7,294 research outputs found
Data-Driven Chance Constrained Optimization under Wasserstein Ambiguity Sets
We present a data-driven approach for distributionally robust chance
constrained optimization problems (DRCCPs). We consider the case where the
decision maker has access to a finite number of samples or realizations of the
uncertainty. The chance constraint is then required to hold for all
distributions that are close to the empirical distribution constructed from the
samples (where the distance between two distributions is defined via the
Wasserstein metric). We first reformulate DRCCPs under data-driven Wasserstein
ambiguity sets and a general class of constraint functions. When the
feasibility set of the chance constraint program is replaced by its convex
inner approximation, we present a convex reformulation of the program and show
its tractability when the constraint function is affine in both the decision
variable and the uncertainty. For constraint functions concave in the
uncertainty, we show that a cutting-surface algorithm converges to an
approximate solution of the convex inner approximation of DRCCPs. Finally, for
constraint functions convex in the uncertainty, we compare the feasibility set
with other sample-based approaches for chance constrained programs.Comment: A shorter version is submitted to the American Control Conference,
201
K-Adaptability in Two-Stage Distributionally Robust Binary Programming
We propose to approximate two-stage distributionally robust programs with binary recourse decisions by their associated K-adaptability problems, which pre-select K candidate secondstage policies here-and-now and implement the best of these policies once the uncertain parameters have been observed. We analyze the approximation quality and the computational complexity of the K-adaptability problem, and we derive explicit mixed-integer linear programming reformulations. We also provide efficient procedures for bounding the probabilities with which each of the K second-stage policies is selected
Meshfree finite differences for vector Poisson and pressure Poisson equations with electric boundary conditions
We demonstrate how meshfree finite difference methods can be applied to solve
vector Poisson problems with electric boundary conditions. In these, the
tangential velocity and the incompressibility of the vector field are
prescribed at the boundary. Even on irregular domains with only convex corners,
canonical nodal-based finite elements may converge to the wrong solution due to
a version of the Babuska paradox. In turn, straightforward meshfree finite
differences converge to the true solution, and even high-order accuracy can be
achieved in a simple fashion. The methodology is then extended to a specific
pressure Poisson equation reformulation of the Navier-Stokes equations that
possesses the same type of boundary conditions. The resulting numerical
approach is second order accurate and allows for a simple switching between an
explicit and implicit treatment of the viscosity terms.Comment: 19 pages, 7 figure
Convex Optimization Methods for Dimension Reduction and Coefficient Estimation in Multivariate Linear Regression
In this paper, we study convex optimization methods for computing the trace
norm regularized least squares estimate in multivariate linear regression. The
so-called factor estimation and selection (FES) method, recently proposed by
Yuan et al. [22], conducts parameter estimation and factor selection
simultaneously and have been shown to enjoy nice properties in both large and
finite samples. To compute the estimates, however, can be very challenging in
practice because of the high dimensionality and the trace norm constraint. In
this paper, we explore a variant of Nesterov's smooth method [20] and interior
point methods for computing the penalized least squares estimate. The
performance of these methods is then compared using a set of randomly generated
instances. We show that the variant of Nesterov's smooth method [20] generally
outperforms the interior point method implemented in SDPT3 version 4.0 (beta)
[19] substantially . Moreover, the former method is much more memory efficient.Comment: 27 page
- …