The paper proposes and justifies a new algorithm of the proximal Newton type to solve a broad class of nonsmooth composite convex optimization problems without strong convexity assumptions. Based on advanced notions and techniques of variational analysis, we establish implementable results on the global convergence of the proposed algorithm as well as its local convergence with superlinear and quadratic rates. For certain structured problems, the obtained local convergence conditions do not require the local Lipschitz continuity of the corresponding Hessian mappings that is a crucial assumption used in the literature to ensure a superlinear convergence of other algorithms of the proximal Newton type. The conducted numerical experiments of solving the $l_1$ regularized logistic regression model illustrate the possibility of applying the proposed algorithm to deal with practically important problems.
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.
In this paper we present a variant of the proximal forward-backward splitting iteration for solving nonsmooth optimization problems in Hilbert spaces, when the objective function is the sum of two nondifferentiable convex functions. The proposed iteration, which will be called Proximal Subgradient Splitting Method, extends the classical subgradient iteration for important classes of problems, exploiting the additive structure of the objective function. The weak convergence of the generated sequence was established using different stepsizes and under suitable assumptions. Moreover, we analyze t…
In the lines of our approach in \cite{Ouorou2019}, where we exploit Nesterov fast gradient concept \cite{Nesterov1983} to the Moreau-Yosida regularization of a convex function, we devise new proximal algorithms for nonsmooth convex optimization. These algorithms need no bundling mechanism to update the stability center while preserving the complexity estimates established in \cite{Ouorou2019}. We report some preliminary computational results on some academic test problem to give a first estimate of their performance in relation with the classical proximal bundle algorithm.
In this paper, two distributed multi-proximal primal-dual algorithms are proposed to deal with a class of distributed nonsmooth resource allocation problems. In these problems, the global cost function is the summation of local convex and nonsmooth cost functions, each of which consists of one twice differentiable function and multiple nonsmooth functions. Communication graphs of underling multi-agent systems are directed and strongly connected but not necessarily weighted-balanced. The multi-proximal splitting is designed to deal with the difficulty caused by the unproximable property of the …
We propose new proximal bundle algorithms for minimizing a nonsmooth convex function. These algorithms are derived from the application of Nesterov fast gradient methods for smooth convex minimization to the so-called Moreau-Yosida regularization $F_μ$ of $f$ w.r.t. some $μ>0$. Since the exact values and gradients of $F_μ$ are difficult to evaluate, we use approximate proximal points thanks to a bundle strategy to get implementable algorithms. One of these algorithms appears as an implementable version of a special case of inertial proximal algorithm. We give their complexity estimates in term…
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 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 first study nonsmooth steepest descent method for nonsmooth functions defined on 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 Lipschitzian continuity. As one application, the established algorithm is used to search the minimizer of lower semicontinuous and convex functions on finite-dimensional space. The convergent theorem, as one extension and improvement of the existing converging result for twice continuously differentiable convex f…
This technical note studies a class of distributed nonsmooth convex consensus optimization problem. The cost function is a summation of local cost functions which are convex but nonsmooth. Each of the local cost functions consists of a twice differentiable convex function and two lower semi-continuous convex functions. We call this problem as single-smooth plus double-nonsmooth (SSDN) problem. Under mild conditions, we propose a distributed double proximal primal-dual optimization algorithm. The double proximal operator is designed to deal with the difficulty caused by the unproximable propert…
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…
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…
Minimax optimization problems are an important class of optimization problems arising from modern machine learning and traditional research areas. While there have been many numerical algorithms for solving smooth convex-concave minimax problems, numerical algorithms for nonsmooth convex-concave minimax problems are very rare. This paper aims to develop an efficient numerical algorithm for a structured nonsmooth convex-concave minimax problem. A majorized semi-proximal alternating coordinate method (mspACM) is proposed, in which a majorized quadratic convex-concave function is adopted for appr…
Asymptotic Analysis of Gradient Mapping-type Stationarity Measure for the Sum of Nonconvex Nonsmooth Functions and Applications to Proximal Gradient-type Algorithms
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 algori…
The problem of minimizing the sum of nonsmooth, convex objective functions defined on a real Hilbert space over the intersection of fixed point sets of nonexpansive mappings, onto which the projections cannot be efficiently computed, is considered. The use of proximal point algorithms that use the proximity operators of the objective functions and incremental optimization techniques is proposed for solving the problem. With the focus on fixed point approximation techniques, two algorithms are devised for solving the problem. One blends an incremental subgradient method, which is a useful algor…
Escaping Saddle Points for Nonsmooth Weakly Convex Functions via Perturbed Proximal Algorithms
Department of Electrical and Computer Engineering, University of California, Davis（加州大学戴维斯分校电气与计算机工程系）；Simon Business School, University of Rochester（罗切斯特大学西蒙商学院）
Many applications in machine learning or signal processing involve nonsmooth optimization problems. This nonsmoothness brings a low-dimensional structure to the optimal solutions. In this paper, we propose a randomized proximal gradient method harnessing this underlying structure. We introduce two key components: i) a random subspace proximal gradient algorithm; ii) an identification-based sampling of the subspaces. Their interplay brings a significant performance improvement on typical learning problems in terms of dimensions explored.
