In this paper, we consider an accelerated method for solving nonconvex and nonsmooth minimization problems. We propose a Bregman Proximal Gradient algorithm with extrapolation(BPGe). This algorithm extends and accelerates the Bregman Proximal Gradient algorithm (BPG), which circumvents the restrictive global Lipschitz gradient continuity assumption needed in Proximal Gradient algorithms (PG). The BPGe algorithm has higher generality than the recently introduced Proximal Gradient algorithm with extrapolation(PGe), and besides, due to the extrapolation step, BPGe converges faster than BPG algorithm. Analyzing the convergence, we prove that any limit point of the sequence generated by BPGe is a stationary point of the problem by choosing parameters properly. Besides, assuming Kurdyka-{Ĺ}ojasiewicz property, we prove the whole sequences generated by BPGe converges to a stationary point. Finally, to illustrate the potential of the new method BPGe, we apply it to two important practical problems that arise in many fundamental applications (and that not satisfy global Lipschitz gradient continuity assumption): Poisson linear inverse problems and quadratic inverse problems. In the tests the accelerated BPGe algorithm shows faster convergence results, giving an interesting new algorithm.
To solve distributed optimization efficiently with various constraints and nonsmooth functions, we propose a distributed mirror descent algorithm with embedded Bregman damping, as a generalization of conventional distributed projection-based algorithms. In fact, our continuous-time algorithm well inherits good capabilities of mirror descent approaches to rapidly compute explicit solutions to the problems with some specific constraint structures. Moreover, we rigorously prove the convergence of our algorithm, along with the boundedness of the trajectory and the accuracy of the solution.
Large-scale nonconvex and nonsmooth problems have attracted considerable attention in the fields of compress sensing, big data optimization and machine learning. Exploring effective methods is still the main challenge of today's research. Stochastic coordinate descent type methods have been widely used to solve large-scale optimization problems. In this paper, we derive the convergence of variable Bregman stochastic coordinate descent (VBSCD) method for a broad class of nonsmooth and nonconvex optimization problems, i.e., any accumulation of the sequence generated by VBSCD is almost surely a c…
Variable Bregman Majorization-Minimization algorithms for nonconvex nonsmooth optimization, with application to Poisson imaging
Université Paris-Saclay, CentraleSupélec, Inria, Centre pour la Vision Numérique（巴黎萨克雷大学、中央苏佩莱克学院、法国国家信息与自动化研究所、数字视觉中心）；Université Paris-Saclay, Laboratoire BioMaps, CEA（巴黎萨克雷大学、BioMaps实验室、法国原子能和替代能源委员会）
In this paper we study the problems of minimizing the sum of two nonconvex functions: one is differentiable and satisfies smooth adaptable property. The smooth adaptable property, also named relatively smooth condition, is weaker than the globally gradient Lipschitz continuity. We analyze an inertial version of the Bregman Proximal Gradient (BPG) algorithm and prove its stationary convergence. Besides, we prove a sublinear convergence of the inertial algorithm. Moreover, if the objective function satisfies Kurdyka--Łojasiewicz (KL) property, its global convergence to a critical point of the ob…
We propose a Bregman inertial forward-reflected-backward (BiFRB) method for nonconvex composite problems. Our analysis relies on a novel approach that imposes general conditions on implicit merit function parameters, which yields a stepsize condition that is independent of inertial parameters. In turn, a question of Malitsky and Tam regarding whether FRB can be equipped with a Nesterov-type acceleration is resolved. Assuming the generalized concave Kurdyka-Łojasiewicz property of a quadratic regularization of the objective, we obtain sequential convergence of BiFRB, as well as convergence rate…
The alternating direction method with multipliers (ADMM) has been one of most powerful and successful methods for solving various convex or nonconvex composite problems that arise in the fields of image & signal processing and machine learning. In convex settings, numerous convergence results have been established for ADMM as well as its varieties. However, due to the absence of convexity, the convergence analysis of nonconvex ADMM is generally very difficult. In this paper we study the Bregman modification of ADMM (BADMM), which includes the conventional ADMM as a special case and often leads…
The alternating direction method with multipliers (ADMM) has been one of most powerful and successful methods for solving various composite problems. The convergence of the conventional ADMM (i.e., 2-block) for convex objective functions has been justified for a long time, and its convergence for nonconvex objective functions has, however, been established very recently. The multi-block ADMM, a natural extension of ADMM, is a widely used scheme and has also been found very useful in solving various nonconvex optimization problems. It is thus expected to establish convergence theory of the mult…
We prove convergence of a single time-scale stochastic subgradient method with subgradient averaging for constrained problems with a nonsmooth and nonconvex objective function having the property of generalized differentiability. As a tool of our analysis, we also prove a chain rule on a path for such functions.
