99,670 research outputs found
Controller Synthesis for Discrete-Time Polynomial Systems via Occupation Measures
In this paper, we design nonlinear state feedback controllers for
discrete-time polynomial dynamical systems via the occupation measure approach.
We propose the discrete-time controlled Liouville equation, and use it to
formulate the controller synthesis problem as an infinite-dimensional linear
programming problem on measures, which is then relaxed as finite-dimensional
semidefinite programming problems on moments of measures and their duals on
sums-of-squares polynomials. Nonlinear controllers can be extracted from the
solutions to the relaxed problems. The advantage of the occupation measure
approach is that we solve convex problems instead of generally non-convex
problems, and the computational complexity is polynomial in the state and input
dimensions, and hence the approach is more scalable. In addition, we show that
the approach can be applied to over-approximating the backward reachable set of
discrete-time autonomous polynomial systems and the controllable set of
discrete-time polynomial systems under known state feedback control laws. We
illustrate our approach on several dynamical systems
Complexity of convex optimization using geometry-based measures and a reference point
Abstract in HTML and working paper for download in PDF available via World Wide Web at the Social Science Research Network.Title from cover. "September 2001."Includes bibliographical references (leaf 29).Our concern lies in solving the following convex optimization problem: minimize cx subject to Ax=b, x \in P, where P is a closed convex set. We bound the complexity of computing an almost-optimal solution of this problem in terms of natural geometry-based measures of the feasible region and the level-set of almost-optimal solutions, relative to a given reference point xr that might be close to the feasible region and/or the almost-optimal level set. This contrasts with other complexity bounds for convex optimization that rely on data-based condition numbers or algebraic measures, and that do not take into account any a priori reference point information. Keywords: Convex Optimization, Complexity, Interior-Point Method, Barrier Method.Robert M. Freund
Non-stationary Stochastic Optimization
We consider a non-stationary variant of a sequential stochastic optimization
problem, in which the underlying cost functions may change along the horizon.
We propose a measure, termed variation budget, that controls the extent of said
change, and study how restrictions on this budget impact achievable
performance. We identify sharp conditions under which it is possible to achieve
long-run-average optimality and more refined performance measures such as rate
optimality that fully characterize the complexity of such problems. In doing
so, we also establish a strong connection between two rather disparate strands
of literature: adversarial online convex optimization; and the more traditional
stochastic approximation paradigm (couched in a non-stationary setting). This
connection is the key to deriving well performing policies in the latter, by
leveraging structure of optimal policies in the former. Finally, tight bounds
on the minimax regret allow us to quantify the "price of non-stationarity,"
which mathematically captures the added complexity embedded in a temporally
changing environment versus a stationary one
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