140 research outputs found
Linear programming on the Stiefel manifold
Linear programming on the Stiefel manifold (LPS) is studied for the first
time. It aims at minimizing a linear objective function over the set of all
-tuples of orthonormal vectors in satisfying additional
linear constraints. Despite the classical polynomial-time solvable case ,
general (LPS) is NP-hard. According to the Shapiro-Barvinok-Pataki theorem,
(LPS) admits an exact semidefinite programming (SDP) relaxation when
, which is tight when . Surprisingly, we can greatly
strengthen this sufficient exactness condition to , which covers the
classical case and . Regarding (LPS) as a smooth nonlinear
programming problem, we reveal a nice property that under the linear
independence constraint qualification, the standard first- and second-order
{\it local} necessary optimality conditions are sufficient for {\it global}
optimality when
Model Uncertainty and Policy Evaluation: Some Theory and Empirics
This paper explores ways to integrate model uncertainty into policy evaluation. We first describe a general framework for the incorporation of model uncertainty into standard econometric calculations. This framework employs Bayesian model averaging methods that have begun to appear in a range of economic studies. Second, we illustrate these general ideas in the context of assessment of simple monetary policy rules for some standard New Keynesian specifications. The specifications vary in their treatment of expectations as well as in the dynamics of output and inflation. We conclude that the Taylor rule has good robustness properties, but may reasonably be challenged in overall quality with respect to stabilization by alternative simple rules that also condition on lagged interest rates, even though these rules employ parameters that are set without accounting for model uncertainty.
The Meta-Model Approach for Simulation-based Design Optimization.
The design of products and processes makes increasing use of computer simulations for the prediction of its performance. These computer simulations are considerably cheaper than their physical equivalent. Finding the optimal design has therefore become a possibility. One approach for finding the optimal design using computer simulations is the meta-model approach, which approximates the behaviour of the computer simulation outcome using a limited number of time-consuming computer simulations. This thesis contains four main contributions, which are illustrated by industrial cases. First, a method is presented for the construction of an experimental design for computer simulations when the design space is restricted by many (nonlinear) constraints. The second contribution is a new approach for the approximation of the simulation outcome. This approximation method is particularly useful when the simulation model outcome reacts highly nonlinear to its inputs. Third, the meta-model based approach is extended to a robust optimization framework. Using this framework, many uncertainties can be taken into account, including uncertainty on the simulation model outcome. The fourth main contribution is the extension of the approach for use in integral design of many parts of complex systems.
An Efficient Alternating Riemannian/Projected Gradient Descent Ascent Algorithm for Fair Principal Component Analysis
Fair principal component analysis (FPCA), a ubiquitous dimensionality
reduction technique in signal processing and machine learning, aims to find a
low-dimensional representation for a high-dimensional dataset in view of
fairness. The FPCA problem involves optimizing a non-convex and non-smooth
function over the Stiefel manifold. The state-of-the-art methods for solving
the problem are subgradient methods and semidefinite relaxation-based methods.
However, these two types of methods have their obvious limitations and thus are
only suitable for efficiently solving the FPCA problem in special scenarios.
This paper aims at developing efficient algorithms for solving the FPCA problem
in general, especially large-scale, settings. In this paper, we first transform
FPCA into a smooth non-convex linear minimax optimization problem over the
Stiefel manifold. To solve the above general problem, we propose an efficient
alternating Riemannian/projected gradient descent ascent (ARPGDA) algorithm,
which performs a Riemannian gradient descent step and an ordinary projected
gradient ascent step at each iteration. We prove that ARPGDA can find an
-stationary point of the above problem within
iterations. Simulation results show that,
compared with the state-of-the-art methods, our proposed ARPGDA algorithm can
achieve a better performance in terms of solution quality and speed for solving
the FPCA problems.Comment: 5 pages, 8 figures, submitted for possible publicatio
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