An iterative method for minimizing a convex nonsmooth function on a convex smooth surface

An iterative algorithm is proposed for the constrained minimization of a convex nonsmooth function on a set given as a convex smooth surface. The convergence of the algorithm in the sense of necessary conditions for a local minimum is proved.     

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.     

An SQP-type algorithm for nonlinear second-order cone programs

We propose an SQP-type algorithm for solving nonlinear second-order cone programming (NSOCP) problems. At every iteration, the algorithm solves a convex SOCP subproblem in which the constraints involve linear approximations of the constraint functions in the original problem and the objective function is a convex quadratic function. Those subproblems can be transformed into linear SOCP problems, for which efficient interior point solvers are available. We establish global convergence and local quadratic convergence of the algorithm under appropriate assumptions. We report numerical results to examine the effectiveness of the algorithm. This work was supported in part by the Scientific Research Grant-in-Aid from Japan Society for the Promotion of Science.     

A coordinate gradient descent method for nonsmooth separable minimization

We consider the problem of minimizing the sum of a smooth function and a separable convex function. This problem includes as special cases bound-constrained optimization and smooth optimization with ?₁-regularization. We propose a (block) coordinate gradient descent method for solving this class of nonsmooth separable problems. We establish global convergence and, under a local Lipschitzian error bound assumption, linear convergence for this method. The local Lipschitzian error bound holds under assumptions analogous to those for constrained smooth optimization, e.g., the convex function is polyhedral and the smooth function is (nonconvex) quadratic or is the composition of a strongly convex function with a linear mapping. We report numerical experience with solving the ?₁-regularization of unconstrained optimization problems from Moré et al. in ACM Trans. Math. Softw. 7, 17–41, 1981 and from the CUTEr set (Gould and Orban in ACM Trans. Math. Softw. 29, 373–394, 2003). Comparison with L-BFGS-B and MINOS, applied to a reformulation of the ?₁-regularized problem as a bound-constrained optimization problem, is also reported.     

Regularized Newton Methods for Convex Minimization Problems with Singular Solutions

This paper studies convergence properties of regularized Newton methods for minimizing a convex function whose Hessian matrix may be singular everywhere. We show that if the objective function is LC², then the methods possess local quadratic convergence under a local error bound condition without the requirement of isolated nonsingular solutions. By using a backtracking line search, we globalize an inexact regularized Newton method. We show that the unit stepsize is accepted eventually. Limited numerical experiments are presented, which show the practical advantage of the method.     

Incrementally Updated Gradient Methods for Constrained and Regularized Optimization

We consider incrementally updated gradient methods for minimizing the sum of smooth functions and a convex function. This method can use a (sufficiently small) constant stepsize or, more practically, an adaptive stepsize that is decreased whenever sufficient progress is not made. We show that if the gradients of the smooth functions are Lipschitz continuous on the space of n-dimensional real column vectors or the gradients of the smooth functions are bounded and Lipschitz continuous over a certain level set and the convex function is Lipschitz continuous on its domain, then every cluster point of the iterates generated by the method is a stationary point. If in addition a local Lipschitz error bound assumption holds, then the method is linearly convergent.     

The method of feasible directions for mathematical programming problems with preconvex constraints

The convergence of the method of feasible directions is proved for the case of the smooth objective function and a constraint in the form of the difference of convex sets (the so-called preconvex set). It is shown that the method converges to the set of stationary points, which generally is narrower than the corresponding set in the case of a smooth function and smooth constraints. The scheme of the proof is similar to that proposed earlier by Karmanov.     

复合凸优化问题的Gauss-Newton法的收敛性

本文研究解决复合凸优化问题:min F(x):=h(f(x)) （P）x∈X的Gauss-Newton法的收敛性.这里f是从Banach空间X到Banach空间Y的具有Frechet导数的非线性映照,h是定义在Y上的凸泛函. 复合凸优化问题近年来一直受到广泛的关注,目前它已成为非线性光滑理论中的一个主流方向.它在非线性包含,最大最小问题,罚函数技巧 [1-5]等许多重要的问题和技巧中得到了广泛的应用.同时它也提供了一个新的统一框架,使优化问题数值解的理论分析得到别开生面的发展.并且它也是研究有限区域内一阶或二阶最优性条件的一个便利工具[3,5,6,7].     

