Survey Descent: A Multipoint Generalization of Gradient Descent for Nonsmooth Optimization

For strongly convex objectives that are smooth, the classical theory of gradient descent ensures linear convergence relative to the number of gradient evaluations. An analogous nonsmooth theory is challenging. Even when the objective is smooth at every iterate, the corresponding local models are unstable and the number of cutting planes invoked by traditional remedies is … 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

Minimax optimization for handling range and setup uncertainties in proton therapy

Purpose: Intensity modulated proton therapy (IMPT) is sensitive to errors, mainly due to high stopping power dependency and steep beam dose gradients. Conventional margins are often insufficient to ensure robustness of treatment plans. In this article, a method is developed that takes the uncertainties into account during the plan optimization. Methods: Dose contributions for a … Read more

Cutting-Set Methods for Robust Convex Optimization with Pessimizing Oracles

We consider a general worst-case robust convex optimization problem, with arbitrary dependence on the uncertain parameters, which are assumed to lie in some given set of possible values. We describe a general method for solving such a problem, which alternates between optimization and worst-case analysis. With exact worst-case analysis, the method is shown to converge … Read more