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We study computable topological spaces and semicomputable and computable sets in these spaces. In particular, we investigate conditions under which semicomputable sets are computable. We prove that a semicomputable compact manifold $M$ is…
In many naturally occurring optimization problems one needs to ensure that the definition of the optimization problem lends itself to solutions that are tractable to compute. In cases where exact solutions cannot be computed tractably, it…
Convex nonsmooth optimization problems, whose solutions live in very high dimensional spaces, have become ubiquitous. To solve them, the class of first-order algorithms known as proximal splitting algorithms is particularly adequate: they…
We consider the problem of designing an optimal quantum detector to minimize the probability of a detection error when distinguishing between a collection of quantum states, represented by a set of density operators. We show that the design…
The mathematical modeling of numerous real-world applications results in hierarchical optimization problems with two decision makers where at least one of them has to solve an optimal control problem of ordinary or partial differential…
Among the major difficulties that one may encounter when estimating parameters in a nonlinear regression model are the nonuniqueness of the estimator, its instability with respect to small perturbations of the observations and the presence…
In this paper, we study some problems with continuously differentiable quasiconvex objective function. We prove that exactly one of the following two alternatives holds: (I) the gradient of the objective function is different from zero over…
A (unit norm) frame is scalable if its vectors can be rescaled so as to result into a tight frame. Tight frames can be considered optimally conditioned because the condition number of their frame operators is unity. In this paper we…
Semidefinite programs (SDPs) are a class of optimisation problems that find application in numerous areas of physics, engineering and mathematics. Semidefinite programming is particularly suited to problems in quantum physics and quantum…
We consider the problem of maximizing a convex function over a closed convex set in a real Hilbert space. For linear functions, we show that a single orthogonal projection suffices to obtain an approximate solution. For continuous convex…
We propose a distributionally robust principal agent formulation, which generalizes some common variants of worst-case and Bayesian principal agent problems. We construct a theoretical framework to certify whether any surjective contract…
In this workshop, we present a compact but rigorous introduction to second-order optimality conditions for mathematical programs with equilibrium constraints (MPECs). We start from the classical nonlinear programming template, then explain…
Submodular functions, as well as the sub-class of decomposable submodular functions, and their optimization appear in a wide range of applications in machine learning, recommendation systems, and welfare maximization. However, optimization…
The paper puts forward sufficient conditions for local controllability of a control dynamical system. The results obtained are meaningful in the case when the linear approximation to this system is not completely controllable. As a…
The paper addresses the study and applications of a broad class of extended-real-valued functions, known as optimal value or marginal functions, which are frequently appeared in variational analysis, parametric optimization, and a variety…
Many problems of theoretical and practical interest involve finding a convex or concave function. For instance, optimization problems such as finding the projection on the convex functions in $H^k(\Omega)$, or some problems in economics. In…
Submodularity is a discrete domain functional property that can be interpreted as mimicking the role of the well-known convexity/concavity properties in the continuous domain. Submodular functions exhibit strong structure that lead to…
Sampling from Gibbs distributions and computing their log-partition function are fundamental tasks in statistics, machine learning, and statistical physics. While efficient algorithms are known for log-concave densities, the worst-case…
Optimal transport is the problem of designing a joint distribution for two random variables with fixed marginals. In virtually the entire literature on this topic, the objective is to minimize expected cost. This paper is the first to study…
The problem of finding the minimizer of a sum of convex functions is central to the field of optimization. Thus, it is of interest to understand how that minimizer is related to the properties of the individual functions in the sum. In this…