## The cost of nonconvexity in deterministic nonsmooth optimization

 We study the impact of nonconvexity on the complexity of nonsmooth optimization, emphasizing objectives such as piecewise linear functions, which may not be weakly convex. We focus on a dimension-independent analysis, slightly modifying a black-box algorithm of Zhang et al. that approximates an $\epsilon$-stationary point of any directionally differentiable Lipschitz objective using $O(\epsilon^{-4})$ calls … Read more

## A Nonmonontone Accelerated Proximal Gradient Method with Variable Stepsize Strategy for Nonsmooth and Nonconvex Minimization Problems

We propose a new nonmonontone accelerated proximal gradient method with variable stepsize strategy for minimizing the sum of a nonsmooth function with a smooth one in the nonconvex setting. In this algorithm, the objective function value be allowed to increase discontinuously, but is decreasing from the overall point of view. The variable stepsize strategy don’t … Read more

## The Landscape of the Proximal Point Method for Nonconvex-Nonconcave Minimax Optimization

Minimax optimization has become a central tool for modern machine learning with applications in generative adversarial networks, robust optimization, reinforcement learning, etc. These applications are often nonconvex-nonconcave, but the existing theory is unable to identify and deal with the fundamental difficulties posed by nonconvex-nonconcave structures. In this paper, we study the classic proximal point method … Read more

## Adversarial Classification via Distributional Robustness with Wasserstein Ambiguity

We study a model for adversarial classification based on distributionally robust chance constraints. We show that under Wasserstein ambiguity, the model aims to minimize the conditional value-at-risk of the distance to misclassification, and we explore links to adversarial classification models proposed earlier and to maximum-margin classifiers. We also provide a reformulation of the distributionally robust … Read more

## On Generalized Surrogate Duality in Mixed-Integer Nonlinear Programming

The most important ingredient for solving mixed-integer nonlinear programs (MINLPs) to global epsilon-optimality with spatial branch and bound is a tight, computationally tractable relaxation. Due to both theoretical and practical considerations, relaxations of MINLPs are usually required to be convex. Nonetheless, current optimization solver can often successfully handle a moderate presence of nonconvexities, which opens … Read more

## Inexact proximal stochastic second-order methods for nonconvex composite optimization

In this paper, we propose a framework of Inexact Proximal Stochastic Second-order (IPSS) methods for solving nonconvex optimization problems, whose objective function consists of an average of finitely many, possibly weakly, smooth functions and a convex but possibly nons- mooth function. At each iteration, IPSS inexactly solves a proximal subproblem constructed by using some positive … Read more

## On the Convergence to Stationary Points of Deterministic and Randomized Feasible Descent Directions Methods

This paper studies the class of nonsmooth nonconvex problems in which the difference between a continuously differentiable function and a convex nonsmooth function is minimized over linear constraints. Our goal is to attain a point satisfying the stationarity necessary optimality condition, defined as the lack of feasible descent directions. Although elementary in smooth optimization, this … Read more

## Finding Second-Order Stationary Points in Constrained Minimization: A Feasible Direction Approach

This paper introduces a method for computing points satisfying the second-order necessary optimality conditions in constrained nonconvex minimization. The method comprises two independent steps corresponding to the first and second order conditions. The first-order step is a generic closed map algorithm which can be chosen from a variety of first-order algorithms, making it The second-order … Read more

## A simple Newton method for local nonsmooth optimization

Superlinear convergence has been an elusive goal for black-box nonsmooth optimization. Even in the convex case, the subgradient method is very slow, and while some cutting plane algorithms, including traditional bundle methods, are popular in practice, local convergence is still sluggish. Faster variants depend either on problem structure or on analyses that elide sequences of … Read more

## Using two-dimensional Projections for Stronger Separation and Propagation of Bilinear Terms

One of the most fundamental ingredients in mixed-integer nonlinear programming solvers is the well- known McCormick relaxation for a product of two variables x and y over a box-constrained domain. The starting point of this paper is the fact that the convex hull of the graph of xy can be much tighter when computed over … Read more