A NONMOTOTONE ALGORITHM FOR MINIMIZING NONSMOOTH COMPOSITE FUNCTIONS

In this paper, we present a nonmonotone algorithm for solving nonsmooth composite optimization problems. The objective function of these problems is composited by a nonsmooth convex function and a differentiable function. The method generates the search directions by solving quadratic programming successively, and makes use of the nonmonotone line search instead of the usual Armijo-type line search. Global convergence is proved under standard assumptions. Numerical results are given.     

A syncro-parallel nonsmooth PGD algorithm for nonsmooth optimization

A nonsmooth PGD scheme for minimizing a nonsmooth convex function is presented. In the parallelization step of the algorithm, a method due to Pang, Han and Pangaraj (1991), [7], is employed to solve a subproblem for constructing search directions. The convergence analysis is given as well.     

A method for minimizing the sum of a convex function and a continuously differentiable function

This paper presents a method for minimizing the sum of a possibly nonsmooth convex function and a continuously differentiable function. As in the convex case developed by the author, the algorithm is a descent method which generates successive search directions by solving quadratic programming subproblems. An inexact line search ensures global convergence of the method to stationary points.     

A descent method with inexact linear search for nonsmooth equilibrium problems

A descent method with respect to the gap function is formulated and justified for the nonsmooth equilibrium problem. It uses the procedure of inexact linear search of the Armijo type. The proposed method converges under the same assumptions as the methods with exact linear search.     

On the global minimum in a balanced circular packing problem

The paper considers balanced packing problem of a given family of circles into a larger circle of the minimal radius as a multiextremal nonlinear programming problem. We reduce the problem to unconstrained minimization problem of a nonsmooth function by means of nonsmooth penalty functions. We propose an efficient algorithm to search for local extrema and an algorithm for improvement of the lower bound of the global minimum value of the objective function. The algorithms employ nonsmooth optimization methods based on Shor’s r-algorithm. Computational results are given.     

A Constrained Optimization Approach for LCP

In this paper, LCP is converted to an equivalent nonsmooth nonlinear equation system H(x,y) = 0 by using the famous NCP function-Fischer-Burmeister function. Note that some equations in H(x, y) = 0 are nonsmooth and nonlinear hence difficult to solve while the others are linear hence easy to solve. Then we further convert the nonlinear equation system H(x, y) = 0 to an optimization problem with linear equality constraints. After that we study the conditions under which the K-T points of the optimization problem are the solutions of the original LCP and propose a method to solve the optimization problem. In this algorithm, the search direction is obtained by solving a strict convex programming at each iterative point, However, our algorithm is essentially different from traditional SQP method. The global convergence of the method is proved under mild conditions. In addition, we can prove that the algorithm is convergent superlinearly under the conditions: M is P0 matrix and the limit point is a strict complementarity solution of LCP. Preliminary numerical experiments are reported with this method.     

无罚函数和滤子的一个新的QP-free方法

通过构造一个等价于原约束问题一阶KKT条件的非光滑方程组, 提出一类新的QP-free方法. 在迭代中采用了无罚函数和无滤子线搜索方法, 在此基础上, 通过牛顿-拟牛顿迭代得到满足KKT最优条件的解, 并证明该算法是可实现、具有全局收敛性. 另外, 在较弱条件下可以证明该方法具有超线性收敛性.     

An algorithm for composite nonsmooth optimization problems

Nonsmooth optimization problems are divided into two categories. The first is composite nonsmooth problems where the generalized gradient can be approximated by information available at the current point. The second is basic nonsmooth problems where the generalized gradient must be approximated using information calculated at previous iterates.Methods for minimizing composite nonsmooth problems where the nonsmooth function is made up from a finite number of smooth functions, and in particular max functions, are considered. A descent method which uses an active set strategy, a nonsmooth line search, and a quasi-Newton approximation to the reduced Hessian of a Lagrangian function is presented. The Theoretical properties of the method are discussed and favorable numerical experience on a wide range of test problems is reported.This work was carried out at the University of Dundee from 1976–1979 and at the University of Kentucky at Lexington from 1979–1980. The provision of facilities in both universities is gratefully acknowledged, as well as the support of NSF Grant No. ECS-79-23272 for the latter period. The first author also wishes to acknowledge financial support from a George Murray Scholarship from the University of Adelaide and a University of Dundee Research Scholarship for the former period.     

A proximal cutting plane method using Chebychev center for nonsmooth convex optimization

An algorithm is developed for minimizing nonsmooth convex functions. This algorithm extends Elzinga–Moore cutting plane algorithm by enforcing the search of the next test point not too far from the previous ones, thus removing compactness assumption. Our method is to Elzinga–Moore’s algorithm what a proximal bundle method is to Kelley’s algorithm. Instead of lower approximations used in proximal bundle methods, the present approach is based on some objects regularizing translated functions of the objective function. We propose some variants and using some academic test problems, we conduct a numerical comparative study with Elzinga–Moore algorithm and two other well-known nonsmooth methods.      

凸约束非光滑方程组一个新的谱梯度投影算法

基于寻找分离超平面的三种经典线搜索技术,本文提出了一种自适应线搜索技术.结合谱梯度投影法,提出了凸约束非光滑单调方程组的一个谱梯度投影算法.该算法不需要计算和存储任何矩阵,因而适合求解大规模非光滑的非线性单调方程组.在较弱的条件下,证明了方法的全局收敛性,并分析了算法的收敛率.数值试验结果表明算法是有效的和鲁棒的.     

A parameterized Newton method and a quasi-Newton method for nonsmooth equations

This paper presents a parameterized Newton method using generalized Jacobians and a Broyden-like method for solving nonsmooth equations. The former ensures that the method is well-defined even when the generalized Jacobian is singular. The latter is constructed by using an approximation function which can be formed for nonsmooth equations arising from partial differential equations and nonlinear complementarity problems. The approximation function method generalizes the splitting function method for nonsmooth equations. Locally superlinear convergence results are proved for the two methods. Numerical examples are given to compare the two methods with some other methods.This work is supported by the Australian Research Council.     

最小化三个凸函数之和的一个简单原始-对偶算法

提出一个简单的原始-对偶算法求解三个凸函数之和的最小化问题, 其中目标函数包含有梯度李普希兹连续的光滑函数, 非光滑函数和含有复合算子的非光滑函数. 在新方法中, 对偶变量迭代使用预估-矫正的方案. 分析了算法的收敛性和收敛速率. 最后, 数值实验说明了算法的有效性.     

A smoothing augmented Lagrangian method for solving simple bilevel programs

In this paper, we design a numerical algorithm for solving a simple bilevel program where the lower level program is a nonconvex minimization problem with a convex set constraint. We propose to solve a combined problem where the first order condition and the value function are both present in the constraints. Since the value function is in general nonsmooth, the combined problem is in general a nonsmooth and nonconvex optimization problem. We propose a smoothing augmented Lagrangian method for solving a general class of nonsmooth and nonconvex constrained optimization problems. We show that, if the sequence of penalty parameters is bounded, then any accumulation point is a Karush-Kuch-Tucker (KKT) point of the nonsmooth optimization problem. The smoothing augmented Lagrangian method is used to solve the combined problem. Numerical experiments show that the algorithm is efficient for solving the simple bilevel program.     

非光滑非线性互补问题的牛顿法

研究了非光滑的非线性互补问题. 首先将非光滑的非线性互补问题转化为一个非光滑方程组,然后用牛顿法求解这个非光滑方程组. 在该牛顿法中,每次迭代只需一个原始函数B-微分中的一个元素. 最后证明了该牛顿法的超线性收敛性.     

Globally Convergent Cutting Plane Method for Nonconvex Nonsmooth Minimization

Nowadays, solving nonsmooth (not necessarily differentiable) optimization problems plays a very important role in many areas of industrial applications. Most of the algorithms developed so far deal only with nonsmooth convex functions. In this paper, we propose a new algorithm for solving nonsmooth optimization problems that are not assumed to be convex. The algorithm combines the traditional cutting plane method with some features of bundle methods, and the search direction calculation of feasible direction interior point algorithm (Herskovits, J. Optim. Theory Appl. 99(1):121–146, 1998). The algorithm to be presented generates a sequence of interior points to the epigraph of the objective function. The accumulation points of this sequence are solutions to the original problem. We prove the global convergence of the method for locally Lipschitz continuous functions and give some preliminary results from numerical experiments.     

数据挖掘中聚类中心问题的光滑化和填充函数方法

本文提出了数据挖掘中求解聚类中心问题的一种新方法.这类问题属于非凸非光滑全局最优化问题.我们首先利用光滑化方法将非光滑聚类函数用光滑函数逼近,然后对光滑化问题利用填充函数搜索其全局最优点.对不同数据库的数值试验表明,本文提出的算法是可行和有效的.     

基于次梯度选取的非光滑优化强次可行方向法

本文结合次梯度选取技术及割平面法和强次可行方向法的思想,提出了一个求解目标函数非光滑约束优化问题的强次可行方向算法.通过设计一个新的寻找搜索方向子问题和构造新型线搜索,算法不仅能接受不可行的初始点,而且能保持迭代点的强次可行性,同时避免在可行域外目标函数值的不适度增加.算法具备全局收敛性,且初步的数值试验表明算法是稳定有效的.     

Nonsmooth Steepest Descent Method by Proximal Subdifferentials in Hilbert Spaces

In this paper, we first study a nonsmooth steepest descent method for nonsmooth functions defined on a Hilbert space and establish the corresponding algorithm by proximal subgradients. Then, we use this algorithm to find stationary points for those functions satisfying prox-regularity and Lipschitz continuity. As an application, the established algorithm is used to search for the minimizer of a lower semicontinuous and convex function on a finite-dimensional space. A convergence theorem, as an extension and improvement of the existing converging result for twice continuously differentiable convex functions, is also presented therein.     

Nonsmooth optimization via quasi-Newton methods

We investigate the behavior of quasi-Newton algorithms applied to minimize a nonsmooth function f, not necessarily convex. We introduce an inexact line search that generates a sequence of nested intervals containing a set of points of nonzero measure that satisfy the Armijo and Wolfe conditions if f is absolutely continuous along the line. Furthermore, the line search is guaranteed to terminate if f is semi-algebraic. It seems quite difficult to establish a convergence theorem for quasi-Newton methods applied to such general classes of functions, so we give a careful analysis of a special but illuminating case, the Euclidean norm, in one variable using the inexact line search and in two variables assuming that the line search is exact. In practice, we find that when f is locally Lipschitz and semi-algebraic with bounded sublevel sets, the BFGS (Broyden–Fletcher–Goldfarb–Shanno) method with the inexact line search almost always generates sequences whose cluster points are Clarke stationary and with function values converging R-linearly to a Clarke stationary value. We give references documenting the successful use of BFGS in a variety of nonsmooth applications, particularly the design of low-order controllers for linear dynamical systems. We conclude with a challenging open question.     

New Bundle Methods for Solving Lagrangian Relaxation Dual Problems

Bundle methods have been used frequently to solve nonsmooth optimization problems. In these methods, subgradient directions from past iterations are accumulated in a bundle, and a trial direction is obtained by performing quadratic programming based on the information contained in the bundle. A line search is then performed along the trial direction, generating a serious step if the function value is improved by or a null step otherwise. Bundle methods have been used to maximize the nonsmooth dual function in Lagrangian relaxation for integer optimization problems, where the subgradients are obtained by minimizing the performance index of the relaxed problem. This paper improves bundle methods by making good use of near-minimum solutions that are obtained while solving the relaxed problem. The bundle information is thus enriched, leading to better search directions and less number of null steps. Furthermore, a simplified bundle method is developed, where a fuzzy rule is used to combine linearly directions from near-minimum solutions, replacing quadratic programming and line search. When the simplified bundle method is specialized to an important class of problems where the relaxed problem can be solved by using dynamic programming, fuzzy dynamic programming is developed to obtain efficiently near-optimal solutions and their weights for the linear combination. This method is then applied to job shop scheduling problems, leading to better performance than previously reported in the literature.