The paper proposes and develops new globally convergent algorithms of the generalized damped Newton type for solving important classes of nonsmooth optimization problems. These algorithms are based on the theory and calculations of second-order subdifferentials of nonsmooth functions with employing the machinery of second-order variational analysis and generalized differentiation. First we develop a globally superlinearly convergent damped Newton-type algorithm for the class of continuously differentiable functions with Lipschitzian gradients, which are nonsmooth of second order. Then we desig…
A new framework for analyzing Fejer convergent algorithms is presented. Using this framework we define a very general class of Fejer convergent algorithms and establish its convergence properties. We also introduce a new definition of approximations of resolvents which preserve some useful features of the exact resolvent, and use this concept to present an unifying view of the Forward-Backward splitting method, Tseng's Modified Forward-Backward splitting method and Korpelevich's method. We show that methods based on families of approximate resolvents fall within the aforementioned class of Fej…
Proximal methods are known to identify the underlying substructure of nonsmooth optimization problems. Even more, in many interesting situations, the output of a proximity operator comes with its structure at no additional cost, and convergence is improved once it matches the structure of a minimizer. However, it is impossible in general to know whether the current structure is final or not; such highly valuable information has to be exploited adaptively. To do so, we place ourselves in the case where a proximal gradient method can identify manifolds of differentiability of the nonsmooth objec…
We investigate the asymptotic properties of the trajectories generated by a second-order dynamical system of proximal-gradient type stated in connection with the minimization of the sum of a nonsmooth convex and a (possibly nonconvex) smooth function. The convergence of the generated trajectory to a critical point of the objective is ensured provided a regularization of the objective function satisfies the Kurdyka-Łojasiewicz property. We also provide convergence rates for the trajectory formulated in terms of the Łojasiewicz exponent.
We develop a trust-region method for minimizing the sum of a smooth term $f$ and a nonsmooth term $h$), both of which can be nonconvex. Each iteration of our method minimizes a possibly nonconvex model of $f + h$ in a trust region. The model coincides with $f + h$ in value and subdifferential at the center. We establish global convergence to a first-order stationary point when $f$ satisfies a smoothness condition that holds, in particular, when it has Lipschitz-continuous gradient, and $h$ is proper and lower semi-continuous. The model of $h$ is required to be proper, lower-semi-continuous and…
In this paper, we present a semi-proximal alternating direction method of multipliers (ADMM) for solving $3$-block separable convex minimization problems with the second block in the objective being a strongly convex function and one coupled linear equation constraint. By choosing the semi-proximal terms properly, we establish the global convergence of the proposed semi-proximal ADMM for the step-length $τ\in (0, (1+\sqrt{5})/2)$ and the penalty parameter $σ\in (0, +\infty)$. In particular, if $σ>0$ is smaller than a certain threshold and the first and third linear operators in the linear equa…
Two distributed algorithms are described that enable all users connected over a network to cooperatively solve the problem of minimizing the sum of all users' objective functions over the intersection of all users' constraint sets, where each user has its own private nonsmooth convex objective function and closed convex constraint set, which is the intersection of a number of simple, closed convex sets. One algorithm enables each user to adjust its estimate by using a proximity operator of its objective function and the metric projection onto one set randomly selected from the simple, closed c…
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…