A globally linearly convergent method for pointwise quadratically supportable convex–concave saddle point problems

We study the Proximal Alternating Predictor–Corrector (PAPC) algorithm introduced recently by Drori, Sabach and Teboulle [8] to solve nonsmooth structured convex–concave saddle point problems consisting of the sum of a smooth convex function, a finite collection of nonsmooth convex functions and bilinear terms. We introduce the notion of pointwise quadratic supportability, which is a relaxation of a standard strong convexity assumption and allows us to show that the primal sequence is R-linearly convergent to an optimal solution and the primal-dual sequence is globally Q-linearly convergent. We illustrate the proposed method on total variation denoising problems and on locally adaptive estimation in signal/image deconvolution and denoising with multiresolution statistical constraints.     

On the convergence rate of a forward-backward type primal-dual splitting algorithm for convex optimization problems

In this paper, we analyse the convergence rate of the sequence of objective function values of a primal-dual proximal-point algorithm recently introduced in the literature for solving a primal convex optimization problem having as objective the sum of linearly composed infimal convolutions, nonsmooth and smooth convex functions and its Fenchel-type dual one. The theoretical part is illustrated by numerical experiments in image processing.     

A Smoothing Newton-Type Algorithm of Stronger Convergence for the Quadratically Constrained Convex Quadratic Programming

In this paper we propose a smoothing Newton-type algorithm for the problem of minimizing a convex quadratic function subject to finitely many convex quadratic inequality constraints. The algorithm is shown to converge globally and possess stronger local superlinear convergence. Preliminary numerical results are also reported. Mathematics Subject Classification (1991): 90C33, 65K10 This author’s work was also partially supported by the Scientific Research Foundation of Tianjin University for the Returned Overseas Chinese Scholars and the Scientific Research Foundation of Liu Hui Center for Applied Mathematics, Nankai University-Tianjin University.     

Proximal-gradient algorithms for fractional programming

In this paper, we propose two proximal-gradient algorithms for fractional programming problems in real Hilbert spaces, where the numerator is a proper, convex and lower semicontinuous function and the denominator is a smooth function, either concave or convex. In the iterative schemes, we perform a proximal step with respect to the nonsmooth numerator and a gradient step with respect to the smooth denominator. The algorithm in case of a concave denominator has the particularity that it generates sequences which approach both the (global) optimal solutions set and the optimal objective value of the underlying fractional programming problem. In case of a convex denominator the numerical scheme approaches the set of critical points of the objective function, provided the latter satisfies the Kurdyka-?ojasiewicz property.     

Local linear convergence analysis of Primal–Dual splitting methods

In this paper, we study the local linear convergence properties of a versatile class of Primal–Dual splitting methods for minimizing composite non-smooth convex optimization problems. Under the assumption that the non-smooth components of the problem are partly smooth relative to smooth manifolds, we present a unified local convergence analysis framework for these methods. More precisely, in our framework, we first show that (i) the sequences generated by Primal–Dual splitting methods identify a pair of primal and dual smooth manifolds in a finite number of iterations, and then (ii) enter a local linear convergence regime, which is characterized based on the structure of the underlying active smooth manifolds. We also show how our results for Primal–Dual splitting can be specialized to cover existing ones on Forward–Backward splitting and Douglas–Rachford splitting/ADMM (alternating direction methods of multipliers). Moreover, based on these obtained local convergence analysis result, several practical acceleration techniques are discussed. To exemplify the usefulness of the obtained result, we consider several concrete numerical experiments arising from fields including signal/image processing, inverse problems and machine learning. The demonstration not only verifies the local linear convergence behaviour of Primal–Dual splitting methods, but also the insights on how to accelerate them in practice.     

