1,225 research outputs found
Algebraic Relaxations and Hardness Results in Polynomial Optimization and Lyapunov Analysis
This thesis settles a number of questions related to computational complexity
and algebraic, semidefinite programming based relaxations in optimization and
control.Comment: PhD Thesis, MIT, September, 201
Neural Lyapunov Control
We propose new methods for learning control policies and neural network
Lyapunov functions for nonlinear control problems, with provable guarantee of
stability. The framework consists of a learner that attempts to find the
control and Lyapunov functions, and a falsifier that finds counterexamples to
quickly guide the learner towards solutions. The procedure terminates when no
counterexample is found by the falsifier, in which case the controlled
nonlinear system is provably stable. The approach significantly simplifies the
process of Lyapunov control design, provides end-to-end correctness guarantee,
and can obtain much larger regions of attraction than existing methods such as
LQR and SOS/SDP. We show experiments on how the new methods obtain high-quality
solutions for challenging control problems.Comment: NeurIPS 201
Lower Bounds on Complexity of Lyapunov Functions for Switched Linear Systems
We show that for any positive integer , there are families of switched
linear systems---in fixed dimension and defined by two matrices only---that are
stable under arbitrary switching but do not admit (i) a polynomial Lyapunov
function of degree , or (ii) a polytopic Lyapunov function with facets, or (iii) a piecewise quadratic Lyapunov function with
pieces. This implies that there cannot be an upper bound on the size of the
linear and semidefinite programs that search for such stability certificates.
Several constructive and non-constructive arguments are presented which connect
our problem to known (and rather classical) results in the literature regarding
the finiteness conjecture, undecidability, and non-algebraicity of the joint
spectral radius. In particular, we show that existence of an extremal piecewise
algebraic Lyapunov function implies the finiteness property of the optimal
product, generalizing a result of Lagarias and Wang. As a corollary, we prove
that the finiteness property holds for sets of matrices with an extremal
Lyapunov function belonging to some of the most popular function classes in
controls
Propensity Score Analysis with Matching Weights
The propensity score analysis is one of the most widely used methods for
studying the causal treatment effect in observational studies. This paper
studies treatment effect estimation with the method of matching weights. This
method resembles propensity score matching but offers a number of new features
including efficient estimation, rigorous variance calculation, simple
asymptotics, statistical tests of balance, clearly identified target population
with optimal sampling property, and no need for choosing matching algorithm and
caliper size. In addition, we propose the mirror histogram as a useful tool for
graphically displaying balance. The method also shares some features of the
inverse probability weighting methods, but the computation remains stable when
the propensity scores approach 0 or 1. An augmented version of the matching
weight estimator is developed that has the double robust property, i.e., the
estimator is consistent if either the outcome model or the propensity score
model is correct. In the numerical studies, the proposed methods demonstrated
better performance than many widely used propensity score analysis methods such
as stratification by quintiles, matching with propensity scores, and inverse
probability weighting
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