Nonsmooth Nonconvex Problems Research Articles

The modified second APG method for a class of nonconvex nonsmooth problems

The modified second APG method for a class of nonconvex nonsmooth problems

A proximal subgradient algorithm with extrapolation for structured nonconvex nonsmooth problems

In this paper, we consider a class of structured nonconvex nonsmooth optimization problems, in which the objective function is formed by the sum of a possibly nonsmooth nonconvex function and a differentiable function with Lipschitz continuous gradient, subtracted by a weakly convex function. This general framework allows us to tackle problems involving nonconvex loss functions and problems with specific nonconvex constraints, and it has many applications such as signal recovery, compressed sensing, and optimal power flow distribution. We develop a proximal subgradient algorithm with extrapolation for solving these problems with guaranteed subsequential convergence to a stationary point. The convergence of the whole sequence generated by our algorithm is also established under the widely used Kurdyka–Łojasiewicz property. To illustrate the promising numerical performance of the proposed algorithm, we conduct numerical experiments on two important nonconvex models. These include a compressed sensing problem with a nonconvex regularization and an optimal power flow problem with distributed energy resources.

On an iteratively reweighted linesearch based algorithm for nonconvex composite optimization

In this paper we propose a new algorithm for solving a class of nonsmooth nonconvex problems, which is obtained by combining the iteratively reweighted scheme with a finite number of forward–backward iterations based on a linesearch procedure. The new method overcomes some limitations of linesearch forward–backward methods, since it can be applied also to minimize functions containing terms that are both nonsmooth and nonconvex. Moreover, the combined scheme can take advantage of acceleration techniques consisting in suitable selection rules for the algorithm parameters. We develop the convergence analysis of the new method within the framework of the Kurdyka–Łojasiewicz property. Finally, we present the results of a numerical experience on microscopy image super resolution, showing that the performances of our method are comparable or superior to those of other algorithms designed for this specific application.

A generalized inertial proximal alternating linearized minimization method for nonconvex nonsmooth problems

We propose a general inertial version of the proximal alternating linearized minimization (PALM) (denoted by NiPALM) for a class of nonconvex and nonsmooth minimization problems, whose objective function is the sum of a smooth function of the entire variables and two nonsmooth functions of each variable. NiPALM is general in the sense that it contains the popular PALM, the inertial PALM (iPALM) and Gauss-Seidel type inertial PALM (GiPALM) as special cases. Under mild assumptions, namely, the underlying functions satisfy the Kurdyka-Łojasiewicz (KL) property and some suitable conditions on the parameters, we prove that each bounded sequence generated by NiPALM globally converges to a critical point. We also apply NiPALM to nonnegative matrix factorization, sparse principal component analysis, and weighted low-rank matrix restoration problems. Comparing results with those by PALM, iPALM, and GiPALM demonstrate the robustness and effectiveness of the proposed algorithm.

Some accelerated alternating proximal gradient algorithms for a class of nonconvex nonsmooth problems

Some accelerated alternating proximal gradient algorithms for a class of nonconvex nonsmooth problems

Distributed Stochastic Inertial-Accelerated Methods with Delayed Derivatives for Nonconvex Problems

Stochastic gradient methods (SGMs) are predominant approaches for solving stochastic optimization. On smooth nonconvex problems, a few acceleration techniques have been applied to improve the convergence rate of SGMs. However, little exploration has been made on applying a certain acceleration technique to a stochastic subgradient method (SsGM) for nonsmooth nonconvex problems. In addition, few efforts have been made to analyze an (accelerated) SsGM with delayed derivatives. The information delay naturally happens in a distributed system, where computing workers do not coordinate with each other. In this paper, we propose an inertial proximal SsGM for solving nonsmooth nonconvex stochastic optimization problems. The proposed method can have guaranteed convergence even with delayed derivative information in a distributed environment. Convergence rate results are established for three classes of nonconvex problems: weakly convex nonsmooth problems with a convex regularizer, composite nonconvex problems with a nonsmooth convex regularizer, and smooth nonconvex problems. For each problem class, the convergence rate is $O(1/K^{\frac{1}{2}})$ in the expected value of the gradient norm square, for $K$ iterations. In a distributed environment, the convergence rate of the proposed method will be slowed down by the information delay. Nevertheless, the slow-down effect will decay with the number of iterations for the latter two problem classes. We test the proposed method on three applications. The numerical results clearly demonstrate the advantages of using the inertial-based acceleration. Furthermore, we observe higher parallelization speed-up in asynchronous updates over the synchronous counterpart, though the former uses delayed derivatives. Our source code is available at https://github.com/RPI-OPT/Inertial-SsGM.

