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In this article we approach a class of stochastic reachability problems with state constraints from an optimal control perspective. Preceding approaches to solving these reachability problems are either confined to the deterministic setting…
This paper presents a partial differential equation framework for deep residual neural networks and for the associated learning problem. This is done by carrying out the continuum limits of neural networks with respect to width and depth.…
We study spaces $\mathcal{CV}^{k}(\Omega,E)$ of $k$-times continuously partially differentiable functions on an open set $\Omega\subset\mathbb{R}^{d}$ with values in a locally convex Hausdorff space $E$. The space…
Let $USC^*_p(X)$ be the topological space of real upper semicontinuous bounded functions defined on $X$ with the subspace topology of the product topology on ${}^X\mathbb{R}$. $\tilde\Phi^{\uparrow},\tilde\Psi^{\uparrow}$ are the sets of…
Motivated by the dynamic assortment offerings and item pricings occurring in e-commerce, we study a general problem of allocating finite inventories to heterogeneous customers arriving sequentially. We analyze this problem under the…
We discuss the problem of optimizing the distance function from a given point, subject to polynomial constraints. A key algebraic invariant that governs its complexity is the Euclidean distance degree, which pertains to first-order…
Optimal values and solutions of empirical approximations of stochastic optimization problems can be viewed as statistical estimators of their true values. From this perspective, it is important to understand the asymptotic behavior of these…
Given a strictly positive measure, we characterize inner semicontinuous solid convex-valued mappings for which continuous functions which are selections almost everywhere are selections. This class contains continuous mappings as well as…
The goal of Feature Selection - comprising filter, wrapper, and embedded approaches - is to find the optimal feature subset for designated downstream tasks. Nevertheless, current feature selection methods are limited by: 1) the selection…
We consider entropically regularized, semi-discrete versions of variational problems on the set of probability measures involving optimal transport as well as other terms. We prove that the solutions can be characterized by well-posed…
We show that hereditarily indecomposable spaces can be characterized by a special instance of the Intermediate Value Theorem in their rings of continuous functions.
We define a natural topology on the collection of (equivalence classes up to scaling of) locally finite measures on a homogeneous space and prove that in this topology, pushforwards of certain infinite volume orbits equidistribute in the…
Variable division and optimization (D\&O) is a frequently utilized algorithm design paradigm in Evolutionary Algorithms (EAs). A D\&O EA divides a variable into partial variables and then optimize them respectively. A complicated problem is…
In this manuscript we consider a class optimal control problem for stochastic differential delay equations. First, we rewrite the problem in a suitable infinite-dimensional Hilbert space. Then, using the dynamic programming approach, we…
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 investigate the computation of the gradient of the value function in parametric convex optimization problems. We derive general expression for the gradient of the value function in terms of the cost function, constraints and Lagrange…
We consider the problem of allocating indivisible goods in a way that is fair, using one of the leading market mechanisms in economics: the competitive equilibrium from equal incomes. Focusing on two major classes of valuations, namely…
We examine robustness of exponential dichotomies of boundary value problems for general linear first-order one-dimensional hyperbolic systems. The boundary conditions are supposed to be of types ensuring smoothing solutions in finite time,…
The optimal control of a mechanical system is of crucial importance in many realms. Typical examples are the determination of a time-minimal path in vehicle dynamics, a minimal energy trajectory in space mission design, or optimal motion…
Under certain initial conditions, we prove the existence of set-valued selectors of univariate compact-valued multifunctions of bounded (Jordan) variation when the notion of variation is defined taking into account only the Pompeiu…