On inexact stochastic splitting methods for a class of nonconvex composite optimization problems with relative error

A solution algorithm for non-convex mixed integer optimization problems with only few continuous variables

A neurodynamic optimization technique based on overestimator and underestimator functions for solving a class of non-convex optimization problems

In this paper, the classical exact absolute value penalty function method is used for solving a new class of nonconvex nonsmooth optimization problems. Nonconvex nondifferentiable optimization problems with both inequality and equality constraints are considered here, in which not all functions constituting them have the fundamental property of convex functions and most classes of generalized convex functions—namely, a stationary point of such a function is its global minimum. It is proved for such nonconvex optimization problems that there exists a finite threshold of penalty parameters equal to the largest absolute value of a Lagrange multiplier such that, for every penalty parameter exceeding this lower bound, there is the equivalence between an optimal solution in the original constrained minimization problem with ‐invex functions and a minimizer in its associated penalized optimization problem with the exact l1 penalty function. Further, under nondifferentiable ‐invexity assumptions, a characterization of a saddle point of the Lagrange function, defined for the considered constrained optimization problem in terms of minimizers of its associated exact penalized optimization problem with the exact l1 penalty function, is presented.

We develop new fast and efficient algorithms for designing single or multiple unimodular waveforms with good auto- and cross-correlation or weighted correlation properties, which are highly desired in radar and communication systems. The waveform design is based on the minimization of the integrated sidelobe level (ISL) and weighted ISL (WISL) of waveforms. As the corresponding problems can quickly grow to a large scale with increasing the code length and the number of waveforms, the main issue turns to be the development of fast large-scale optimization techniques. The difficulty is also that the corresponding optimization problems are nonconvex, but the required accuracy is high. Therefore, we formulate the ISL and WISL minimization problems as nonconvex quartic optimization problems in frequency domain, and then simplify them into quadratic problems via majorization-minimization technique, which is one of the basic techniques for addressing large-scale and/or nonconvex optimization problems. While designing our fast algorithms, we explore and use the inherent algebraic structures in objective functions to rewrite them into quartic forms, and in the case of WISL minimization, to derive additionally an alternative quartic form that allows us to apply the quartic-quadratic transformation. Our algorithms are applicable to large-scale unimodular waveform design problems as they are proved to have lower or comparable computational burden (analyzed theoretically) and faster convergence speed (confirmed by comprehensive simulations) than the state-of-the-art algorithms. In addition, the waveforms designed by our algorithms demonstrate better correlation properties compared to their counterparts.

Trajectory optimization for asteroid landing with two-phase free final time

On Nonconvex Optimization Problems with D.C. Equality and Inequality Constraints

Optimal insurance contract specification in the upstream sector of the oil and gas industry

A vast majority of machine learning algorithms train their models and perform inference by solving optimization problems. In order to capture the learning and prediction problems accurately, structural constraints such as sparsity or low rank are frequently imposed or else the objective itself is designed to be a non-convex function. This is especially true of algorithms that operate in high-dimensional spaces or that train non-linear models such as tensor models and deep networks. The freedom to express the learning problem as a non-convex optimization problem gives immense modeling power to the algorithm designer, but often such problems are NP-hard to solve. A popular workaround to this has been to relax non-convex problems to convex ones and use traditional methods to solve the (convex) relaxed optimization problems. However this approach may be lossy and nevertheless presents significant challenges for large scale optimization. On the other hand, direct approaches to non-convex optimization have met with resounding success in several domains and remain the methods of choice for the practitioner, as they frequently outperform relaxation-based techniques - popular heuristics include projected gradient descent and alternating minimization. However, these are often poorly understood in terms of their convergence and other properties. This monograph presents a selection of recent advances that bridge a long-standing gap in our understanding of these heuristics. We hope that an insight into the inner workings of these methods will allow the reader to appreciate the unique marriage of task structure and generative models that allow these heuristic techniques to (provably) succeed. The monograph will lead the reader through several widely used nonconvex optimization techniques, as well as applications thereof. The goal of this monograph is to both, introduce the rich literature in this area, as well as equip the reader with the tools and techniques needed to analyze these simple procedures for non-convex problems.

In this article, we introduce gradient descent and stochastic gradient descent for solving large‐scale statistical and machine learning problems. We introduce in detail how to derive gradient descent and stochastic gradient descent algorithms, briefly highlight their convergence properties, and show how to apply them to solve statistical and machine learning problems. In the end, we also discuss some open problems when applying stochastic gradient descent to train large‐scale machine learning models.