Robust projection twin support vector machine via DC programming

In this paper, we propose a robust projection twin support vector machine (RPTSVM), where a new truncated -norm distance measure is applied to the with-class scatter to boost the robustness of the classifier when encountering a large number of outliers. In order to further improve the robustness of the model, chance constraints are employed to specify the lower bound of the probability that the distance from the projected samples to the projection of the other class centre is at least one. RPTSVM considers a pair of non-convex non-smooth problems with chance constraints. To solve these difficult problems, a newly designed method based on the difference of convex functions (DC) programs approach is presented. Extensive experiments on artificial datasets and benchmark datasets demonstrate the robustness and feasibility of our proposed RPTSVM.

Адаптивные методы градиентного типа для задач оптимизации с относительной точностью и острым минимумом

In this paper, we consider gradient-type methods for convex positively homogeneous optimization problems with relative accuracy. An analogue of the accelerated universal gradient-type method for positively homogeneous optimization problems with relative accuracy is investigated. The second approach is related to subgradient methods with B. T. Polyak stepsize. Result on the linear convergence rate for some methods of this type with adaptive step adjustment is obtained for some class of non-smooth problems. Some generalization to a special class of non-convex non-smooth problems is also considered.

Reliable shot identification for complex event detection via visual-semantic embedding

Multimedia event detection is the task of detecting a specific event of interest in an user-generated video on websites. The most fundamental challenge facing this task lies in the enormously varying quality of the video as well as the high-level semantic abstraction of event inherently. In this paper, we decompose the video into several segments and intuitively model the task of complex event detection as a multiple instance learning problem by representing each video as a “bag” of segments in which each segment is referred to as an instance. Instead of treating the instances equally, we associate each instance with a reliability variable to indicate its importance and then select reliable instances for training. To measure the reliability of the varying instances precisely, we propose a visual-semantic guided loss by exploiting low-level feature from visual information together with instance-event similarity based high-level semantic feature. Motivated by curriculum learning, we introduce a negative elastic-net regularization term to start training the classifier with instances of high reliability and gradually taking the instances with relatively low reliability into consideration. An alternative optimization algorithm is developed to solve the proposed challenging non-convex non-smooth problem. Experimental results on standard datasets, i.e., TRECVID MEDTest 2013 and TRECVID MEDTest 2014, demonstrate the effectiveness and superiority of the proposed method to the baseline algorithms.

Smoothing neural network for [formula omitted] regularized optimization problem with general convex constraints

In this paper, we propose a neural network modeled by a differential inclusion to solve a class of discontinuous and nonconvex sparse regression problems with general convex constraints, whose objective function is the sum of a convex but not necessarily differentiable loss function and L0 regularization. We construct a smoothing relaxation function of L0 regularization and propose a neural network to solve the considered problem. We prove that the solution of proposed neural network with any initial point satisfying linear equality constraints is global existent, bounded and reaches the feasible region in finite time and remains there thereafter. Moreover, the solution of proposed neural network is its slow solution and any accumulation point of it is a Clarke stationary point of the brought forward nonconvex smoothing approximation problem. In the box-constrained case, all accumulation points of the solution own a unified lower bound property and have a common support set. Except for a special case, any accumulation point of the solution is a local minimizer of the considered problem. In particular, the proposed neural network has a simple structure than most existing neural networks for solving the locally Lipschitz continuous but nonsmooth nonconvex problems. Finally, we give some numerical experiments to show the efficiency of proposed neural network.

