Linear programming solutions of periodic optimization problems: approximation of the optimal control

Finlay, Luke; Gaitsgory, Vladimir; Lebedev, Ivan

doi:10.3934/jimo.2007.3.399

Cited by 10 publications

(5 citation statements)

References 22 publications

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“…Proof. The validity of (5.9) was proved in Theorem 5.2 (ii) of [9]. Let us prove (5.10) and (5.11) (note that the argument we are using is similar to that used in [17]).…”

Section: Corollary 52mentioning

confidence: 79%

Use of Approximations of Hamilton-Jacobi-Bellman Inequality for Solving Periodic Optimization Problems

Gaitsgory

Manic

2014

Springer Proceedings in Mathematics &Amp; Statistics

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We show that necessary and sufficient conditions of optimality in periodic optimization problems can be stated in terms of a solution of the corresponding HJB inequality, the latter being equivalent to a max-min type variational problem considered on the space of continuously differentiable functions. We approximate the latter with a maximin problem on a finite dimensional subspace of the space of continuously differentiable functions and show that a solution of this problem (existing under natural controllability conditions) can be used for construction of near optimal controls. We illustrate the construction with a numerical example.

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“…Proof. The validity of (5.9) was proved in Theorem 5.2 (ii) of [9]. Let us prove (5.10) and (5.11) (note that the argument we are using is similar to that used in [17]).…”

Section: Corollary 52mentioning

confidence: 79%

Use of Approximations of Hamilton-Jacobi-Bellman Inequality for Solving Periodic Optimization Problems

Gaitsgory

Manic

2014

Springer Proceedings in Mathematics &Amp; Statistics

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“…However, the speed of convergence is given by the estimates in Lemma 23 and are less explicit than the Hölder ones exhibited in the cited papers. Second, having stated the equivalent problem on a linear space of measures should prove useful for optimality issues (see [20] or, more recently, [19] in a general Markovian framework or [25] in a Brownian one).…”

Section: Conclusion and Commentsmentioning

confidence: 99%

“…Note also that for any t ≤ t collision ∧ t control , we have y ρε γ (t + t * ; x, α) , e 1 > 0. For every 0 < t ≤ t collision ∧ t control , one uses (20) to get…”

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confidence: 99%

A Piecewise Deterministic Markov Toy Model for Traffic/Maintenance and Associated Hamilton–Jacobi Integrodifferential Systems on Networks

2015

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We study optimal control problems in infinite horizon when the dynamics belong to a specific class of piecewise deterministic Markov processes constrained to star-shaped networks (corresponding to a toy traffic model). We adapt the results in [35] to prove the regularity of the value function and the dynamic programming principle. Extending the networks and Krylov's "shaking the coefficients" method, we prove that the value function can be seen as the solution to a linearized optimization problem set on a convenient set of probability measures. The approach relies entirely on viscosity arguments. As a by-product, the dual formulation guarantees that the value function is the pointwise supremum over regular subsolutions of the associated Hamilton-Jacobi integrodifferential system. This ensures that the value function satisfies Perron's preconization for the (unique) candidate to viscosity solution.Mathematics Subject Classification. 49L25, 93E20, 60J25, 49L20 Acknowledgement. The authors would like to thank the anonymous referees for constructive remarks allowing to improve the manuscript. v δ (x, γ) := inf α,X x,γ,α · ∈network E ∞ 0 e −δt l Γ x,γ,α t (X x,γ,α t , α t ) dt . Tel. : +33 (0)1 60 95 75 27, Fax : +33 (0)1 60 95 75 45 ‡ Acknowledgement. )) ds ,where α 2 ∈ L 0 (R + × R m × E; A). We set (X x 0 ,γ 0 ,α t , Γ x 0 ,γ 0 ,α t ) = (y Υ 1 (t; τ 1 , Y 1 , α 2 ) , Υ 1 ) , if t ∈ [τ 1 , τ 2 ) .The post-jump location (Y 2 , Υ 2 ) satisfies

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“…Some optimal control problems can, in fact, be viewed as infinite-dimensional programming. See, for example, Teo, Goh and Wong [16], Finlay, Gaitsgory and Lebedev [7], Wu and Teo [18], Gerdts and Kunkel [10], Gong and Xiang [11], Dahleh and Pearson [4] and Dahleh and Diaz-Bobillo [5]. Recently Vanderbei [17] investigated an optimization problem for the best high-contrast apodization, this is an infinite-dimensional linear programming problem in which the decision variable has a lower bound and an upper bound.…”

Section: G(y) Dν(y)mentioning

confidence: 99%

A numerical approach to infinite-dimensional linear programming in $L_1$ spaces

Ito¹,

Wu²,

Shiu³

et al. 2010

Journal of Industrial &Amp; Management Optimization

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An infinite-dimensional linear programming formulated on L 1 spaces, problem (P), is studied in this paper. A related optimization problem, general capacity problem (GCAP), is also mentioned in this paper. But we find that the optimal solution does not exist in problem (P). Thus, we approach the optimal value for problem (P) via solving the problem (GCAP). A proposed algorithm is shown that we solve a sequence of semi-infinite subproblems to approach the optimal value of problem (P). The error bound for the difference between the optimal value for problem (P) and optimal value for semi-infinite subproblem is also given in this paper. Finally, numerical examples are implemented and compared with discretization method to show our computational efficiency.

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Linear programming solutions of periodic optimization problems: approximation of the optimal control

Cited by 10 publications

References 22 publications

Use of Approximations of Hamilton-Jacobi-Bellman Inequality for Solving Periodic Optimization Problems

Use of Approximations of Hamilton-Jacobi-Bellman Inequality for Solving Periodic Optimization Problems

A Piecewise Deterministic Markov Toy Model for Traffic/Maintenance and Associated Hamilton–Jacobi Integrodifferential Systems on Networks

A numerical approach to infinite-dimensional linear programming in $L_1$ spaces

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