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Convex optimization problems with staged structure appear in several contexts, including optimal control, verification of deep neural networks, and isotonic regression. Off-the-shelf solvers can solve these problems but may scale poorly. We…

For a quadratic matrix polynomial dependent on parameters and a given tolerance $\epsilon > 0$, the minimization of the $\epsilon$-pseudospectral abscissa over the set of permissible parameter values is discussed, with applications in…

Semidefinite Optimization has become a standard technique in the landscape of Mathematical Programming that has many applications in finite dimensional Quantum Information Theory. This paper presents a way for finite-dimensional relaxations…

This paper addresses the problem of solving a class of nonlinear optimal control problems (OCP) with infinite-dimensional linear state constraints involving Riesz-spectral operators. Each instance within this class has time/control…

In this paper, we propose an unconstrained framework for eigenvalue problems in both discrete and continuous settings. We begin our discussion to solve a generalized eigenvalue problem $A{\bf x} = \lambda B{\bf x}$ with two $N\times N$ real…

We propose a general theory for studying the \xl{landscape} of nonconvex \xl{optimization} with underlying symmetric structures \tz{for a class of machine learning problems (e.g., low-rank matrix factorization, phase retrieval, and deep…

We investigate how to solve smooth matrix optimization problems with general linear inequality constraints on the eigenvalues of a symmetric matrix. We present solution methods to obtain exact global minima for linear objective functions,…

We consider semidefinite programs (SDPs) with equality constraints. The variable to be optimized is a positive semidefinite matrix $X$ of size $n$. Following the Burer--Monteiro approach, we optimize a factor $Y$ of size $n \times p$…

This work focuses on minimizing the eigenvalue of a noncommutative polynomial subject to a finite number of noncommutative polynomial inequality constraints. Based on the Helton-McCullough Positivstellensatz, the noncommutative analog of…

We study binary optimization problems of the form \( \min_{x\in\{-1,1\}^n} f(Ax-b) \) with possibly nonsmooth loss \(f\). Following the lifted rank-one semidefinite programming (SDP) approach\cite{qian2023matrix}, we develop a…

We construct a convergent family of outer approximations for the problem of optimizing polynomial functions over convex bodies subject to polynomial constraints. This is achieved by generalizing the polarization hierarchy, which has…

We consider a hierarchy of upper approximations for the minimization of a polynomial $f$ over a compact set $K \subseteq \mathbb{R}^n$ proposed recently by Lasserre (arXiv:1907.097784, 2019). This hierarchy relies on using the push-forward…

We investigate exact semidefinite programming (SDP) relaxations for the problem of minimizing a nonconvex quadratic objective function over a feasible region defined by both finitely and infinitely many nonconvex quadratic inequality…

It is well-known that by adding integrality constraints to the semidefinite programming (SDP) relaxation of the max-cut problem, the resulting integer semidefinite program is an exact formulation of the problem. In this paper we show…

The multi-objective optimization is to optimize several objective functions over a common feasible set. Since the objectives usually do not share a common optimizer, people often consider (weakly) Pareto points. This paper studies…

We investigate an alternative solution method to the joint signal-beamformer optimization problem considered by Setlur and Rangaswamy[1]. First, we directly demonstrate that the problem, which minimizes the received noise, interference, and…

We introduce a generic technique to obtain linear relaxations of semidefinite programs with provable guarantees based on the commutativity of the constraint and the objective matrices. We study conditions under which the optimal value of…

A semidefinite program (SDP) is a particular kind of convex optimization problem with applications in operations research, combinatorial optimization, quantum information science, and beyond. In this work, we propose variational quantum…

This document introduces a strategy to solve linear optimization problems. The strategy is based on the bounding condition each constraint produces on each one of the problem's dimension. The solution of a linear optimization problem is…

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…