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This paper studies the complexity for finding approximate stationary points of nonconvex-strongly-concave (NC-SC) smooth minimax problems, in both general and averaged smooth finite-sum settings. We establish nontrivial lower complexity…

In this paper, we consider a finite-dimensional optimization problem minimizing a continuous objective on a compact domain subject to a multi-dimensional constraint function. For the latter, we assume the availability of a global Lipschitz…

Optimization models with non-convex constraints arise in many tasks in machine learning, e.g., learning with fairness constraints or Neyman-Pearson classification with non-convex loss. Although many efficient methods have been developed…

We analyze stochastic algorithms for optimizing nonconvex, nonsmooth finite-sum problems, where the nonconvex part is smooth and the nonsmooth part is convex. Surprisingly, unlike the smooth case, our knowledge of this fundamental problem…

We study minimization of a structured objective function, being the sum of a smooth function and a composition of a weakly convex function with a linear operator. Applications include image reconstruction problems with regularizers that…

We study the min-max optimization problem where each function contributing to the max operation is strongly-convex and smooth with bounded gradient in the search domain. By smoothing the max operator, we show the ability to achieve an…

Nonlinear convex problems arise in various areas of applied mathematics and engineering. Classical techniques such as the relaxed proximal point algorithm (PPA) and the prediction correction (PC) method were proposed for linearly…

We propose a stochastic optimization method for the minimization of the sum of three convex functions, one of which has Lipschitz continuous gradient as well as restricted strong convexity. Our approach is most suitable in the setting where…

This paper develops the proximal method of multipliers for a class of nonsmooth convex optimization. The method generates a sequence of minimization problems (subproblems). We show that the sequence of approximations to the solutions of the…

It is well-known that the lower bound of iteration complexity for solving nonconvex unconstrained optimization problems is $\Omega(1/\epsilon^2)$, which can be achieved by standard gradient descent algorithm when the objective function is…

This paper considers the problem of minimizing a convex expectation function over a closed convex set, coupled with a set of inequality convex expectation constraints. We present a new stochastic approximation type algorithm, namely the…

This paper studies consensus-based decentralized stochastic optimization for minimizing possibly non-convex expected objectives with convex non-smooth regularizers and nonlinear functional inequality constraints. We reformulate the…

This paper presents a piecewise convexification method to approximate the whole approximate optimal solution set of non-convex optimization problems with box constraints. In the process of box division, we first classify the sub-boxes and…

In this paper, we propose new accelerated methods for smooth convex optimization, called contracting proximal methods. At every step of these methods, we need to minimize a contracted version of the objective function augmented by a…

This paper presents a novel method for reformulating non-differentiable collision avoidance constraints into smooth nonlinear constraints using strong duality of convex optimization. We focus on a controlled object whose goal is to avoid…

In this two-part paper, we propose a general algorithmic framework for the minimization of a nonconvex smooth function subject to nonconvex smooth constraints. The algorithm solves a sequence of (separable) strongly convex problems and…

This article proposes a new discrete framework for approximating solutions to shape optimization problems under convexity constraints. The numerical method, based on the support function or the gauge function, is guaranteed to generate…

In this paper, we propose an inexact block coordinate descent algorithm for large-scale nonsmooth nonconvex optimization problems. At each iteration, a particular block variable is selected and updated by inexactly solving the original…

We consider the fundamental problem in non-convex optimization of efficiently reaching a stationary point. In contrast to the convex case, in the long history of this basic problem, the only known theoretical results on first-order…

In this paper, we consider a nonconvex optimization problem with nonlinear equality constraints. We assume that both, the objective function and the functional constraints are locally smooth. For solving this problem, we propose a…