A regularized minimization model with $l_1$-norm penalty (RP) is introduced for training the autoencoders that belong to a class of two-layer neural networks. We show that the RP can act as an exact penalty model which shares the same global minimizers, local minimizers, and d(irectional)-stationary points with the original regularized model under mild conditions. We construct a bounded box region that contains at least one global minimizer of the RP, and propose a linearly constrained regularized minimization model with $l_1$-norm penalty (LRP) for training autoencoders. A smoothing proximal gradient algorithm is designed to solve the LRP. Convergence of the algorithm to a generalized d-stationary point of the RP and LRP is delivered. Comprehensive numerical experiments convincingly illustrate the efficiency as well as the robustness of the proposed algorithm.
An important aspect in the solution process of constraint satisfaction problems is to identify exclusion boxes which are boxes that do not contain feasible points. This paper presents a certificate of infeasibility for finding such boxes by solving a linearly constrained nonsmooth optimization problem. Furthermore, the constructed certificate can be used to enlarge an exclusion box by solving a nonlinearly constrained nonsmooth optimization problem.
It is well known that there have been many numerical algorithms for solving nonsmooth minimax problems, numerical algorithms for nonsmooth minimax problems with joint linear constraints are very rare. This paper aims to discuss optimality conditions and develop practical numerical algorithms for minimax problems with joint linear constraints. First of all, we use the properties of proximal mapping and KKT system to establish optimality conditions. Secondly, we propose a framework of alternating coordinate algorithm for the minimax problem and analyze its convergence properties. Thirdly, we dev…
We study the \emph{Proximal Alternating Predictor-Corrector} (PAPC) algorithm introduced recently by Drori, Sabach and Teboulle 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…
Motivated by variational models in continuum mechanics, we introduce a novel algorithm to perform nonsmooth and nonconvex minimizations with linear constraints in Euclidean spaces. We show how this algorithm is actually a natural generalization of the well-known non-stationary augmented Lagrangian method for convex optimization. The relevant features of this approach are its applicability to a large variety of nonsmooth and nonconvex objective functions, its guaranteed convergence to critical points of the objective energy independently of the choice of the initial value, and its simplicity of…
In this paper, we consider a proximal linearized alternating direction method of multipliers (PL-ADMM) for solving linearly constrained nonconvex and possibly nonsmooth optimization problems. The algorithm is generalized by using variable metric proximal terms in the primal updates and an over-relaxation stepsize in the multiplier update. We prove that the sequence generated by this method is bounded and its limit points are critical points. Under the powerful Kurdyka-{Łojasiewicz} properties we prove that the sequence has a finite length thus converges, and we drive its convergence rates.
In this paper, we study the stochastic gradient descent (SGD) method for the nonconvex nonsmooth optimization, and propose an accelerated SGD method by combining the variance reduction technique with Nesterov's extrapolation technique. Moreover, based on the local error bound condition, we establish the linear convergence of our method to obtain a stationary point of the nonconvex optimization. In particular, we prove that not only the sequence generated linearly converges to a stationary point of the problem, but also the corresponding sequence of objective values is linearly convergent. Fina…
The generalized alternating direction method of multipliers (ADMM) of Xiao et al. [{\tt Math. Prog. Comput., 2018}] aims at the two-block linearly constrained composite convex programming problem, in which each block is in the form of "nonsmooth + quadratic". However, in the case of non-quadratic (but smooth), this method may fail unless the favorable structure of 'nonsmooth + smooth' is no longer used. This paper aims to remedy this defect by using a majorized technique to approximate the augmented Lagrangian function, so that the corresponding subprobllem can be decomposed into some smaller …
The proximal gradient algorithm for minimizing the sum of a smooth and a nonsmooth convex function often converges linearly even without strong convexity. One common reason is that a multiple of the step length at each iteration may linearly bound the "error" -- the distance to the solution set. We explain the observed linear convergence intuitively by proving the equivalence of such an error bound to a natural quadratic growth condition. Our approach generalizes to linear convergence analysis for proximal methods (of Gauss-Newton type) for minimizing compositions of nonsmooth functions with s…
Convergence analysis of accelerated algorithms via a mixed-order dynamical system for separable nonsmooth convex optimization
Chongqing Key Laboratory of Statistical Intelligent Computing and Monitoring, School of Mathematics and Statistics, Chongqing Technology and Business University（重庆工商大学数学与统计学院，统计智能计算与监测重庆市重点实验室）
In this paper, we study the proximal gradient algorithm with extrapolation for minimizing the sum of a Lipschitz differentiable function and a proper closed convex function. Under the error bound condition used in [19] for analyzing the convergence of the proximal gradient algorithm, we show that there exists a threshold such that if the extrapolation coefficients are chosen below this threshold, then the sequence generated converges $R$-linearly to a stationary point of the problem. Moreover, the corresponding sequence of objective values is also $R$-linearly convergent. In addition, the thre…
We introduce the Asynchronous PALM algorithm, a new extension of the Proximal Alternating Linearized Minimization (PALM) algorithm for solving nonsmooth, nonconvex optimization problems. Like the PALM algorithm, each step of the Asynchronous PALM algorithm updates a single block of coordinates; but unlike the PALM algorithm, the Asynchronous PALM algorithm eliminates the need for sequential updates that occur one after the other. Instead, our new algorithm allows each of the coordinate blocks to be updated asynchronously and in any order, which means that any number of computing cores can comp…
