Related papers: Simple Approximations of Semialgebraic Sets and th…
This paper investigates the minimization of the expectation of piecewise polynomial loss functions over Wasserstein balls. This optimization problem often appears as a key sub-problem of distributionally robust optimization problems. We…
We study quasi-convex optimization problems, where only a subset of the constraints can be sampled, and yet one would like a probabilistic guarantee on the obtained solution with respect to the initial (unknown) optimization problem. Even…
The matrix logarithm, when applied to Hermitian positive definite matrices, is concave with respect to the positive semidefinite order. This operator concavity property leads to numerous concavity and convexity results for other matrix…
An algorithm which computes a solution of a set optimization problem is provided. The graph of the objective map is assumed to be given by finitely many linear inequalities. A solution is understood to be a set of points in the domain…
In this paper we propose a scalarization proximal point method to solve multiobjective unconstrained minimization problems with locally Lipschitz and quasiconvex vector functions. We prove, under natural assumptions, that the sequence…
Semidefinite programming (SDP) is a fundamental class of convex optimization problems with diverse applications in mathematics, engineering, machine learning, and related disciplines. This paper investigates the application of the…
We consider estimating a compact set from finite data by approximating the support function of that set via sublinear regression. Support functions uniquely characterize a compact set up to closure of convexification, and are sublinear…
In this work, we propose a new local optimization method to solve a class of nonconvex semidefinite programming (SDP) problems. The basic idea is to approximate the feasible set of the nonconvex SDP problem by inner positive semidefinite…
Bilevel optimization problems are receiving increasing attention in machine learning as they provide a natural framework for hyperparameter optimization and meta-learning. A key step to tackle these problems is the efficient computation of…
We construct least squares formulations of PDEs with inhomogeneous essential boundary conditions, where boundary residuals are not measured in unpractical fractional Sobolev norms, but which formulations nevertheless are shown to yield a…
Low rank regularization, in essence, involves introducing a low rank or approximately low rank assumption for matrix we aim to learn, which has achieved great success in many fields including machine learning, data mining and computer…
This paper addresses problems on the structural design of control systems taking explicitly into consideration the possible application to large-scale systems. We provide an efficient and unified framework to solve the following major…
We propose and discuss a new computational method for the numerical approximation of reachable sets for nonlinear control systems. It is based on the support vector machine algorithm and represents the set approximation as a sublevel set of…
We consider the problem of computing the maximal invariant set of discrete-time linear systems subject to a class of non-convex constraints that admit quadratic relaxations. These non-convex constraints include semialgebraic sets and other…
This paper presents a selected tour through the theory and applications of lifts of convex sets. A lift of a convex set is a higher-dimensional convex set that projects onto the original set. Many convex sets have lifts that are…
A polynomial matrix inequality is a formula asserting that a polynomial matrix is positive semidefinite. Polynomial matrix optimization concerns minimizing the smallest eigenvalue of a symmetric polynomial matrix subject to a tuple of…
Validating and controlling safety-critical systems in uncertain environments necessitates probabilistic reachable sets of future state evolutions. The existing methods of computing probabilistic reachable sets normally assume that…
We study the role of the constraint set in determining the solution to low-rank, positive semidefinite (PSD) matrix sensing problems. The setting we consider involves rank-one sensing matrices: In particular, given a set of rank-one…
For finite-dimensional problems, stochastic approximation methods have long been used to solve stochastic optimization problems. Their application to infinite-dimensional problems is less understood, particularly for nonconvex objectives.…
In this paper, we present a geometric approach for computing controlled invariant sets for hybrid control systems. While the problem is well studied in the ellipsoidal case, this family is quite conservative for constrained or switched…