New Bregman proximal type algorithms for solving DC optimization problems

Difference of Convex (DC) optimization problems have objective functions that are differences between two convex functions. Representative ways of solving these problems are the proximal DC algorithms, which require that the convex part of the objective function have L-smoothness. In this article, we propose the Bregman Proximal DC Algorithm (BPDCA) for solving large-scale DC optimization … Read more

Factorization of completely positive matrices using iterative projected gradient steps

We aim to factorize a completely positive matrix by using an optimization approach which consists in the minimization of a nonconvex smooth function over a convex and compact set. To solve this problem we propose a projected gradient algorithm with parameters that take into account the effects of relaxation and inertia. Both projection and gradient … Read more

A Proximal Quasi-Newton Trust-Region Method for Nonsmooth Regularized Optimization

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 apossibly 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 … Read more

Further developments of methods for traversing regions of non-convexity in optimization problems

This paper continues to address one of its author’s obsession with the well- known problem of dealing with non-convexity during the minimization of a nonlinear function f(x) by Newton-like methods. It builds on some proposals made by the present authors in “A Comparison of methods for traversing regions of non-convexity in optimization problems”. (Numerical Algorithms … Read more

A Computational Study of Perspective Cuts

The benefits of cutting planes based on the perspective function are well known for many specific classes of mixed-integer nonlinear programs with on/off structures. However, we are not aware of any empirical studies that evaluate their applicability and computational impact over large, heterogeneous test sets in general-purpose solvers. This paper provides a detailed computational study … Read more

A Framework of Inertial Alternating Direction Method of Multipliers for Non-Convex Non-Smooth Optimization

In this paper, we propose an algorithmic framework dubbed inertial alternating direction methods of multipliers (iADMM), for solving a class of nonconvex nonsmooth multiblock composite optimization problems with linear constraints. Our framework employs the general minimization-majorization (MM) principle to update each block of variables so as to not only unify the convergence analysis of previous … Read more

Cutting Plane Generation Through Sparse Principal Component Analysis

Quadratically-constrained quadratic programs (QCQPs) are optimization models whose remarkable expressiveness has made them a cornerstone of methodological research for nonconvex optimization problems. However, modern methods to solve a general QCQP fail to scale, encountering computational challenges even with just a few hundred variables. Specifically, a semidefinite programming (SDP) relaxation is typically employed, which provides strong … Read more

Implicit Regularization of Sub-Gradient Method in Robust Matrix Recovery: Don’t be Afraid of Outliers

It is well-known that simple short-sighted algorithms, such as gradient descent, generalize well in the over-parameterized learning tasks, due to their implicit regularization. However, it is unknown whether the implicit regularization of these algorithms can be extended to robust learning tasks, where a subset of samples may be grossly corrupted with noise. In this work, … Read more

A Matrix-Free Trust-Region Newton Algorithm for Convex-Constrained Optimization

We describe a matrix-free trust-region algorithm for solving convex-constrained optimization problems that uses the spectral projected gradient method to compute trial steps. To project onto the intersection of the feasible set and the trust region, we reformulate and solve the dual projection problem as a one-dimensional root finding problem. We demonstrate our algorithm’s performance on … Read more

Stochastic Dual Dynamic Programming And Its Variants – A Review

We provide a tutorial-type review on stochastic dual dynamic programming (SDDP), as one of the state-of-the-art solution methods for large-scale multistage stochastic programs. Since introduced about 30 years ago for solving large-scale multistage stochastic linear programming problems in energy planning, SDDP has been applied to practical problems from several fields and is enriched by various … Read more