Related papers: A Continuous Nonlinear Optimization Perspective on…
In this thesis we develop a novel framework to study smooth and strongly convex optimization algorithms, both deterministic and stochastic. Focusing on quadratic functions we are able to examine optimization algorithms as a recursive…
Combinatorial optimization problems are pervasive across science and industry. Modern deep learning tools are poised to solve these problems at unprecedented scales, but a unifying framework that incorporates insights from statistical…
This work presents a unified framework that combines global approximations with locally built models to handle challenging nonconvex and nonsmooth composite optimization problems, including cases involving extended real-valued functions. We…
We study efficient optimization of the Hamiltonians of multi-species spherical spin glasses. Our results characterize the maximum value attained by algorithms that are suitably Lipschitz with respect to the disorder through a variational…
In this work, we aim to explore connections between dynamical systems techniques and combinatorial optimization problems. In particular, we construct heuristic approaches for the traveling salesman problem (TSP) based on embedding the…
We propose the first general and scalable framework to design certifiable algorithms for robust geometric perception in the presence of outliers. Our first contribution is to show that estimation using common robust costs, such as truncated…
We consider the Scenario Convex Program (SCP) for two classes of optimization problems that are not tractable in general: Robust Convex Programs (RCPs) and Chance-Constrained Programs (CCPs). We establish a probabilistic bridge from the…
In this paper, we consider a well-known sparse optimization problem that aims to find a sparse solution of a possibly noisy underdetermined system of linear equations. Mathematically, it can be modeled in a unified manner by minimizing…
Mirror descent (MD) is a powerful first-order optimization technique that subsumes several optimization algorithms including gradient descent (GD). In this work, we develop a semi-definite programming (SDP) framework to analyze the…
In this paper, we study the numerical method for stochastic optimal control problems (SOCPs). By reducing the optimal control problem to the discrete case, we derive a discrete stochastic maximum principle (SMP). With the help of this SMP,…
In this paper we investigate how standard nonlinear programming algorithms can be used to solve constrained optimization problems in a distributed manner. The optimization setup consists of a set of agents interacting through a…
This paper investigates numerical methods for solving stochastic linear quadratic (SLQ) optimal control problems governed by stochastic partial differential equations (SPDEs). Two distinct approaches, the open-loop and closed-loop ones, are…
In this paper we study the convex problem of optimizing the sum of a smooth function and a compactly supported non-smooth term with a specific separable form. We analyze the block version of the generalized conditional gradient method when…
We develop a class of algorithms, as variants of the stochastically controlled stochastic gradient (SCSG) methods (Lei and Jordan, 2016), for the smooth non-convex finite-sum optimization problem. Assuming the smoothness of each component,…
We present an efficient framework for solving algebraically-constrained global non-convex polynomial optimization problems over subsets of the hypercube. We prove the existence of an equivalent nonlinear reformulation of such problems that…
Inverse problems are paramount in Science and Engineering. In this paper, we consider the setup of Statistical Inverse Problem (SIP) and demonstrate how Stochastic Gradient Descent (SGD) algorithms can be used in the linear SIP setting. We…
Semi-Infinite Programming (SIP) has emerged as a powerful framework for modeling problems with infinite constraints, however, its theoretical development in the context of nonconvex and large-scale optimization remains limited. In this…
In this article, we present an extension of the formulation recently developed by the authors (A Framework for Data-Driven Computational Mechanics Based on Nonlinear Optimization, arXiv:1910.12736 [math.NA]) to the structural dynamics…
Max-min bilinear optimization models, where one agent maximizes and an adversary minimizes a common bilinear objective, serve as canonical saddle-point formulations in optimization theory. They capture, among others, two-player zero-sum…
We study the conditions under which the convex relaxation of a mixed-integer linear programming formulation for ordered optimization problems, where sorting is part of the decision process, yields integral optimal solutions. Thereby solving…