A Block Inertial Bregman Proximal Algorithm for Nonsmooth Nonconvex Problems with Application to Symmetric Nonnegative Matrix Tri-Factorization

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 show that the cluster points of the sequence generated by BIBPA are critical points of the objective under standard assumptions, and this sequence converges globally when a regularization of the objective function satisfies the Kurdyka-Łojasiewicz (KL) property. We also provide the convergence rate when a regularization of the objective function satisfies the Łojasiewicz inequality. We apply BIBPA to the symmetric nonnegative matrix tri-factorization (SymTriNMF) problem, where we propose kernel functions for SymTriNMF and provide closed-form solutions for subproblems of BIBPA.

On the superiority of PGMs to PDCAs in nonsmooth nonconvex sparse regression

This paper conducts a comparative study of proximal gradient methods (PGMs) and proximal DC algorithms (PDCAs) for sparse regression problems which can be cast as Difference-of-two-Convex-functions (DC) optimization problems. It has been shown that for DC optimization problems, both General Iterative Shrinkage and Thresholding algorithm (GIST), a modified version of PGM, and PDCA converge to critical points. Recently some enhanced versions of PDCAs are shown to converge to d-stationary points, which are stronger necessary condition for local optimality than critical points. In this paper we claim that without any modification, PGMs converge to a d-stationary point not only to DC problems but also to more general nonsmooth nonconvex problems under some technical assumptions. While the convergence to d-stationary points is known for the case where the step size is small enough, the finding of this paper is valid also for extended versions such as GIST and its alternating optimization version, which is to be developed in this paper. Numerical results show that among several algorithms in the two categories, modified versions of PGM perform best among those not only in solution quality but also in computation time.

Block-coordinate and incremental aggregated proximal gradient methods for nonsmooth nonconvex problems

This paper analyzes block-coordinate proximal gradient methods for minimizing the sum of a separable smooth function and a (nonseparable) nonsmooth function, both of which are allowed to be nonconvex. The main tool in our analysis is the forward-backward envelope, which serves as a particularly suitable continuous and real-valued Lyapunov function. Global and linear convergence results are established when the cost function satisfies the Kurdyka–Łojasiewicz property without imposing convexity requirements on the smooth function. Two prominent special cases of the investigated setting are regularized finite sum minimization and the sharing problem; in particular, an immediate byproduct of our analysis leads to novel convergence results and rates for the popular Finito/MISO algorithm in the nonsmooth and nonconvex setting with very general sampling strategies.

Distributed Stochastic Consensus Optimization With Momentum for Nonconvex Nonsmooth Problems

While many distributed optimization algorithms have been proposed for solving smooth or convex problems over the networks, few of them can handle non-convex and non-smooth problems. Based on a proximal primal-dual approach, this paper presents a new (stochastic) distributed algorithm with Nesterov momentum for accelerated optimization of non-convex and non-smooth problems. Theoretically, we show that the proposed algorithm can achieve an $\epsilon$-stationary solution under a constant step size with $\mathcal{O}(1/\epsilon^2)$ computation complexity and $\mathcal{O}(1/\epsilon)$ communication complexity. When compared to the existing gradient tracking based methods, the proposed algorithm has the same order of computation complexity but lower order of communication complexity. To the best of our knowledge, the presented result is the first stochastic algorithm with the $\mathcal{O}(1/\epsilon)$ communication complexity for non-convex and non-smooth problems. Numerical experiments for a distributed non-convex regression problem and a deep neural network based classification problem are presented to illustrate the effectiveness of the proposed algorithms.

Kurdyka–Łojasiewicz Property of Zero-Norm Composite Functions

This paper focuses on a class of zero-norm composite optimization problems. For this class of nonconvex nonsmooth problems, we establish the Kurdyka–Łojasiewicz property of exponent being a half for its objective function under a suitable assumption and provide some examples to illustrate that such an assumption is not very restricted which, in particular, involve the zero-norm regularized or constrained piecewise linear–quadratic function, the zero-norm regularized or constrained logistic regression function, the zero-norm regularized or constrained quadratic function over a sphere.