We consider optimization algorithms that successively minimize simple Taylor-like models of the objective function. Methods of Gauss-Newton type for minimizing the composition of a convex function and a smooth map are common examples. Our main result is an explicit relationship between the step-size of any such algorithm and the slope of the function at a nearby point. Consequently, we (1) show that the step-sizes can be reliably used to terminate the algorithm, (2) prove that as long as the step-sizes tend to zero, every limit point of the iterates is stationary, and (3) show that conditions,…
In this paper, we consider large-scale linearly constrained composite convex optimization problem, whose objective is a sum of a smooth function and a possibly nonsmooth function. We propose a scalable \textbf{F}rank-\textbf{W}olfe based \textbf{A}ugmented \textbf{L}agrangian (FW-AL) method for solving this problem. At each iteration, the proposed FW-AL method employs the FW method (or its variants) to approximately solve the AL subproblem {(with fixed Lagrange multiplier)} within a preselected tolerance and then updates the Lagrange multiplier. The proposed FW-AL method is well suitable for s…
Consider the problem of minimizing the sum of a smooth convex function and a separable nonsmooth convex function subject to linear coupling constraints. Problems of this form arise in many contemporary applications including signal processing, wireless networking and smart grid provisioning. Motivated by the huge size of these applications, we propose a new class of first order primal-dual algorithms called the block successive upper-bound minimization method of multipliers (BSUM-M) to solve this family of problems. The BSUM-M updates the primal variable blocks successively by minimizing local…
King Abdullah University of Science and Technology（阿卜杜拉国王科技大学）
We consider minimizing the sum of three convex functions, where the first one F is smooth, the second one is nonsmooth and proximable and the third one is the composition of a nonsmooth proximable function with a linear operator L. This template problem has many applications, for instance, in image processing and machine learning. First, we propose a new primal-dual algorithm, which we call PDDY, for this problem. It is constructed by applying Davis-Yin splitting to a monotone inclusion in a primal-dual product space, where the operators are monotone under a specific metric depending on L. We …
We develop randomized (block) coordinate descent (CD) methods for linearly constrained convex optimization. Unlike most CD methods, we do not assume the constraints to be separable, but let them be coupled linearly. To our knowledge, ours is the first CD method that allows linear coupling constraints, without making the global iteration complexity have an exponential dependence on the number of constraints. We present algorithms and analysis for four key problem scenarios: (i) smooth; (ii) smooth + nonsmooth separable; (iii) asynchronous parallel; and (iv) stochastic. We illustrate empirical b…
The augmented Lagrangian method (ALM) is one of the most useful methods for constrained optimization. Its convergence has been well established under convexity assumptions or smoothness assumptions, or under both assumptions. ALM may experience oscillations and divergence when the underlying problem is simultaneously nonconvex and nonsmooth. In this paper, we consider the linearly constrained problem with a nonconvex (in particular, weakly convex) and nonsmooth objective. We modify ALM to use a Moreau envelope of the augmented Lagrangian and establish its convergence under conditions that are …
Randomized Coordinate Subgradient Method for Nonsmooth Composite Optimization
Institute of Translational Medicine（转化医学研究院）；National Center for Translational Medicine（国家转化医学中心）；Shanghai Jiao Tong University（上海交通大学）；Alibaba Group（阿里巴巴集团）；Antai College of Economics and Management（安泰经济与管理学院）；The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳））；School of Data Science（数据科学学院）；Xiangfu Laboratory（祥符实验室）
Linearized alternating direction method of multipliers (ADMM) as an extension of ADMM has been widely used to solve linearly constrained problems in signal processing, machine leaning, communications, and many other fields. Despite its broad applications in nonconvex optimization, for a great number of nonconvex and nonsmooth objective functions, its theoretical convergence guarantee is still an open problem. In this paper, we propose a two-block linearized ADMM and a multi-block parallel linearized ADMM for problems with nonconvex and nonsmooth objectives. Mathematically, we present that the …
This paper focuses on the quadratic optimization over two classes of nonnegative zero-norm constraints: nonnegative zero-norm sphere constraint and zero-norm simplex constraint, which have important applications in nonnegative sparse eigenvalue problems and sparse portfolio problems, respectively. We establish the KL property of exponent 1/2 for the extended-valued objective function of these nonconvex and nonsmooth optimization problems, and use this crucial property to develop a globally and linearly convergent projection gradient descent (PGD) method. Numerical results are included for none…
Southwest Jiaotong University（西南交通大学）；Zhejiang University City College（浙江大学城市学院）
Square-root Lasso problems are proven robust regression problems. Furthermore, square-root regression problems with structured sparsity also plays an important role in statistics and machine learning. In this paper, we focus on the numerical computation of large-scale linearly constrained sparse group square-root Lasso problems. In order to overcome the difficulty that there are two nonsmooth terms in the objective function, we propose a dual semismooth Newton (SSN) based augmented Lagrangian method (ALM) for it. That is, we apply the ALM to the dual problem with the subproblem solved by the S…
Stochastic (sub)gradient methods require step size schedule tuning to perform well in practice. Classical tuning strategies decay the step size polynomially and lead to optimal sublinear rates on (strongly) convex problems. An alternative schedule, popular in nonconvex optimization, is called \emph{geometric step decay} and proceeds by halving the step size after every few epochs. In recent work, geometric step decay was shown to improve exponentially upon classical sublinear rates for the class of \emph{sharp} convex functions. In this work, we ask whether geometric step decay similarly impro…
