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We provide a new proof of the linear convergence of the alternating direction method of multipliers (ADMM) when one of the objective terms is strongly convex. Our proof is based on a framework for analyzing optimization algorithms…

Modern statistical applications often involve minimizing an objective function that may be nonsmooth and/or nonconvex. This paper focuses on a broad Bregman-surrogate algorithm framework including the local linear approximation, mirror…

We consider the problem of approximating the solution of variational problems subject to the constraint that the admissible functions must be convex. This problem is at the interface between convex analysis, convex optimization, variational…

This paper introduces a general framework for solving constrained convex quaternion optimization problems in the quaternion domain. To soundly derive these new results, the proposed approach leverages the recently developed generalized…

Constrained non-convex optimization is fundamentally challenging, as global solutions are generally intractable and constraint qualifications may not hold. However, in many applications, including safe policy optimization in control and…

Non-convex optimization is a critical tool in advancing machine learning, especially for complex models like deep neural networks and support vector machines. Despite challenges such as multiple local minima and saddle points, non-convex…

We propose a new framework for deriving screening rules for convex optimization problems. Our approach covers a large class of constrained and penalized optimization formulations, and works in two steps. First, given any approximate point,…

Non-probabilistic convex model utilizes a convex set to quantify the uncertainty domain of uncertain-but-bounded parameters, which is very effective for structural uncertainty analysis with limited or poor-quality experimental data. To…

Lossless Convexification (LCvx) is a modeling approach that transforms a class of nonconvex optimal control problems, where nonconvexity primarily arises from control constraints, into convex problems through convex relaxations. These…

In this paper, we study the local convergence of the standard ADMM scheme for a class of nonconvex composite problems arising from modern imaging and machine learning models. This problem is constrained by a closed convex set, while its…

Outer approximation methods have long been employed to tackle a variety of optimization problems, including linear programming, in the 1960s, and continue to be effective for solving variational inequalities, general convex problems, as…

In this paper, we propose a new Fully Composite Formulation of convex optimization problems. It includes, as a particular case, the problems with functional constraints, max-type minimization problems, and problems of Composite…

This paper introduces an abstract framework for randomized subspace correction methods for convex optimization, which unifies and generalizes a broad class of existing algorithms, including domain decomposition, multigrid, and block…

The nonlinear conjugate gradient methods are known to be an effective approach for standard unconstrained optimization problems especially for large-scale problems. This paper proposes a proximal nonlinear conjugate gradient method, which…

Convex optimization is a well-established research area with applications in almost all fields. Over the decades, multiple approaches have been proposed to solve convex programs. The development of interior-point methods allowed solving a…

The aim of this paper is to study the characteristics of a general method to produce a new approximation sequence from a given one, by using suitable convex combinations.

Non-convex optimization problems are pervasive across mathematical programming, engineering design, and scientific computing, often posing intractable challenges for traditional solvers due to their complex objective functions and…

The alternating direction method with multipliers (ADMM) has been one of most powerful and successful methods for solving various composite problems. The convergence of the conventional ADMM (i.e., 2-block) for convex objective functions…

Motivated by recent increased interest in optimization algorithms for non-convex optimization in application to training deep neural networks and other optimization problems in data analysis, we give an overview of recent theoretical…

This paper proposes and develops new Newton-type methods to solve structured nonconvex and nonsmooth optimization problems with justifying their fast local and global convergence by means of advanced tools of variational analysis and…