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We consider a variational convex relaxation of a class of optimal partitioning and multiclass labeling problems, which has recently proven quite successful and can be seen as a continuous analogue of Linear Programming (LP) relaxation…
We consider integer-restricted optimal control of systems governed by abstract semilinear evolution equations. This includes the problem of optimal control design for certain distributed parameter systems endowed with multiple actuators,…
Finding efficient and provable methods to solve non-convex optimization problems is an outstanding challenge in machine learning and optimization theory. A popular approach used to tackle non-convex problems is to use convex relaxation…
Convexification is a core technique in global polynomial optimization. Currently, there are two main approaches competing in theory and practice: the approach of nonlinear programming and the approach based on positivity certificates from…
Optimal power flow (OPF) is a key problem in power system operations. OPF problems that use the nonlinear AC power flow equations to accurately model the network physics have inherent challenges associated with non-convexity. To address…
A convex envelope for the problem of finding the best approximation to a given matrix with a prescribed rank is constructed. This convex envelope allows the usage of traditional optimization techniques when additional constraints are added…
We consider the constrained assortment optimization problem under the mixed multinomial logit model. Even moderately sized instances of this problem are challenging to solve directly using standard mixed-integer linear optimization…
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…
Artificial neural networks (ANNs) are used in various applications for data-driven black-box modeling and subsequent optimization. Herein, we present an efficient method for deterministic global optimization of ANN embedded optimization…
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…
Minimax optimization has become a central tool in machine learning with applications in robust optimization, reinforcement learning, GANs, etc. These applications are often nonconvex-nonconcave, but the existing theory is unable to identify…
This article investigates the numerical approximation of shape optimization problems with PDE constraint on classes of convex domains. The convexity constraint provides a compactness property which implies well posedness of the problem.…
Minimax optimization has been central in addressing various applications in machine learning, game theory, and control theory. Prior literature has thus far mainly focused on studying such problems in the continuous domain, e.g.,…
Convex hulls of monomials have been widely studied in the literature, and monomial convexifications are implemented in global optimization software for relaxing polynomials. However, there has been no study of the error in the global…
This thesis focuses on the intersection of mathematical and computational optimization and quantum information. Main contributions are open-source software code: A hybrid approach mixing "traditional" nonconvex and convex methods can make…
Many high dimensional sparse learning problems are formulated as nonconvex optimization. A popular approach to solve these nonconvex optimization problems is through convex relaxations such as linear and semidefinite programming. In this…
Mixed-integer nonlinear optimization encompasses a broad class of problems that present both theoretical and computational challenges. We propose a new type of method to solve these problems based on a branch-and-bound algorithm with convex…
We propose an approach to trajectory optimization for piecewise polynomial systems based on the recently proposed graphs of convex sets framework. We instantiate the framework with a convex relaxation of optimal control based on occupation…
In this work, we develop an adaptive, multivariate partitioning algorithm for solving mixed-integer nonlinear programs (MINLP) with multi-linear terms to global optimality. This iterative algorithm primarily exploits the advantages of…
The affine inverse eigenvalue problem consists of identifying a real symmetric matrix with a prescribed set of eigenvalues in an affine space. Due to its ubiquity in applications, various instances of the problem have been widely studied in…