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17
CurseofComplexity Attenuation in the CurseofDimensionalityFree Method for HJB PDEs,
 Proc. ACC
, 2008
"... AbstractRecently, a curseofdimensionalityfree method was developed for solution of HamiltonJacobiBellman partial differential equations (HJB PDEs) for nonlinear control problems, using semiconvex duality and maxplus analysis. The curseofdimensionalityfree method may be applied to HJB PDEs ..."
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Cited by 15 (7 self)
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AbstractRecently, a curseofdimensionalityfree method was developed for solution of HamiltonJacobiBellman partial differential equations (HJB PDEs) for nonlinear control problems, using semiconvex duality and maxplus analysis. The curseofdimensionalityfree method may be applied to HJB PDEs where the Hamiltonian is given as (or wellapproximated by) a pointwise maximum of quadratic forms. Such HJB PDEs also arise in certain switched linear systems. The method constructs the correct solution of an HJB PDE from a maxplus linear combination of quadratics. The method completely avoids the curseofdimensionality, and is subject to cubic computational growth as a function of space dimension. However, it is subject to a curseofcomplexity. In particular, the number of quadratics in the approximation grows exponentially with the number of iterations. Efficacy of such a method depends on the pruning of quadratics to keep the complexity growth at a reasonable level. Here we apply a pruning algorithm based on semidefinite programming. Computational speeds are exceptional, with an example HJB PDE in sixdimensional Euclidean space solved to the indicated quality in approximately 30 minutes on a typical desktop machine.
Approximate dynamic programming via iterated Bellman inequalities
, 2010
"... In this paper we introduce new methods for finding functions that lower bound the value function of a stochastic control problem, using an iterated form of the Bellman inequality. Our method is based on solving linear or semidefinite programs, and produces both a bound on the optimal objective, as w ..."
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Cited by 11 (4 self)
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In this paper we introduce new methods for finding functions that lower bound the value function of a stochastic control problem, using an iterated form of the Bellman inequality. Our method is based on solving linear or semidefinite programs, and produces both a bound on the optimal objective, as well as a suboptimal policy that appears to works very well. These results extend and improve bounds obtained by authors in a previous paper using a single Bellman inequality condition. We describe the methods in a general setting, and show how they can be applied in specific cases including the finite state case, constrained linear quadratic control, switched affine control, and multiperiod portfolio investment.
Curse of dimensionality reduction in maxplus based approximation methods: Theoretical estimates and improved pruning algorithms
 In CDCECE
, 2011
"... Abstract — Maxplus based methods have been recently developed to approximate the value function of possibly high dimensional optimal control problems. A critical step of these methods consists in approximating a function by a supremum of a small number of functions (maxplus “basis functions”) tak ..."
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Cited by 9 (2 self)
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Abstract — Maxplus based methods have been recently developed to approximate the value function of possibly high dimensional optimal control problems. A critical step of these methods consists in approximating a function by a supremum of a small number of functions (maxplus “basis functions”) taken from a prescribed dictionary. We study several variants of this approximation problem, which we show to be continuous versions of the facility location and kcenter combinatorial optimization problems, in which the connection costs arise from a Bregman distance. We give theoretical error estimates, quantifying the number of basis functions needed to reach a prescribed accuracy. We derive from our approach a refinement of the curse of dimensionality free method introduced previously by McEneaney, with a higher accuracy for a comparable computational cost. I.
Convergence rate for a curseofdimensionalityfree method for a class of HJB PDEs
 SIAM J. Control Optim
"... Abstract. In previous work of the first author and others, maxplus methods have been explored for solution of firstorder, nonlinear HamiltonJacobiBellman partial differential equations (HJB PDEs) and corresponding nonlinear control problems. Although maxplus basis expansion and maxplus finite ..."
