Neural network architectures using min plus algebra for solving certain high dimensional optimal control problems and Hamilton-Jacobi PDEs

Darbon, Jérôme; Dower, Peter M.; Meng, Tingwei

doi:10.48550/arxiv.2105.03336

Cited by 4 publications

(21 citation statements)

References 134 publications

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“…In this case, Proposition 21 shows that the function (x, t) → V (x, t; p, a, b) defined in (9) and (18) (where p is the slope of Φ) is the unique continuously differentiable solution in the solution set G to the HJ PDE (8) with this linear initial data Φ. Moreover, Proposition 22 shows that the trajectory s → γ(s; x, t, p, a, b) defined by (12), ( 13), ( 14), ( 16), (17), and (19) for different cases is the unique optimal trajectory of the optimal control problem (6), whose optimal value equals V (x, t; p, a, b).…”

Section: One-dimensional Casementioning

confidence: 98%

“…where the right-hand side is well-defined using ( 12), ( 13), ( 14), (16), and (17) for different cases. Now, we consider a general convex initial cost Φ : R → R. The corresponding HJ PDE is solved using the following Hopf-type formula:…”

Section: One-dimensional Casementioning

confidence: 99%

“…where the function γ(s; x, t, p, a, b) on the right-hand side is defined by ( 12), ( 13), ( 14), ( 16), (17), and (19) for different cases of x, t and p. Next, we provide some theoretical properties for the functions V and γ defined above. To be specific, in Proposition 21, we prove that under some assumptions, the function V defined above is indeed the unique viscosity solution to the HJ PDE (8).…”

Section: One-dimensional Casementioning

confidence: 99%

“…where the i-th element γ(s; x i , t, p * i , a i , b i ) on the right-hand side is the onedimensional trajectory defined in ( 12), ( 13), ( 14), ( 16), (17), and (19) for different cases of x i , t and p * i . Note that the components of γ(s; x, t) are independent from each other, and hence they can be computed in parallel as long as p * is known.…”

Section: Separable High-dimensional Casementioning

confidence: 99%

“…Due to the curse of dimensionality, these grid-based methods are infeasible for solving high-dimensional problems, e.g., for dimensions greater than five. Several grid-free methods have been proposed to overcome or mitigate the curse of dimensionality, which include, but are not limited to, optimization methods [16,22,19,21,81,59], max-plus methods [1,2,29,34,37,65,64,66,67], tensor decomposition techniques [27,42,80], sparse grids [11,36,53], polynomial approximation [51,52], model order reduction [4,57], dynamic programming and reinforcement learning [3,10,82], and neural networks [5,6,26,46,39,44,45,58,71,75,76,79,62,18,20,17,69,70,49,…”

Section: Introductionmentioning

confidence: 99%

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Hopf-type representation formulas and efficient algorithms for certain high-dimensional optimal control problems

Chen¹,

Darbon²,

Meng³

2021

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Solving high-dimensional optimal control problems and their corresponding Hamilton-Jacobi partial differential equations is an important but challenging problem. In particular, handling optimal control problems with state-dependent running costs presents an additional challenge. In this paper, we consider a class of optimal control problems whose running costs consist of a quadratic on the control variable and a convex, non-negative, piecewise affine function on the state variable. We provide the analytical solution for this class of optimal control problems as well as a Hopf-type representation formula for the corresponding Hamilton-Jacobi partial differential equations. Finally, we propose efficient numerical algorithms based on our Hopf-type representation formula, convex optimization algorithms, and min-plus techniques. We also present several high-dimensional numerical examples, which demonstrate that our algorithms overcome the curse of dimensionality. As such, our proposed methods have the potential to serve as a building block for solving more complicated high-dimensional optimal control problems in real-time. KeywordsOptimal control • Hamilton-Jacobi partial differential equations • Grid-free numerical methods • High dimensions Research supported by NSF 1820821 and AFOSR MURI FA9550-20-1-0358. P.C. is supported by the SMART Scholarship, which is funded by USD/R&E (The Under Secretary of Defense-Research and Engineering), National Defense Education Program (NDEP) / BA-1, Basic Research. Authors' names are given in last/family name alphabetical order.

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Section: One-dimensional Casementioning

confidence: 98%

Section: One-dimensional Casementioning

confidence: 99%

Section: One-dimensional Casementioning

confidence: 99%

Section: Separable High-dimensional Casementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Hopf-type representation formulas and efficient algorithms for certain high-dimensional optimal control problems

Chen¹,

Darbon²,

Meng³

2021

Preprint

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Terminologi Dasar Pada Himpunan Simetrisasi Aljabar Min-Plus

Suroto

Wardayani

Sugihandardji

2022

j. Matematika Sains dan Teknologi

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This paper discusses some of the terminology and basic properties of the set of min-plus algebra symmetrization. The results are algebraic structure of the positive or zero part of the set of min-plus algebra symmetrization is a semi-field, and the min-plus algebra can be viewed as the positive or zero part in the context of the set of min-plus algebra symmetrization. Furthermore, the form of the result of the subtraction of the elements in the set of min-plus algebra symmetrization, the multiplicative inverse of element in the non zero-positive and negative part, and some basic properties of balance as an analogy to the similarity of linear algebra are also obtained.

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Lax-Oleinik-type formulas and efficient algorithms for certain high-dimensional optimal control problems

Chen¹,

Darbon²,

Meng³

2021

Preprint

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We provide analytical solutions to certain optimal control problems whose running cost depends on the state variable and with constraints on the control. We also provide Lax-Oleinik-type representation formulas for the corresponding Hamilton-Jacobi partial differential equations with statedependent Hamiltonians. Additionally, we present efficient, grid-free numerical solvers based on our representation formulas, which are shown, in some cases, to scale linearly with the state dimension, and thus, to overcome the curse of dimensionality. Using existing optimization methods and the min-plus technique, we extend our numerical solvers to address more general classes of convex and nonconvex initial costs. As a result, our numerical methods have the potential to serve as a building block for solving broader classes of highdimensional optimal control problems in real-time.Keywords Optimal control • Hamilton-Jacobi partial differential equations • Grid-free numerical methods • High dimensions Research supported by NSF 1820821 and AFOSR MURI FA9550-20-1-0358. Authors' names are given in last/family name alphabetical order.

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Neural network architectures using min plus algebra for solving certain high dimensional optimal control problems and Hamilton-Jacobi PDEs

Cited by 4 publications

References 134 publications

Hopf-type representation formulas and efficient algorithms for certain high-dimensional optimal control problems

Hopf-type representation formulas and efficient algorithms for certain high-dimensional optimal control problems

Terminologi Dasar Pada Himpunan Simetrisasi Aljabar Min-Plus

Lax-Oleinik-type formulas and efficient algorithms for certain high-dimensional optimal control problems

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