An inexact multiple proximal bundle algorithm for nonsmooth nonconvex multiobjective optimization problems

For a class of nonsmooth nonconvex multiobjective problems, we develop an inexact multiple proximal bundle method. In our approach instead of scalarization, we find descent direction for every objective function separately by utilizing the inexact proximal bundle method. Then we attempt to find a common descent direction for all objective functions. We study the effect of the inexactness of the objective and subgradient values on the new proposed method and obtain the reasonable convergence properties. We further consider a class of difficult nonsmooth nonconvex problems, made even more difficult by inserting the inexactness in the available information. At the end, to demonstrate the efficiency of the proposed algorithm, some encouraging numerical experiments are provided.

Relax-and-split method for nonconvex inverse problems

We develop and analyze a new ‘relax-and-split’ (RS) approach for inverse problems modeled using nonsmooth nonconvex optimization formulations. 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 robustify any such inverse problem through trimming, a mechanism that robustifies inverse problems to measurement outliers. We then show practical performance of RS and trimmed RS (TRS) on a diverse set of problems, including: (1) phase retrieval, (2) semi-supervised classification, (3) stochastic shortest path problems, and (4) nonconvex clustering. RS/TRS are easy to implement, competitive with existing methods, and show promising results on difficult inverse problems with nonsmooth and nonconvex features.

Reduced subgradient bundle method for linearly constrained non-smooth non-convex problems

ABSTRACT In this paper, we propose an algorithm for solving linearly constrained non-smooth, non-convex optimization problems. The objective functions in these problems are, in general, upper semidifferentiable locally Lipschitz functions. The method is based on the idea of adapting, to the non-smooth setting, the variant of the reduced gradient algorithm proposed by Luenberger, and on bundle techniques which are aimed at building an approximation of the subdifferential. It may be thought of as an extension of reduced gradient methods for dealing with both non-smoothness and non-convexity of the objective function. Under the non-degeneracy assumption, the termination of the proposed algorithm at a stationary point is proved. Numerical results and comparisons with some existing methods are reported to show the efficiency of our algorithm.

On the Convergence to Stationary Points of Deterministic and Randomized Feasible Descent Directions Methods

This paper studies the class of nonsmooth nonconvex problems in which the difference between a continuously differentiable function and a convex nonsmooth function is minimized over linear constraints. Our goal is to attain a point satisfying the stationarity necessary optimality condition, defined as the lack of feasible descent directions. Although elementary in smooth optimization, this condition is nontrivial when the objective function is nonsmooth, and, correspondingly, there are very few methods that obtain stationary points in such settings. We prove that stationarity in our model can be characterized by a finite number of directions and develop two methods, one deterministic and one random, that use these directions to obtain stationary points. Numerical experiments illustrate the benefit of obtaining a stationary point and the advantage of using the random method to do so.

A Scaled Conjugate Gradient Method Based on New BFGS Secant Equation with Modified Nonmonotone Line Search

In this paper, we provide and analyze a new scaled conjugate gradient method and its performance, based on the modified secant equation of the Broyden-Fletcher-Goldfarb-Shanno (BFGS) method and on a new modified nonmonotone line search technique. The method incorporates the modified BFGS secant equation in an effort to include the second order information of the objective function. The new secant equation has both gradient and function value information, and its update formula inherits the positive definiteness of Hessian approximation for general convex function. In order to improve the likelihood of finding a global optimal solution, we introduce a new modified nonmonotone line search technique. It is shown that, for nonsmooth convex problems, the proposed algorithm is globally convergent. Numerical results show that this new scaled conjugate gradient algorithm is promising and efficient for solving not only convex but also some large scale nonsmooth nonconvex problems in the sense of the Dolan-Moré performance profiles.