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This work proposes an implementable proximal-type method for a broad class of optimization problems involving nonsmooth and nonconvex objective and constraint functions. In contrast to existing methods that rely on an ad hoc model…

Majorization-minimization algorithms consist of successively minimizing a sequence of upper bounds of the objective function. These upper bounds are tight at the current estimate, and each iteration monotonically drives the objective…

Min-max optimization problems (i.e., min-max games) have been attracting a great deal of attention because of their applicability to a wide range of machine learning problems. Although significant progress has been made recently, the…

In this paper, we show how to transform any optimization problem that arises from fitting a machine learning model into one that (1) detects and removes contaminated data from the training set while (2) simultaneously fitting the trimmed…

Batch policy optimization considers leveraging existing data for policy construction before interacting with an environment. Although interest in this problem has grown significantly in recent years, its theoretical foundations remain…

Adversarial training, a special case of multi-objective optimization, is an increasingly prevalent machine learning technique: some of its most notable applications include GAN-based generative modeling and self-play techniques in…

Trace norm regularization is a widely used approach for learning low rank matrices. A standard optimization strategy is based on formulating the problem as one of low rank matrix factorization which, however, leads to a non-convex problem.…

We present successive convexification, a real-time-capable solution method for nonconvex trajectory optimization, with continuous-time constraint satisfaction and guaranteed convergence, that only requires first-order information. The…

Majorization-minimization algorithms consist of iteratively minimizing a majorizing surrogate of an objective function. Because of its simplicity and its wide applicability, this principle has been very popular in statistics and in signal…

Many machine learning problems can be formulated as minimax problems such as Generative Adversarial Networks (GANs), AUC maximization and robust estimation, to mention but a few. A substantial amount of studies are devoted to studying the…

This paper presents the first experimental evaluation of four previously untested modifications of Unbounded Best-First Minimax algorithm. This algorithm explores the game tree by iteratively expanding the most promising sequences of…

Making the gradients small is a fundamental optimization problem that has eluded unifying and simple convergence arguments in first-order optimization, so far primarily reserved for other convergence criteria, such as reducing the…

We propose a framework for modeling and solving low-rank optimization problems to certifiable optimality. We introduce symmetric projection matrices that satisfy $Y^2=Y$, the matrix analog of binary variables that satisfy $z^2=z$, to model…

In this paper, an inexact proximal-point penalty method is studied for constrained optimization problems, where the objective function is non-convex, and the constraint functions can also be non-convex. The proposed method approximately…

In this work, we reveal a rich combinatorial structure underlying exact minimax optimal algorithms for classical nonexpansive fixed-point problems. This viewpoint unifies all extremal optimal methods and provides a systematic and practical…

We consider regression problems with binary weights. Such optimization problems are ubiquitous in quantized learning models and digital communication systems. A natural approach is to optimize the corresponding Lagrangian using variants of…

We consider joint optimization and learning problems arising in real-time decision systems. While most existing work focuses primarily on convex, revenue-based objectives, we extend this line of research to multi-objective formulations. In…

The study of nonconvex minimax games has gained significant momentum in machine learning and decision science communities due to their fundamental connections to adversarial training scenarios. This work develops a primal-dual alternating…

The TREX is a recently introduced method for performing sparse high-dimensional regression. Despite its statistical promise as an alternative to the lasso, square-root lasso, and scaled lasso, the TREX is computationally challenging in that…

Bilevel optimization is one of the fundamental problems in machine learning and optimization. Recent theoretical developments in bilevel optimization focus on finding the first-order stationary points for nonconvex-strongly-convex cases. In…