圆锥规划问题的光滑牛顿方法

给出求解圆锥规划问题的一种新光滑牛顿方法.基于圆锥互补函数的一个新光滑函数,将圆锥规划问题转化成一个非线性方程组,然后用光滑牛顿方法求解该方程组.该算法可从任意初始点开始,且不要求中间迭代点是内点.运用欧几里得代数理论,证明算法具有全局收敛性和局部超线性收敛速度.数值算例表明算法的有效性.     

一个求解广义圆锥互补问题的无导数光滑算法

本文研究了一个求解广义圆锥互补问题的无导数光滑算法.利用光滑函数将广义圆锥互补问题等价转化成一个光滑方程组,然后再利用牛顿法求解此方程组.该算法采用了一种新的非单调无导数线搜索技术,并且在适当条件下具有全局和局部二次收敛性质.数值实验结果表明算法是非常有效的.     

一个求解二阶锥规划的光滑牛顿算法

本文研究了二阶锥规划问题.利用新的最小值函数的光滑函数,给出一个求解二阶锥规划的光滑牛顿算法.算法可以从任意点出发,在每一步迭代只需求解一个线性方程组并进行一次线性搜索.在不需要满足严格互补假设条件下,证明了算法是全局收敛和局部二阶收敛的.数值试验表明算法是有效的.     

二次锥规划的预估-校正光滑方法

基于光滑Fischer-Burmeister函数,给出一个求解二次锥规划的预估-校正光滑牛顿法.该算法构造一个等价于最优性条件的非线性方程组,再用牛顿法求解此方程组的扰动.在适当的假设下,证明算法是全局收敛且是局部二阶收敛的.数值试验表明算法的有效性.     

基于新光滑函数的P0映射非线性互补问题的光滑牛顿法(英文)

非线性互补问题(NCP)可以重新表述为一个非光滑方程组的解.通过引入一个新的光滑函数,将问题近似为参数化光滑方程组.基于这个光滑函数,我们提出了一个求解P₀映射和R₀映射非线性互补问题的光滑牛顿法.该算法每次迭代只求解一个线性方程和一次线搜索.在适当的条件下,证明了该方法是全局和局部二次收敛的.数值结果表明,该算法是有效的.     

一种双层规划的光滑化目标罚函数算法

论文研究了一种双层规划的光滑化目标罚函数算法,在一些条件下,证明了光滑化罚优化问题等价于原双层规划问题,而且,当下层规划问题是凸规划问题时, 给出了一个求解算法和收敛性证明.     

线性二阶锥规划的一个光滑化方法及其收敛性

首先讨论了用Chen-Harker-Kanzow-Smale光滑函数刻画线性二阶锥规划的中心路径条件;基于此,提出了求解线性二阶锥规划的一个光滑化算法,然后分析了该算法的全局及其局部二次收敛性质.     

非光滑凸优化问题的一个非精确梯度镜面下降算法

本文提出一个求解非光滑凸优化问题非精确梯度镜面下降算法.该算法是Allen-Zhu2016年提出求解光滑凸优化问题梯度镜面下降算法的推广,而且该算法允许目标函数中光滑部分梯度计算和非光滑部分邻近算子计算都存在误差,并且在适当条件下分析了该算法函数值序列的O(1/(k²))收敛速度,这里k表示迭代数.最后关于Lasso问题和Logistic问题的数值结果表明该算法是有效的.     

非线性均衡问题一个超线性收敛的光滑逼近SQP算法

研究非线性均衡问题,引入一个磨光算子将原问题转化为光滑问题,并用此光滑问题来逼近原来的问题而求解.在每步迭代中,通过转轴运算,求解一个线性约束二次规划问题和显式修正方向来得到主方向,并通过一个显式公式来得到高阶修正方向使得算法避免Maratos效应.在不需要上层互补条件下证明了算法具有全局收敛性和强收敛性且具有超线性收敛速度.     

基于凝聚函数的互补问题的光滑化算法

对于不可微的"极大值"形式的函数,可以利用凝聚函数对其进行光滑逼近.借助这个技术,给出了求解线性互补问题的光滑方程组算法.首先是将互补问题转化为等价的非光滑方程组,再利用凝聚函数进行光滑逼近,从而转化为光滑方程组的求解问题.通过一些考题对这个算法进行了数值试验,结果显示了该算法的有效性和稳定性.     

A nonmonotone smoothing Newton method for circular cone programming

The circular cone programming (CCP) problem is to minimize or maximize a linear function over the intersection of an affine space with the Cartesian product of circular cones. In this paper, we study nondegeneracy and strict complementarity for the CCP, and present a nonmonotone smoothing Newton method for solving the CCP. We reformulate the CCP as a second-order cone programming (SOCP) problem using the algebraic relation between the circular cone and the second-order cone. Then based on a one parametric class of smoothing functions for the SOCP, a smoothing Newton method is developed for the CCP by adopting a new nonmonotone line search scheme. Without restrictions regarding its starting point, our algorithm solves one linear system of equations approximately and performs one line search at each iteration. Under mild assumptions, our algorithm is shown to possess global and local quadratic convergence properties. Some preliminary numerical results illustrate that our nonmonotone smoothing Newton method is promising for solving the CCP.     