We develop a new variational approach on level sets aiming towards convergence rate analysis of a variable Bregman proximal gradient (VBPG) method for a broad class of nonsmooth and nonconvex optimization problems. With this new approach, we are able to extend the concepts of Bregman proximal mapping and their corresponding Bregman proximal envelops, Bregman proximal gap function to nonconvex setting. Properties of these mappings and functions are examined. An aim of this work is to provide a solid foundation on which further design and analysis of VBPG for more general nonconvex optimization …
A general class of nonconvex optimization problems is considered, where the penalty is the composition of a linear operator with a nonsmooth nonconvex mapping, which is concave on the positive real line. The necessary optimality condition of a regularized version of the original problem is solved by means of a monotonically convergent scheme. Such problems arise in continuum mechanics, as for instance cohesive fractures, where singular behaviour is usually modelled by nonsmooth nonconvex energies. The proposed algorithm is successfully tested for fracture mechanics problems. Its performance is…
Distributed consensus optimization has received considerable attention in recent years; several distributed consensus-based algorithms have been proposed for (nonsmooth) convex and (smooth) nonconvex objective functions. However, the behavior of these distributed algorithms on {\it nonconvex, nonsmooth and stochastic} objective functions is not understood. This class of functions and distributed setting are motivated by several applications, including problems in machine learning and signal processing. This paper presents the first convergence analysis of the decentralized stochastic subgradie…
We provide an overview of primal-dual algorithms for nonsmooth and non-convex-concave saddle-point problems. This flows around a new analysis of such methods, using Bregman divergences to formulate simplified conditions for convergence.
This work is concerned with the optimization of nonconvex, nonsmooth composite optimization problems, whose objective is a composition of a nonlinear mapping and a nonsmooth nonconvex function, that can be written as an infimal convolution (inf-conv). To tackle this problem class we propose to reformulate the problem exploiting its inf-conv structure and derive a block coordinate descent scheme on a Bregman augmented Lagrangian, that can be implemented asynchronously. We prove convergence of our scheme to stationary points of the original model for a specific choice of the penalty parameter. O…
We investigate the convergence of an application of a proximal gradient method to control problems with nonsmooth and nonconvex control cost. Here, we focus on control cost functionals that promote sparsity, which includes functionals of $L^p$-type for $p\in [0,1)$. We prove stationarity properties of weak limit points of the method. These properties are weaker than those provided by Pontryagin's maximum principle and weaker than $L$-stationarity.
An adaptive regularization algorithm using inexact function and derivatives evaluations is proposed for the solution of composite nonsmooth nonconvex optimization. It is shown that this algorithm needs at most $O(|\log(ε)|\,ε^{-2})$ evaluations of the problem's functions and their derivatives for finding an $ε$-approximate first-order stationary point. This complexity bound therefore generalizes that provided by [Bellavia, Gurioli, Morini and Toint, 2018] for inexact methods for smooth nonconvex problems, and is within a factor $|\log(ε)|$ of the optimal bound known for smooth and nonsmooth no…
In this paper, nonconvex and nonsmooth models for compressed sensing (CS) and low rank matrix completion (MC) is studied. The problem is formulated as a nonconvex regularized leat square optimization problems, in which the l0-norm and the rank function are replaced by l1-norm and nuclear norm, and adding a nonconvex penalty function respectively. An alternating minimization scheme is developed, and the existence of a subsequence, which generate by the alternating algorithm that converges to a critical point, is proved. The NSP, RIP, and RIP condition for stable recovery guarantees also be anal…
We introduce the Stochastic Asynchronous Proximal Alternating Linearized Minimization (SAPALM) method, a block coordinate stochastic proximal-gradient method for solving nonconvex, nonsmooth optimization problems. SAPALM is the first asynchronous parallel optimization method that provably converges on a large class of nonconvex, nonsmooth problems. We prove that SAPALM matches the best known rates of convergence --- among synchronous or asynchronous methods --- on this problem class. We provide upper bounds on the number of workers for which we can expect to see a linear speedup, which match t…
A class of exact penalty-type local search methods for optimal control problems with nonsmooth cost functional, nonsmooth (but continuous) dynamics, and nonsmooth state and control constraints is presented, in which the the penalty parameter and several line search parameters are adaptively adjusted during the optimisation process. This class of methods is applicable to problems having a known DC (Difference-of-Convex functions) structure and in its core is based on the classical DCA method, combined with the steering exact penalty rules for updating the penalty parameter and an adaptive nonmo…
