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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
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.
Minplus techniques for setvalued state estimation. Arxiv preprint arXiv:1203.2846 (under review
, 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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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).
Path Integral Formulation of Stochastic Optimal Control with Generalized Costs?
"... Abstract: Path integral control solves a class of stochastic optimal control problems with a Monte Carlo (MC) method for an associated HamiltonJacobiBellman (HJB) equation. The MC approach avoids the need for a global grid of the domain of the HJB equation and, therefore, path integral control is ..."
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Cited by 1 (1 self)
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Abstract: Path integral control solves a class of stochastic optimal control problems with a Monte Carlo (MC) method for an associated HamiltonJacobiBellman (HJB) equation. The MC approach avoids the need for a global grid of the domain of the HJB equation and, therefore, path integral control is in principle applicable to control problems of moderate to large dimension. The class of problems path integral control can solve, however, is defined by requirements on the cost function, the noise covariance matrix and the control input matrix. We relax the requirements on the cost function by introducing a new state that represents an augmented running cost. In our new formulation the cost function can contain stochastic integral terms and linear control costs, which are important in applications in engineering, economics and finance. We find an efficient numerical implementation of our gridfree MC approach and demonstrate its performance and usefulness in examples from hierarchical electric load management. The dimension of one of our examples is large enough to make classical gridbased HJB solvers impractical. 1.
Idempotent Method for Dynamic Games and Complexity Reduction in MinMax Expansions
"... Abstract — In recent years, idempotent methods (specifically, maxplus methods) have been developed for solution of nonlinear control problems. It was thought that idempotent linearity of the associated semigroup was required for application of these techniques. It is now known that application of ..."
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Abstract — In recent years, idempotent methods (specifically, maxplus methods) have been developed for solution of nonlinear control problems. It was thought that idempotent linearity of the associated semigroup was required for application of these techniques. It is now known that application of the maxplus distributive property allows one to apply the maxplus curseofdimensionalityfree approach to stochastic control problems. Here, we see that a similar, albeit more abstract, approach can be applied to deterministic game problems. The main difficulty is a curseofcomplexity growth in the computational cost. Attenuation of this effect requires finding reducedcomplexity approximations to minmax sums of maxplus affine functions. We demonstrate that that problem can be reduced to a pruning problem. I.
function approximation for Markov decision processes.
, 2014
"... Approximate dynamic programming with (min;+) linear function approximation for Markov decision processes. ..."
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Approximate dynamic programming with (min;+) linear function approximation for Markov decision processes.
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Public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing this collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing