A Trust-region Method for Nonsmooth Nonconvex Optimization

We propose a trust-region type method for a class of nonsmooth nonconvex optimization problems where the objective function is a summation of a (probably nonconvex) smooth function and a (probably nonsmooth) convex function. The model function of our trust-region subproblem is always quadratic and the linear term of the model is generated using abstract descent directions. Therefore, the trust-region subproblems can be easily constructed as well as efficiently solved by cheap and standard methods. When the accuracy of the model function at the solution of the subproblem is not sufficient, we add a safeguard on the stepsizes for improving the accuracy. For a class of functions that can be "truncated'', an additional truncation step is defined and a stepsize modification strategy is designed. The overall scheme converges globally and we establish fast local convergence under suitable assumptions. In particular, using a connection with a smooth Riemannian trust-region method, we prove local quadratic convergence for partly smooth functions under a strict complementary condition. Preliminary numerical results on a family of $\ell_1$-optimization problems are reported and demonstrate the efficiency of our approach.     

On Approximations with Finite Precision in Bundle Methods for Nonsmooth Optimization

Journal of Optimization Theory and Applications - We consider the proximal form of a bundle algorithm for minimizing a nonsmooth convex function, assuming that the function and subgradient values...     

Constrained Nonconvex Nonsmooth Optimization via Proximal Bundle Method

In this paper, we consider a constrained nonconvex nonsmooth optimization, in which both objective and constraint functions may not be convex or smooth. With the help of the penalty function, we transform the problem into an unconstrained one and design an algorithm in proximal bundle method in which local convexification of the penalty function is utilized to deal with it. We show that, if adding a special constraint qualification, the penalty function can be an exact one, and the sequence generated by our algorithm converges to the KKT points of the problem under a moderate assumption. Finally, some illustrative examples are given to show the good performance of our algorithm.     

SOS-Convex Semialgebraic Programs and its Applications to Robust Optimization: A Tractable Class of Nonsmooth Convex Optimization

In this paper, we introduce a new class of nonsmooth convex functions called SOS-convex semialgebraic functions extending the recently proposed notion of SOS-convex polynomials. This class of nonsmooth convex functions covers many common nonsmooth functions arising in the applications such as the Euclidean norm, the maximum eigenvalue function and the least squares functions with ? ₁-regularization or elastic net regularization used in statistics and compressed sensing. We show that, under commonly used strict feasibility conditions, the optimal value and an optimal solution of SOS-convex semialgebraic programs can be found by solving a single semidefinite programming problem (SDP). We achieve the results by using tools from semialgebraic geometry, convex-concave minimax theorem and a recently established Jensen inequality type result for SOS-convex polynomials. As an application, we show that robust SOS-convex optimization proble ms under restricted spectrahedron data uncertainty enjoy exact SDP relaxations. This extends the existing exact SDP relaxation result for restricted ellipsoidal data uncertainty and answers an open question in the literature on how to recover a robust solution of uncertain SOS-convex polynomial programs from its semidefinite programming relaxation in this broader setting.     

Comparing Nonsmooth Nonconvex Bundle Methods in Solving Hemivariational Inequalities

Hemivariational inequalities can be considered as a generalization of variational inequalities. Their origin is in nonsmooth mechanics of solid, especially in nonmonotone contact problems. The solution of a hemivariational inequality proves to be a substationary point of some functional, and thus can be found by the nonsmooth and nonconvex optimization methods. We consider two type of bundle methods in order to solve hemivariational inequalities numerically: proximal bundle and bundle-Newton methods. Proximal bundle method is based on first order polyhedral approximation of the locally Lipschitz continuous objective function. To obtain better convergence rate bundle-Newton method contains also some second order information of the objective function in the form of approximate Hessian. Since the optimization problem arising in the hemivariational inequalities has a dominated quadratic part the second order method should be a good choice. The main question in the functioning of the methods is how remarkable is the advantage of the possible better convergence rate of bundle-Newton method when compared to the increased calculation demand.     

A Feasible Point Method with Bundle Modification for Nonsmooth Convex Constrained Optimization

In this paper, a bundle modification strategy is proposed for nonsmooth convex constrained minimization problems. As a result, a new feasible point bundle method is presented by applying this strategy. Whenever the stability center is updated, some points in the bundle will be substituted by new ones which have lower objective values and/or constraint values, aiming at getting a better bundle. The method generates feasible serious iterates on which the objective function is monotonically decreasing. Global convergence of the algorithm is established, and some preliminary numerical results show that our method performs better than the standard feasible point bundle method.     

解非线性约束优化问题的依赖域算法

本文对一般非线性约束优化问题提出了一个信赖域算法,导出了等价的KKT条件.在试探步满足适当条件下,证明了算法的全局收敛性,并进行了数值试验.     