We develop and analyze a new `relax-and-split' (RS) approach for compositions of separable nonconvex nonsmooth functions with linear maps. RS uses a relaxation technique together with partial minimization, and brings classic techniques including direct factorization, matrix decompositions, and fast iterative methods to bear on nonsmooth nonconvex problems. We also extend the approach to trimmed nonconvex-composite formulations; the resulting Trimmed RS (TRS) can fit models while detecting outliers in the data. We then test RS and TRS on a diverse set of applications: (1) phase retrieval, (2) s…
In this paper we study nonconvex and nonsmooth optimization problems with semi-algebraic data, where the variables vector is split into several blocks of variables. The problem consists of one smooth function of the entire variables vector and the sum of nonsmooth functions for each block separately. We analyze an inertial version of the Proximal Alternating Linearized Minimization (PALM) algorithm and prove its global convergence to a critical point of the objective function at hand. We illustrate our theoretical findings by presenting numerical experiments on blind image deconvolution, on sp…
We consider a class of nonsmooth and nonconvex optimization problems over the Stiefel manifold where the objective function is the summation of a nonconvex smooth function and a nonsmooth Lipschitz continuous convex function composed with an linear mapping. We propose three numerical algorithms for solving this problem, by combining smoothing methods and some existing algorithms for smooth optimization over the Stiefel manifold. In particular, we approximate the aforementioned nonsmooth convex function by its Moreau envelope in our smoothing methods, and prove that the Moreau envelope has many…
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.
This paper considers a class of nonsmooth nonconvex-nonconcave min-max problems in machine learning and games. We first provide sufficient conditions for the existence of global minimax points and local minimax points. Next, we establish the first-order and second-order optimality conditions for local minimax points by using directional derivatives. These conditions reduce to smooth min-max problems with Fr{é}chet derivatives. We apply our theoretical results to generative adversarial networks (GANs) in which two neural networks contest with each other in a game. Examples are used to illustrat…
Algorithms for solving nonconvex, nonsmooth, finite-sum optimization problems are proposed and tested. In particular, the algorithms are proposed and tested in the context of an optimization problem formulation arising in semi-supervised machine learning. The common feature of all algorithms is that they employ an incremental quasi-Newton (IQN) strategy, specifically an incremental BFGS (IBFGS) strategy. One applies an IBFGS strategy to the problem directly, whereas the others apply an IBFGS strategy to a difference-of-convex reformulation, smoothed approximation, or (strongly) convex local ap…
We propose a single time-scale stochastic subgradient method for constrained optimization of a composition of several nonsmooth and nonconvex functions. The functions are assumed to be locally Lipschitz and differentiable in a generalized sense. Only stochastic estimates of the values and generalized derivatives of the functions are used. The method is parameter-free. We prove convergence with probability one of the method, by associating with it a system of differential inclusions and devising a nondifferentiable Lyapunov function for this system. For problems with functions having Lipschitz …
We propose BIBPA, a block inertial Bregman proximal algorithm for minimizing the sum of a block relatively smooth function (that is, relatively smooth concerning each block) and block separable nonsmooth nonconvex functions. We prove that the sequence generated by BIBPA subsequentially converges to critical points of the objective under standard assumptions, and globally converges when the objective function is additionally assumed to satisfy the Kurdyka-Łojasiewicz (KŁ) property. We also provide the convergence rate when the objective satisfies the Łojasiewicz inequality. We apply BIBPA to th…
In this paper, we investigate the non-asymptotic stationary convergence behavior of Stochastic Mirror Descent (SMD) for nonconvex optimization. We focus on a general class of nonconvex nonsmooth stochastic optimization problems, in which the objective can be decomposed into a relatively weakly convex function (possibly non-Lipschitz) and a simple non-smooth convex regularizer. We prove that SMD, without the use of mini-batch, is guaranteed to converge to a stationary point in a convergence rate of $ \mathcal{O}(1/\sqrt{t}) $. The efficiency estimate matches with existing results for stochastic…
We present a detailed set of performance comparisons of two state-of-the-art solvers for the application of designing time-delay compensators, an important problem in the field of robust control. Formulating such robust control mechanics as constrained optimization problems often involves objective and constraint functions that are both nonconvex and nonsmooth, both of which present significant challenges to many solvers and their end-users hoping to obtain good solutions to these problems. In our particular engineering task, the main difficulty in the optimization arises in a nonsmooth and no…
This paper extends the SQP-approach of the well-known bundle-Newton method for nonsmooth unconstrained minimization to the nonlinearly constrained case. Instead of using a penalty function or a filter or an improvement function to deal with the presence of constraints, the search direction is determined by solving a convex quadratically constrained quadratic program to obtain good iteration points. Issues arising with this procedure concerning the bundling process as well as in the line search are discussed. Furthermore, global convergence of the method is shown under certain mild assumptions.