混合互补问题的光滑算法及收敛性

利用Fischer-Burmeister函数将混合互补问题转化为非线性方程组,由光滑函数逼近FB函数来求解非线性方程组．文中将信赖域方法和梯度法相结合,提出了Jacobian光滑化方法．算法在一定条件下的全局收敛性得到了证明,数值试验表明算法切实有效,有一定的优越性．     

Convergence property of the Iri-Imai algorithm for some smooth convex programming problems

In this paper, the Iri-Imai algorithm for solving linear and convex quadratic programming is extended to solve some other smooth convex programming problems. The globally linear convergence rate of this extended algorithm is proved, under the condition that the objective and constraint functions satisfy a certain type of convexity, called the harmonic convexity in this paper. A characterization of this convexity condition is given. The same convexity condition was used by Mehrotra and Sun to prove the convergence of a path-following algorithm.The Iri-Imai algorithm is a natural generalization of the original Newton algorithm to constrained convex programming. Other known convergent interior-point algorithms for smooth convex programming are mainly based on the path-following approach.     

On the convergence property of the DFP algorithm

The DFP algorithm of unconstrained optimization possesses excellent properties of convergence for convex functions. However, a convergence theory of the DFP algorithm without the convexity assumption has not yet been established. This paper gives the following result: If the objective function is suitably smooth, and if the DFP algorithm produces a convergent point sequence, then the limit point of the sequence is a critical point of the objective function. Also, some open questions are mentioned.Supported by the National Science Foundation of China.     

Smoothing Trust Region Methods for Nonlinear Complementarity Problems with P 0-Functions

By using the Fischer–Burmeister function to reformulate the nonlinear complementarity problem (NCP) as a system of semismooth equations and using Kanzow’s smooth approximation function to construct the smooth operator, we propose a smoothing trust region algorithm for solving the NCP with P ₀ functions. We prove that every accumulation point of the sequence generated by the algorithm is a solution of the NCP. Under a nonsingularity condition, local Q-superlinear/Q-quadratic convergence of the algorithm is established without the strict complementarity condition. This work was partially supported by the Research Grant Council of Hong Kong and the National Natural Science Foundation of China (Grant 10171030).     

A trust region algorithm for minimization of locally Lipschitzian functions

The classical trust region algorithm for smooth nonlinear programs is extended to the nonsmooth case where the objective function is only locally Lipschitzian. At each iteration, an objective function that carries both first and second order information is minimized over a trust region. The term that carries the first order information is an iteration function that may not explicitly depend on subgradients or directional derivatives. We prove that the algorithm is globally convergent. This convergence result extends the result of Powell for minimization of smooth functions, the result of Yuan for minimization of composite convex functions, and the result of Dennis, Li and Tapia for minimization of regular functions. In addition, compared with the recent model of Pang, Han and Rangaraj for minimization of locally Lipschitzian functions using a line search, this algorithm has the same convergence property without assuming positive definiteness and uniform boundedness of the second order term. Applications of the algorithm to various nonsmooth optimization problems are discussed.This author's work was supported in part by the Australian Research Council.This author's work was carried out while he was visiting the Department of Applied Mathematics at the University of New South Wales.     

凸约束优化问题的带记忆模型信赖域算法

本文我们考虑求解凸约束优化问题的信赖域方法 .与传统的方法不同 ,我们信赖域子问题的逼近模型中包括过去迭代点的信息 ,该模型使我们可以从更全局的角度来求得信赖域试探步 ,从而避免了传统信赖域方法中试探步的求取完全依赖于当前点的信息而过于局部化的困难 .全局收敛性的获得是依靠非单调技术来保证的     

求解凸不等式问题的熵光滑化方法

The maximal entropy principle is applied to solve convex inequality problems. An inequality problem can be transformed into a minmax problem.Then it can be transformed into an unconstrained parameterized min problem,using the entropic function to smooth the minmax problem. The solution of the inequality problem can be obtained, by solving the parameterized min problems and adjusting the parameter to zero, under a certain principle. However, it is sufficient to solve a parameterized inequality problem each time, from the propositions of the aggregate function. In the article, some propositions of the aggregate function are discussed, the algorithm and its convergence are obtained.