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Many problems of theoretical and practical interest involve finding an optimum over a family of convex functions. For instance, finding the projection on the convex functions in $H^k(\Omega)$, and optimizing functionals arising from some…
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 paper we propose a variant of the random coordinate descent method for solving linearly constrained convex optimization problems with composite objective functions. If the smooth part of the objective function has Lipschitz…
We study differentially private (DP) algorithms for stochastic convex optimization: the problem of minimizing the population loss given i.i.d. samples from a distribution over convex loss functions. A recent work of Bassily et al. (2019)…
We present an algorithm (CoDeFi) which overcomes the curse of dimensionality (CoD) in scientific computations and, especially, in mathematical finance (Fi). Our method applies a broad class of partial differential equations such as…
Classically, the optimal control problem in the presence of an adversary is formulated as a two-player zero-sum differential game or an $H_\infty$ control problem. The solution to these problems can be obtained by solving the…
We propose an unconstrained optimization method based on the well-known primal-dual hybrid gradient (PDHG) algorithm. We first formulate the optimality condition of the unconstrained optimization problem as a saddle point problem. We then…
We study a selection problem for degenerate viscous Hamilton--Jacobi equations with convex Hamiltonians, in which the approximation procedure combines a nonlinear discounted approximation with a small potential perturbation. A key question…
This paper presents a novel convex optimization-based method for finding the globally optimal solutions of a class of mixed-integer non-convex optimal control problems. We consider problems with non-convex constraints that restrict the…
This paper presents Hamilton-Jacobi (HJ) formulations for two classes of two-player zero-sum games: one with a maximum cost value over time, and one with a minimum cost value over time. In the zero-sum game setting, player A minimizes the…
The min-max problem, also known as the saddle point problem, is a class of optimization problems which minimizes and maximizes two subsets of variables simultaneously. This class of problems can be used to formulate a wide range of signal…
We present and analyze a new generalized Frank-Wolfe method for the composite optimization problem $(P):{\min}_{x\in\mathbb{R}^n}\; f(\mathsf{A} x) + h(x)$, where $f$ is a $\theta$-logarithmically-homogeneous self-concordant barrier,…
An elegant characterization of the complexity of constraint satisfaction problems has emerged in the form of the the algebraic dichotomy conjecture of [BKJ00]. Roughly speaking, the characterization asserts that a CSP {\Lambda} is tractable…
Non-dominated sorting is a fundamental combinatorial problem in multiobjective optimization, and is equivalent to the longest chain problem in combinatorics and random growth models for crystals in materials science. In a previous work, we…
We investigate feedback control for infinite horizon optimal control problems for partial differential equations. The method is based on the coupling between Hamilton-Jacobi-Bellman (HJB) equations and model reduction techniques. It is…
In this work, we investigate a stochastic control framework for global optimization over both Euclidean spaces and the Wasserstein space of probability measures, where the objective function may be non-convex and/or non-differentiable. In…
Optimal control of diffusion processes is intimately connected to the problem of solving certain Hamilton-Jacobi-Bellman equations. Building on recent machine learning inspired approaches towards high-dimensional PDEs, we investigate the…
Computing the Jacobian of the solution of an optimization problem is a central problem in machine learning, with applications in hyperparameter optimization, meta-learning, optimization as a layer, and dataset distillation, to name a few.…
Mathematical programs with complementarity constraints are notoriously difficult to solve due to their nonconvexity and lack of constraint qualifications in every feasible point. This work focuses on the subclass of quadratic programs with…
This paper proposes a homogeneous second-order descent framework (HSODF) for nonconvex and convex optimization based on the generalized homogeneous model (GHM). In comparison to the Newton steps, the GHM can be solved by extremal symmetric…