5,815 research outputs found
From Infinite to Finite Programs: Explicit Error Bounds with Applications to Approximate Dynamic Programming
We consider linear programming (LP) problems in infinite dimensional spaces
that are in general computationally intractable. Under suitable assumptions, we
develop an approximation bridge from the infinite-dimensional LP to tractable
finite convex programs in which the performance of the approximation is
quantified explicitly. To this end, we adopt the recent developments in two
areas of randomized optimization and first order methods, leading to a priori
as well as a posterior performance guarantees. We illustrate the generality and
implications of our theoretical results in the special case of the long-run
average cost and discounted cost optimal control problems for Markov decision
processes on Borel spaces. The applicability of the theoretical results is
demonstrated through a constrained linear quadratic optimal control problem and
a fisheries management problem.Comment: 30 pages, 5 figure
Linear conic optimization for nonlinear optimal control
Infinite-dimensional linear conic formulations are described for nonlinear
optimal control problems. The primal linear problem consists of finding
occupation measures supported on optimal relaxed controlled trajectories,
whereas the dual linear problem consists of finding the largest lower bound on
the value function of the optimal control problem. Various approximation
results relating the original optimal control problem and its linear conic
formulations are developed. As illustrated by a couple of simple examples,
these results are relevant in the context of finite-dimensional semidefinite
programming relaxations used to approximate numerically the solutions of the
infinite-dimensional linear conic problems.Comment: Submitted for possible inclusion as a contributed chapter in S.
Ahmed, M. Anjos, T. Terlaky (Editors). Advances and Trends in Optimization
with Engineering Applications. MOS-SIAM series, SIAM, Philadelphi
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