Global Convergence of Sub-gradient Method for Robust Matrix Recovery: Small Initialization, Noisy Measurements, and Over-parameterization

In this work, we study the performance of sub-gradient method (SubGM) on a natural nonconvex and nonsmooth formulation of low-rank matrix recovery with $\ell_1$-loss, where the goal is to recover a low-rank matrix from a limited number of measurements, a subset of which may be grossly corrupted with noise. We study a scenario where the … Read more

A deterministic solver for multiobjective mixed-integer convex and nonconvex optimization

This paper proposes a general framework for solving multiobjective nonconvex optimization problems, i.e., optimization problems in which multiple objective functions have to be optimized simultaneously. Thereby, the nonconvexity might come from the objective or constraint functions, or from integrality conditions for some of the variables. In particular, multiobjective mixed-integer convex and nonconvex optimization problems are … Read more

A nonlinear conjugate gradient method with complexity guarantees and its application to nonconvex regression

Nonlinear conjugate gradients are among the most popular techniques for solving continuous optimization problems. Although these schemes have long been studied from a global convergence standpoint, their worst-case complexity properties have yet to be fully understood, especially in the nonconvex setting. In particular, it is unclear whether nonlinear conjugate gradient methods possess better guarantees than … Read more

A Globally Convergent Distributed Jacobi Scheme for Block-Structured Nonconvex Constrained Optimization Problems

Motivated by the increasing availability of high-performance parallel computing, we design a distributed parallel algorithm for linearly-coupled block-structured nonconvex constrained optimization problems. Our algorithm performs Jacobi-type proximal updates of the augmented Lagrangian function, requiring only local solutions of separable block nonlinear programming (NLP) problems. We provide a cheap and explicitly computable Lyapunov function that allows … Read more

An adaptive regularization algorithm for unconstrained optimization with inexact function and derivatives values

An adaptive regularization algorithm for unconstrained nonconvex optimization is proposed that is capable of handling inexact objective-function and derivative values, and also of providing approximate minimizer of arbitrary order. In comparison with a similar algorithm proposed in Cartis, Gould, Toint (2022), its distinguishing feature is that it is based on controlling the relative error between … Read more

Quadratic Regularization Methods with Finite-Difference Gradient Approximations

This paper presents two quadratic regularization methods with finite-difference gradient approximations for smooth unconstrained optimization problems. One method is based on forward finite-difference gradients, while the other is based on central finite-difference gradients. In both methods, the accuracy of the gradient approximations and the regularization parameter in the quadratic models are jointly adjusted using a … Read more

Global Complexity Bound of a Proximal ADMM for Linearly-Constrained Nonseperable Nonconvex Composite Programming

This paper proposes and analyzes a dampened proximal alternating direction method of multipliers (DP.ADMM) for solving linearly-constrained nonconvex optimization problems where the smooth part of the objective function is nonseparable. Each iteration of DP.ADMM consists of: (ii) a sequence of partial proximal augmented Lagrangian (AL) updates, (ii) an under-relaxed Lagrange multiplier update, and (iii) a … Read more

Nonlinear matrix recovery using optimization on the Grassmann manifold

We investigate the problem of recovering a partially observed high-rank matrix whose columns obey a nonlinear structure such as a union of subspaces, an algebraic variety or grouped in clusters. The recovery problem is formulated as the rank minimization of a nonlinear feature map applied to the original matrix, which is then further approximated by … Read more

An abstract convergence framework with application to inertial inexact forward-backward methods

In this paper we introduce a novel abstract descent scheme suited for the minimization of proper and lower semicontinuous functions. The proposed abstract scheme generalizes a set of properties that are crucial for the convergence of several first-order methods designed for nonsmooth nonconvex optimization problems. Such properties guarantee the convergence of the full sequence of … Read more

On obtaining the convex hull of quadratic inequalities via aggregations

A classical approach for obtaining valid inequalities for a set involves weighted aggregations of the inequalities that describe such set. When the set is described by linear inequalities, thanks to the Farkas lemma, we know that every valid inequality can be obtained using aggregations. When the inequalities describing the set are two quadratics, Yildiran showed … Read more