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We present a finite-time analysis of two smoothed functional stochastic approximation algorithms for simulation-based optimization. The first is a two time-scale gradient-based method, while the second is a three time-scale Newton-based…
Given a graph $G = (V, E)$, we wish to compute a spanning tree whose maximum vertex degree, i.e. tree degree, is as small as possible. Computing the exact optimal solution is known to be NP-hard, since it generalizes the Hamiltonian path…
Hamilton-Jacobi reachability (HJR) is an exciting framework used for control of safety-critical systems with nonlinear and possibly uncertain dynamics. However, HJR suffers from the curse of dimensionality, with computation times growing…
We consider the computation of free energy-like quantities for diffusions in high dimension, when resorting to Monte Carlo simulation is necessary. Such stochastic computations typically suffer from high variance, in particular in a low…
The subpath planning problem is a branch of the path planning problem, which has widespread applications in automated manufacturing process as well as vehicle and robot navigation. This problem is to find the shortest path or tour subject…
Two-stage methods addressing continuous shortest path problems start local minimization from discrete shortest paths in a spatial graph. The convergence of such hybrid methods to global minimizers hinges on the discretization error induced…
We present a proximal gradient method for solving convex multiobjective optimization problems, where each objective function is the sum of two convex functions, with one assumed to be continuously differentiable. The algorithm incorporates…
In this paper, we consider a new approach for semi-discretization in time and spatial discretization of a class of semi-linear stochastic partial differential equations (SPDEs) with multiplicative noise. The drift term of the SPDEs is only…
We study the convergence rate of the proximal-gradient homotopy algorithm applied to norm-regularized linear least squares problems, for a general class of norms. The homotopy algorithm reduces the regularization parameter in a series of…
This work considers the non-convex finite sum minimization problem. There are several algorithms for such problems, but existing methods often work poorly when the problem is badly scaled and/or ill-conditioned, and a primary goal of this…
This paper analyzes a space-time finite element method for fractional wave problems. The method uses a Petrov-Galerkin type time-stepping scheme to discretize the time fractional derivative of order $ \gamma $ ($1<\gamma<2$). We establish…
This paper investigates the convergence properties of the hypergradient descent method (HDM), a 25-year-old heuristic originally proposed for adaptive stepsize selection in stochastic first-order methods. We provide the first rigorous…
We address the problem of optimal path planning for a simple nonholonomic vehicle in the presence of obstacles. Most current approaches are either split hierarchically into global path planning and local collision avoidance, or neglect some…
We present a variant of accelerated gradient descent algorithms, adapted from Nesterov's optimal first-order methods, for weakly-quasi-convex and weakly-quasi-strongly-convex functions. We show that by tweaking the so-called estimate…
The multigrid-reduction-in-time (MGRIT) technique has proven to be successful in achieving higher run-time speedup by exploiting parallelism in time. The goal of this article is to develop and analyze a MGRIT algorithm, using FCF-relaxation…
This work proposes a scheme for significantly reducing the computational complexity of discretized problems involving the non-smooth forward propagation of uncertainty by combining the adaptive hierarchical sparse grid stochastic…
Stochastic variance reduced optimization methods are known to be globally convergent while they suffer from slow local convergence, especially when moderate or high accuracy is needed. To alleviate this problem, we propose an optimization…
Many applications require efficient methods for solving continuous shortest path problems. Such paths can be viewed as characteristics of static Hamilton-Jacobi equations. Several fast numerical algorithms have been developed to solve such…
To sidestep the curse of dimensionality when computing solutions to Hamilton-Jacobi-Bellman partial differential equations (HJB PDE), we propose an algorithm that leverages a neural network to approximate the value function. We show that…
This paper considers the hyperparameter optimization problem of mathematical techniques that arise in the numerical solution of differential and integral equations. The well-known approaches grid and random search, in a parallel algorithm…