results of the algorithm for two test problems.     

一种具有非线性约束线性规划全局优化算法

本文提出了一种新的适用于处理非线性约束下线性规划问题的全局优化算法。该算法通过构造子问题来寻找优于当前局部最优解的可行解。该子问题可通过模拟退火算法来解决。通过求解一系列的子问题,当前最优解被不断地更新,最终求得全局最优解。最后,本算法应用于几个典型例题,并与罚函数法相比较,数值结果表明该算法是可行的,有效的。     

A linear programming-based optimization algorithm for solving nonlinear programming problems

In this paper a linear programming-based optimization algorithm called the Sequential Cutting Plane algorithm is presented. The main features of the algorithm are described, convergence to a Karush–Kuhn–Tucker stationary point is proved and numerical experience on some well-known test sets is showed. The algorithm is based on an earlier version for convex inequality constrained problems, but here the algorithm is extended to general continuously differentiable nonlinear programming problems containing both nonlinear inequality and equality constraints. A comparison with some existing solvers shows that the algorithm is competitive with these solvers. Thus, this new method based on solving linear programming subproblems is a good alternative method for solving nonlinear programming problems efficiently. The algorithm has been used as a subsolver in a mixed integer nonlinear programming algorithm where the linear problems provide lower bounds on the optimal solutions of the nonlinear programming subproblems in the branch and bound tree for convex, inequality constrained problems.     

Computational acceleration of projection algorithms for the linear best approximation problem

This is an experimental computational account of projection algorithms for the linear best approximation problem. We focus on the sequential and simultaneous versions of Dykstra’s algorithm and the Halpern-Lions-Wittmann-Bauschke algorithm for the best approximation problem from a point to the intersection of closed convex sets in the Euclidean space. These algorithms employ different iterative approaches to reach the same goal but no mathematical connection has yet been found between their algorithmic schemes. We compare these algorithms on linear best approximation test problems that we generate so that the solution will be known a priori and enable us to assess the relative computational merits of these algorithms. For the simultaneous versions we present a new component-averaging variant that substantially accelerates their initial behavior for sparse systems.     

An implicit least squares algorithm for nonlinear rational model parameter estimation

In this study a new insight into least squares regression is identified and immediately applied to estimating the parameters of nonlinear rational models. From the beginning the ordinary explicit expression for linear in the parameters model is expanded into an implicit expression. Then a generic algorithm in terms of least squares error is developed for the model parameter estimation. It has been proved that a nonlinear rational model can be expressed as an implicit linear in the parameters model, therefore, the developed algorithm can be comfortably revised for estimating the parameters of the rational models. The major advancement of the generic algorithm is its conciseness and efficiency in dealing with the parameter estimation problems associated with nonlinear in the parameters models. Further, the algorithm can be used to deal with those regression terms which are subject to noise. The algorithm is reduced to an ordinary least square algorithm in the case of linear or linear in the parameters models. Three simulated examples plus a realistic case study are used to test and illustrate the performance of the algorithm.     

A novel piecewise linear classifier based on polyhedral conic and max–min separabilities

In this paper, an algorithm for finding piecewise linear boundaries between pattern classes is developed. This algorithm consists of two main stages. In the first stage, a polyhedral conic set is used to identify data points which lie inside their classes, and in the second stage we exclude those points to compute a piecewise linear boundary using the remaining data points. Piecewise linear boundaries are computed incrementally starting with one hyperplane. Such an approach allows one to significantly reduce the computational effort in many large data sets. Results of numerical experiments are reported. These results demonstrate that the new algorithm consistently produces a good test set accuracy on most data sets comparing with a number of other mainstream classifiers.     

Globally solving nonconvex quadratic programming problems via completely positive programming

Nonconvex quadratic programming (QP) is an NP-hard problem that optimizes a general quadratic function over linear constraints. This paper introduces a new global optimization algorithm for this problem, which combines two ideas from the literature—finite branching based on the first-order KKT conditions and polyhedral-semidefinite relaxations of completely positive (or copositive) programs. Through a series of computational experiments comparing the new algorithm with existing codes on a diverse set of test instances, we demonstrate that the new algorithm is an attractive method for globally solving nonconvex QP.