This paper studies a structured compound stochastic program (SP) involving multiple expectations coupled by nonconvex and nonsmooth functions. We present a successive convex-programming based sampling algorithm and establish its subsequential convergence. We describe stationarity properties of the limit points for several classes of the compound SP. We further discuss probabilistic stopping rules based on the computable error-bound for the algorithm. We present several risk measure minimization problems that can be formulated as such a compound stochastic program; these include generalized dev…
This paper presents a concrete implementation of the feasible second order bundle algorithm for nonsmooth, nonconvex optimization problems with inequality constraints \cite{HannesPaperB}. It computes the search direction by solving a convex quadratically constrained quadratic program. Furthermore, certain versions of the search direction problem are discussed and the applicability of this approach is justified numerically by using different solvers for the computation of the search direction. Finally, the good performance of the second order bundle algorithm is demonstrated by comparison with …
We propose a proximal algorithm for minimizing objective functions consisting of three summands: the composition of a nonsmooth function with a linear operator, another nonsmooth function, each of the nonsmooth summands depending on an independent block variable, and a smooth function which couples the two block variables. The algorithm is a full splitting method, which means that the nonsmooth functions are processed via their proximal operators, the smooth function via gradient steps, and the linear operator via matrix times vector multiplication. We provide sufficient conditions for the bou…
In this paper we propose a proximal algorithm for minimizing an objective function of two block variables consisting of three terms: 1) a smooth function, 2) a nonsmooth function which is a composition between a strictly increasing, concave, differentiable function and a convex nonsmooth function, and 3) a smooth function which couples the two block variables. We propose a variable metric composite proximal alternating linearized minimization (CPALM) to solve this class of problems. Building on the powerful Kurdyka-Łojasiewicz property, we derive the convergence analysis and establish that eac…
This paper presents a set of complete solutions of a nonconvex variational problem with a double-well potential. Based on the canonical duality-triality theory, the associated nonlinear differential equation with either Dirichlet/Neumann or mixed boundary conditions can be converted into an algebraic equation, which can be solved analytically to obtain all solutions in the dual space. Both global and local extremality conditions are identified by the triality theory. In the application part, typical mechanical models with specificsources and boundary conditions in $\mathbb{R}^2$ are exhibited.
Modern statistical applications often involve minimizing an objective function that may be nonsmooth and/or nonconvex. This paper focuses on a broad Bregman-surrogate algorithm framework including the local linear approximation, mirror descent, iterative thresholding, DC programming and many others as particular instances. The recharacterization via generalized Bregman functions enables us to construct suitable error measures and establish global convergence rates for nonconvex and nonsmooth objectives in possibly high dimensions. For sparse learning problems with a composite objective, under …
The linearized Bregman iterations (LBreI) and its variants have received considerable attention in signal/image processing and compressed sensing. Recently, LBreI has been extended to a larger class of nonconvex functions, along with several theoretical issues left for further investigation. In particular, the gradient Lipschitz continuity assumption precludes its use in many practical applications. In this study, we propose a generalized algorithmic framework to unify LBreI-type methods. Our main discovery is that the gradient Lipschitz continuity assumption can be replaced by a Lipschitz-lik…