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Cited by 8 (3 self)
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Abstract. In previous work of the first author and others, maxplus methods have been explored for solution of firstorder, nonlinear HamiltonJacobiBellman partial differential equations (HJB PDEs) and corresponding nonlinear control problems. Although maxplus basis expansion and maxplus finiteelement methods can provide substantial computationalspeed advantages, they still generally suffer from the curseofdimensionality. Here we consider HJB PDEs where the Hamiltonian takes the form of a (pointwise) maximum of linear/quadratic forms. The approach to solution will be rather general, but in order to ground the work, we consider only constituent Hamiltonians corresponding to longrun averagecostperunittime optimal control problems for the development. We consider a previously obtained numerical method not subject to the curseofdimensionality. The method is based on construction of the dualspace semigroup corresponding to the HJB PDE. This dualspace semigroup is constructed from the dualspace semigroups corresponding to the constituent linear/quadratic Hamiltonians. The dualspace semigroup is particularly useful due to its form as a maxplus integral operator with kernel obtained from the originating semigroup. One considers repeated application of the dualspace semigroup to obtain the solution. Although previous work indicated that the method was not subject to the curseofdimensionality, it did not indicate any error bounds or convergence rate. Here, we obtain specific error bounds. Key words. partial differential equations, curseofdimensionality, dynamic programming, maxplus algebra, Legendre transform, Fenchel transform, semiconvexity, HamiltonJacobiBellman equations, idempotent analysis. AMS subject classifications. 49LXX, 93C10, 35B37, 35F20, 65N99, 47D99 1. Introduction. A robust approach to the solution of nonlinear control problems is through the general method of dynamic programming. For the typical class of problems in continuous time and continuous space, with the dynamics governed by finitedimensional, ordinary differential equations, this leads to a representation of the problem as a firstorder, nonlinear partial differential equation, the HamiltonJacobiBellman equation or the HJB PDE. If one has an infinite timehorizon problem, then the HJB PDE is a steadystate equation, and this PDE is over a space (or some subset thereof) whose dimension is the dimension of the state variable of the control problem. Due to the nonlinearity, the solutions are generally nonsmooth, and one must use the theory of viscosity solutions
Linear hamilton jacobi bellman equations in high dimensions
 in Conference on Decision and Control (CDC), 2014, arXiv preprint arXiv:1404.1089
"... provides the globally optimal solution to large classes of control problems. Unfortunately, this generality comes at a price, the calculation of such solutions is typically intractible for systems with more than moderate state space size due to the curse of dimensionality. This work combines recent ..."
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Cited by 7 (3 self)
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provides the globally optimal solution to large classes of control problems. Unfortunately, this generality comes at a price, the calculation of such solutions is typically intractible for systems with more than moderate state space size due to the curse of dimensionality. This work combines recent results in the structure of the HJB, and its reduction to a linear Partial Differential Equation (PDE), with methods based on low rank tensor representations, known as a separated representations, to address the curse of dimensionality. The result is an algorithm to solve optimal control problems which scales linearly with the number of states in a system, and is applicable to systems that are nonlinear with stochastic forcing in finitehorizon, average cost, and firstexit settings. The method is demonstrated on inverted pendulum, VTOL aircraft, and quadcopter models, with system dimension two, six, and twelve respectively. I.
Complexity Reduction, Cornices and Pruning
 Proc. of the International Conference on Tropical and Idempotent Mathematics, G.L. Litvinov and S.N. Sergeev (Eds.), AMS
"... Abstract. In maxplus based algorithms for curseofdimensionalityfree solution of HamiltonJacobiBellman partial differential equations, and in sensor tasking algorithms, one is faced with a certain computationalcomplexity growth that must be attenuated. At each step of these algorithms, the s ..."
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Cited by 5 (4 self)
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Abstract. In maxplus based algorithms for curseofdimensionalityfree solution of HamiltonJacobiBellman partial differential equations, and in sensor tasking algorithms, one is faced with a certain computationalcomplexity growth that must be attenuated. At each step of these algorithms, the solutions are represented as maxplus (or minplus) sums of simple functions. Our problem is: Given an approximate solution representation as a maxplus sum of M functions, find the best approximation as a maxplus sum of N functions (with N < M). The main result of the paper is that for certain classes of problems, the optimal reducedcomplexity representation is comprised of a subset of the original set of functions. 1.
THE CONTRACTION RATE IN THOMPSON METRIC OF Orderpreserving Flows On A Cone  Application To GENERALIZED RICCATI EQUATIONS
, 2012
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Certification of inequalities involving transcendental functions: combining sdp and maxplus approximation
 the Proceedings of the European Control Conference, ECC’13
, 2013
"... AbstractWe consider the problem of certifying an inequality of the form f (x) 0, ∀x ∈ K, where f is a multivariate transcendental function, and K is a compact semialgebraic set. We introduce a certification method, combining semialgebraic optimization and maxplus approximation. We assume that f i ..."
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Cited by 2 (2 self)
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AbstractWe consider the problem of certifying an inequality of the form f (x) 0, ∀x ∈ K, where f is a multivariate transcendental function, and K is a compact semialgebraic set. We introduce a certification method, combining semialgebraic optimization and maxplus approximation. We assume that f is given by a syntaxic tree, the constituents of which involve semialgebraic operations as well as some transcendental functions like cos, sin, exp, etc. We bound some of these constituents by suprema or infima of quadratic forms (maxplus approximation method, initially introduced in optimal control), leading to semialgebraic optimization problems which we solve by semidefinite relaxations. The maxplus approximation is iteratively refined and combined with branch and bound techniques to reduce the relaxation gap. Illustrative examples of application of this algorithm are provided, explaining how we solved tight inequalities issued from the Flyspeck project (one of the main purposes of which is to certify numerical inequalities used in the proof of the Kepler conjecture by Thomas Hales).