A new one‐step smoothing newton method for the second‐order cone complementarity problem

In this paper, we present a new one‐step smoothing Newton method for solving the second‐order cone complementarity problem (SOCCP). Based on a new smoothing function, the SOCCP is approximated by a family of parameterized smooth equations. At each iteration, the proposed algorithm only need to solve one system of linear equations and perform only one Armijo‐type line search. The algorithm is proved to be convergent globally and superlinearly without requiring strict complementarity at the SOCCP solution. Moreover, the algorithm has locally quadratic convergence under mild conditions. Numerical experiments demonstrate the feasibility and efficiency of the new algorithm. Copyright © 2010 John Wiley & Sons, Ltd.     

A one-step smoothing Newton method for second-order cone programming

A new smoothing function for the second-order cone programming is given by smoothing the symmetric perturbed Fischer–Burmeister function. Based on this new function, a one-step smoothing Newton method is presented for solving the second-order cone programming. The proposed algorithm solves only one linear system of equations and performs only one line search at each iteration. This algorithm does not have restrictions regarding its starting point and is Q-quadratically convergent. Numerical results suggest the effectiveness of our algorithm.     

SMOOTHING NEWTON ALGORITHM FOR THE CIRCULAR CONE PROGRAMMING WITH A NONMONOTONE LINE SEARCH

In this paper, we present a nonmonotone smoothing Newton algorithm for solving the circular cone programming(CCP) problem in which a linear function is minimized or maximized over the intersection of an affine space with the circular cone. Based on the relationship between the circular cone and the second-order cone(SOC), we reformulate the CCP problem as the second-order cone problem(SOCP). By extending the nonmonotone line search for unconstrained optimization to the CCP, a nonmonotone smoothing Newton method is proposed for solving the CCP. Under suitable assumptions, the proposed algorithm is shown to be globally and locally quadratically convergent. Some preliminary numerical results indicate the effectiveness of the proposed algorithm for solving the CCP.     

Nonsingularity in second-order cone programming via the smoothing metric projector

Based on the differential properties of the smoothing metric projector onto the second-order cone,we prove that,for a locally optimal solution to a nonlinear second-order cone programming problem,the nonsingularity of the Clarke's generalized Jacobian of the smoothing Karush-Kuhn-Tucker system,constructed by the smoothing metric projector,is equivalent to the strong second-order sufficient condition and constraint nondegeneracy,which is in turn equivalent to the strong regularity of the Karush-Kuhn-Tucker p...     

Superlinear/quadratic smoothing Broyden-like method for the generalized nonlinear complementarity problem

In this article, we first reformulate the generalized nonlinear complementarity problem (GNCP) over a polyhedral cone as a smoothing system of equations and then suggest a smoothing Broyden-like method for solving it. The proposed algorithm has to solve only one system of nonhomogeneous linear equations, perform only one line search and update only one matrix per iteration. We show that the iteration sequence generated by the proposed algorithm converges globally and superlinearly under suitable conditions. Furthermore, the algorithm has local quadratic convergence under mild assumptions. Some numerical examples are given to illustrate the performance and efficiency of the presented algorithm.     

Analysis of a smoothing Newton method for second-order cone complementarity problem

In this paper, we consider the second-order cone complementarity problem with P ₀-property. By introducing a smoothing parameter into the Fischer-Burmeister function, we present a smoothing Newton method for the second-order cone complementarity problem. The proposed algorithm solves only a linear system of equations and performs only one line search at each iteration. At the same time, the algorithm does not have restrictions on its starting point and has global convergence. Under the assumption of nonsingularity, we establish the locally quadratic convergence of the algorithm without strict complementarity condition. Preliminary numerical results show that the algorithm is promising.     

A new smoothing Newton-type method for second-order cone programming problems

A new smoothing function of the well-known Fischer–Burmeister function is given. Based on this new function, a smoothing Newton-type method is proposed for solving second-order cone programming. At each iteration, the proposed algorithm solves only one system of linear equations and performs only one line search. This algorithm can start from an arbitrary point and it is Q-quadratically convergent under a mild assumption. Numerical results demonstrate the effectiveness of the algorithm.     

Sub-quadratic convergence of a smoothing Newton method for second-order cone programming

In this paper, we present a smoothing Newton method for solving the second-order cone programming (SOCP) based on the Chen–Harker–Kanzow–Smale (CHKS) smoothing function. Our smoothing method reformulates SOCP as a nonlinear system of equations and then applies Newton’s method to the system. The proposed method solves only one linear system of equations and performs only one line search at each iteration. It is shown that the method is globally and locally sub-quadratically convergent under a nonsingularity assumption. Numerical results suggest that the method is promising.