Related papers: Curse of dimensionality reduction in max-plus base…
It is well known that time dependent Hamilton-Jacobi-Isaacs partial differential equations (HJ PDE), play an important role in analyzing continuous dynamic games and control theory problems. An important tool for such problems when they…
This document is due to appear as a chapter of the forthcoming Handbook of Approximate Bayesian Computation (ABC) edited by S. Sisson, Y. Fan, and M. Beaumont. Since the earliest work on ABC, it has been recognised that using summary…
The purpose of this article is to develop a machinery to study the capacity of deep neural networks (DNNs) to approximate high-dimensional functions. In particular, we show that DNNs have the expressive power to overcome the curse of…
Maximum surjective constraint satisfaction problems (Max-Sur-CSPs) are computational problems where we are given a set of variables denoting values from a finite domain B and a set of constraints on the variables. A solution to such a…
Solving optimal stopping problems by backward induction in high dimensions is often very complex since the computation of conditional expectations is required. Typically, such computations are based on regression, a method that suffers from…
Computational approaches to PDE-constrained optimization under uncertainty may involve finite-dimensional approximations of control and state spaces, sample average approximations of measures of risk and reliability, smooth approximations…
We study the use of approximate Lagrange multipliers and discrete actions in solving convex optimisation problems. We observe that descent, which can be ensured using a wide range of approaches (gradient, subgradient, Newton, etc.), is…
A new exact projective penalty method is proposed for the equivalent reduction of constrained optimization problems to nonsmooth unconstrained ones. In the method, the original objective function is extended to infeasible points by summing…
Numerical experiments indicate that deep learning algorithms overcome the curse of dimensionality when approximating solutions of semilinear PDEs. For certain linear PDEs and semilinear PDEs with gradient-independent nonlinearities this has…
Penalty functions are widely used to enforce constraints in optimization problems and reinforcement leaning algorithms. Softplus and algebraic penalty functions are proposed to overcome the sensitivity of the Courant-Beltrami method to…
We study a new penalty reformulation of constrained convex optimization based on the softplus penalty function. We develop novel and tight upper bounds on the objective value gap and the violation of constraints for the solutions to the…
In this paper, we introduce a method for approximating the solution to inference and optimization tasks in uncertain and deterministic reasoning. Such tasks are in general intractable for exact algorithms because of the large number of…
In this paper, an inexact proximal-point penalty method is studied for constrained optimization problems, where the objective function is non-convex, and the constraint functions can also be non-convex. The proposed method approximately…
Euclidean Markov decision processes are a powerful tool for modeling control problems under uncertainty over continuous domains. Finite state imprecise, Markov decision processes can be used to approximate the behavior of these infinite…
We are concerned with the problem of scheduling $n$ jobs onto $m$ identical machines. Each machine has to be in operation for a prescribed time, and the objective is to minimize the total machine working time. Precisely, let $c_i$ be the…
We introduce a max-plus analogue of the Petrov-Galerkin finite element method, to solve finite horizon deterministic optimal control problems. The method relies on a max-plus variational formulation, and exploits the properties of…
Designing efficient and accurate numerical solvers for high-dimensional partial differential equations (PDEs) remains a challenging and important topic in computational science and engineering, mainly due to the "curse of dimensionality" in…
The generation of decision-theoretic Bayesian optimal designs is complicated by the significant computational challenge of minimising an analytically intractable expected loss function over a, potentially, high-dimensional design space. A…
We present a dimension-incremental method for function approximation in bounded orthonormal product bases to learn the solutions of various differential equations. Therefore, we decompose the source function of the differential equation…
In this work, we study the problem of finding approximate, with minimum support set, solutions to matrix max-plus equations, which we call sparse approximate solutions. We show how one can obtain such solutions efficiently and in polynomial…