参数非光滑优化的微分稳定性

本文研究了含有向量参数的非光滑优化问题的极值函数或叫做边缘函数的连续性及某种意义下的微分性质。给出了目标函数及不等式约束为李普希兹函数,等式约束为连续可微函数,并且带有闭凸约束集C的非凸非光滑问题的最优值函数的几种方向导数的界,把[4],[1]中关于一个参数的单边扰动推广到向量参数的扰动,亦可认为是把[2]由光滑函数类推广到李普希兹函数类。     

非光滑凸规划的割平面法及其在组合优化中的应用

本文利用次梯度构造了一种割平面 ,将非光滑凸规划松驰为光滑规划 ,给出了一种非光滑凸规划的割平面法 ,并证明了其收敛性 ,通过在组合优化中的应用说明该算法是有效的 .     

A Robust Replacement Model with Applications to Medical Equipment

Perceived shortcomings in the applicability of capital equipment replacement modelling, identified in a 1987 survey within the UK are addressed and a robust replacement model formulated. First, however, a comparison between the 1987 survey and a similar 1988 survey undertaken within the USA is made and explanations for apparent differences in conclusions are presented. A replacement model is then developed in the context of medical equipment where factors such as service and risk play a role in replacement decision-making. A mechanism for quantitatively allowing for qualitative and for political type factors within a short time horizon replacement model is introduced by means of a penalty factor. A case example is presented for medical ventilator equipment.     

Stability of the Picard Bundle

Let X be a non-singular algebraic curve of genus g 2, n 2an integer, a line bundle over X of degree d > 2n(g –1) with (n,d) = 1 and M the moduli space of stable bundles ofrank n and determinant over X. It is proved that the Picardbundle W is stable with respect to the unique polarisation ofM. 2000 Mathematics Subject Classification 14H60, 14J60.     

Inequality problems with Nonlocally Lipschitz Energy Functional: Existence Results and Applications to Nonsmooth Mechanics

We give an existence result for a double eigenvalue problem in Hemivariational Inequalities whose energetic functional is not locally Lipschitz. It is used a finite dimensional approach based on Kakutani's fixed point theorem.     

Differences of Polyhedra in Matrix Space and Their Applications to Nonsmooth Analysis

Formulas of the differences of polyhedra in matrix space are proposed. Based on these formulas, the differences of polyhedra can be calculated by solving systems of linear inequalities. A modified algorithm for calculating one element of the differences is presented also. The motivation for this work is to compute the Clarke generalized Jacobian, the B-differential, and one of their elements via the quasidifferential. Applications to Newton methods for solving nonsmooth equations are discussed.This project was sponsored by the Shanghai Education Committee, Grant 04EA01, by the Education Ministry of China, and by the Shanghai Government, Grant T0502. The author thanks two anonymous referees and Professor F. Giannessi for valuable suggestions and comments.     

A Bundle Algorithm Applied to Bilevel Programming Problems with Non-Unique Lower Level Solutions

In the paper, the question is investigated if a bundle algorithm can be used to compute approximate solutions for bilevel programming problems where the lower level optimal solution is in general not uniquely determined. To give a positive answer to this question, an appropriate regularization approach is used in the lower level. In the general case, the resulting algorithm computes an approximate solution. If the problem proves to have strongly stable lower level solutions for all parameter values in a certain neighborhood of the stationary solutions of the bilevel problem, convergence to stationary solutions can be shown.     

无约束不可微观划的信赖域算法

提出了一种易实施的求无约束不可微规划的信赖域算法,并在一定条件下证明了该算法所产生的点列的任何聚点都是原问题的稳定点。     

Unified Robust Necessary Optimality Conditions for Nonconvex Nonsmooth Uncertain Multiobjective Optimization

Journal of Optimization Theory and Applications - This paper is devoted to presenting new error bounds of regularized gap functions for polynomial variational inequalities with exponents explicitly...     

Active-Set Projected Trust-Region Algorithm for Box-Constrained Nonsmooth Equations

In this paper, by means of an active-set strategy, we present a trust-region method for solving box-constrained nonsmooth equations. Nice properties of the proposed method include: (a) all iterates remain feasible; (b) the search direction, as adequate combination of the projected gradient direction and the trust-region direction, is an asymptotic Newton direction under mild conditions; (c) the subproblem of the proposed method, possessing the form of an unconstrained trust-region subproblem, can be solved by existing methods; (d) the subproblem of the proposed method is of reduced dimension, which is potentially cheaper when applied to solve large-scale problems. Under appropriate conditions, we establish global and local superlinear/quadratic convergence of the method. Preliminary numerical results are given.     

一种约束非光滑优化问题的信赖域算法

提出了一种易实施的求解带线性约束的非光滑优化问题的信赖域算法，并在一定的条件下证明了该算法所产生的迭代序列的任何聚点都是原问题的稳定点．有限的数值例子表明，该方法是行之有效的．     

自治非光滑时滞系统的有限时间稳定

主要讨论右端非光滑的自治时滞系统在Filippov解意义下的有限时间稳定问题.基于Filippov微分包含和非光滑的Lyapunov-Krasovskii泛函,提出自治非光滑时滞系统有限时间稳定的定义和比较原理,并给出有限时间稳定的Lyapunov定理.