We study non-convex optimization problems which appear in machine learning and statistics. Recently, convexity became an important feature to develop high-performance learning algorithms. On the other hand, non-convex optimization is also required in machine learning. In this paper, especially we deal with non-convex optimization problems on the Stiefel manifold. The Stiefel manifold consists of rectangular matrices, and many important problems in machine learning can be expressed in this framework. We compare some non-linear optimization methods which are applicable to machine learning algorithms.

In this chapter, we will study an important class of convex optimization problems whose objective function is given by the summation of many components. With wide applications in machine learning and distributed optimization, these problems can be viewed either as deterministic optimization problems with a special finite-sum structure or stochastic optimization problems with a discrete distribution. Accordingly, we will study two typical classes of randomized algorithms for solving them. While the first class incorporates random block decomposition into the dual space of the primal–dual methods for deterministic convex optimization, the second one employs variance reduction techniques into stochastic gradient descent methods for stochastic optimization.

The symmetric alternating direction method of multipliers is an efficient algorithm, which updates the Lagrange multiplier twice at each iteration and the variables are treated in a symmetric manner. Considering that the convergence range of the parameters plays an important role in the implementation of the algorithm. In this paper, we analyze the convergence rate of the symmetric ADMM with a more relaxed parameter range for solving the two block nonconvex separable optimization problem under the assumption that the generated sequence is bounded. Two cases are considered. In the first case, both components of the objective function are nonconvex, we prove the convergence of the augmented Lagrangian function sequence, and establish the $ O(1/\sqrt{k}) $ worst-case complexity measured by the difference of two consecutive iterations. In the second case, one component of the objective function is convex and the error bound condition is assumed, then we can prove that the iterative sequence converges locally to a KKT point in a R-linear rate; and an auxiliary sequence converges in a Q-linear rate. Furthermore, a practical inexact symmetric ADMM with relative error criteria is proposed, and the associated convergence analysis is established under the same conditions.

ABSTRACTIn this paper, we propose a generic algorithmic framework for stochastic proximal quasi-Newton (SPQN) methods to solve non-convex composite optimization problems. Stochastic second-order information is explored to construct proximal subproblem. Under mild conditions we show the non-asympotic convergence of the proposed algorithm to stationary point of original problems and analyse its computational complexity. Besides, we extend the proximal form of Polyak-Łojasiewicz (PL) inequality to constrained settings and obtain the constrained proximal PL (CP-PL) inequality. Under CP-PL inequality linear convergence rate of the proposed algorithm is achieved. Moreover, we propose a modified self-scaling symmetric rank one incorporated in the framework for SPQN method, which is called stochastic symmetric rank one method. Finally, we report some numerical experiments to reveal the effectiveness of the proposed algorithm.

Dictionary learning is a prototypical machine learning problem with broad applications throughout computer vision and signal processing. Given a set of training data, the goal is to learn a dictionary of prototype patterns that provide a sparse (parsimonious) representation of the data. Such a dictionary then enables a variety of problems to be solved by exploiting sparse structure; example problems include noise removal, super-resolution, deblurring, imputing missing data, and classification. Like many problems in machine learning, dictionary learning can be formulated as an optimization problem. Unfortunately, this optimization problem is non-convex, making it potentially difficult to solve to global optimality. Additional questions arise in how to deal with multi-modal data and distributed data, such as might be collected by multiple satellites, UAVs, etc. This presentation will summarize our team’s research on theory and algorithms for solving non-convex matrix-valued optimization problems such as dictionary learning, particularly in multi-modal, distributed, and streaming contexts. We welcome the opportunity to discuss further transitioning this technology for Naval applications.

In this work, we introduce a novel stochastic proximal alternating linearized minimization algorithm [J. Bolte, S. Sabach, and M. Teboulle, Math. Program., 146 (2014), pp. 459--494] for solving a class of nonsmooth and nonconvex optimization problems. Large-scale imaging problems are becoming increasingly prevalent due to the advances in data acquisition and computational capabilities. Motivated by the success of stochastic optimization methods, we propose a stochastic variant of proximal alternating linearized minimization. We provide global convergence guarantees, demonstrating that our proposed method with variance-reduced stochastic gradient estimators, such as SAGA [A. Defazio, F. Bach, and S. Lacoste-Julien, Advances in Neural Information Processing Systems, 2014, pp. 1646--1654] and SARAH [L. M. Nguyen, J. Liu, K. Scheinberg, and M. Takáĉ, Proceedings of the 34th International Conference on Machine Learning, PMLR 70, 2017, pp. 2613--2621], achieves state-of-the-art oracle complexities. We also demonstrate the efficacy of our algorithm via several numerical examples including sparse nonnegative matrix factorization, sparse principal component analysis, and blind image-deconvolution.