A Computationally Efficient FPTAS for Convex Stochastic Dynamic Programs

Halman, Nir; Nannicini, Giacomo; Orlin, James B.

doi:10.1007/978-3-642-40450-4_49

Cited by 5 publications

(14 citation statements)

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“…The framework of [13] deals with stochastic discrete DPs in which the r.v.s are described explicitly as lists of scenarios (d, Pr(D = d)), and the single period cost functions possess either monotone or convex structure. [16] studies a subclass of the DP model of [13], in which the single period cost functions are assumed to possess convex structure, and provides a faster FPTAS from both theoretical worst-case upper bounds and practical standpoint. An extension of [13] to continuous state and action spaces is given in [15]: however, [15] still deals with scalar state and action spaces, and discrete (scalar) r.v.s.…”

Section: Relevance To Existing Literaturementioning

confidence: 99%

“…Algorithm 6 Procedure APXSCHEME1( ) for Condition 3(i) 1: K ← 2T √ 1 + ,ẑ T +1 ← g T +1 2: for t := T downto 1 do 3:F g ← COMPRESSCONVOLUTION( D t , σ D , K) 4:F z ← COMPRESSCONVOLUTION( D t , θ D , K) 5: For fixed I t , defineG D t ← COMPRESSEXPVAL(g D t , (1, σ I I t , 1),F g ) /*G D t (·) is an oracle for a K-approximation of E[g D t (· + σ I I t + σ D · D t )] */ 6: For fixed I t , defineZ t+1 ← COMPRESSEXPVAL(ẑ t+1 , (1, θ I I t , 1),F z ) /*Z t+1 (·) is an oracle for a K-approximation of E[ẑ t+1 (· + θ I I t + θ D · D t )] */ 7:ẑ t ← SCALEDCOMPRESSCONV(z t , [S min t , S max t ], K), wherez t is defined as in (16) 8: returnẑ 1 by approximations. The fact that the resulting value is a max{K 1 K 2 , K 3 } approximation of z t (I t ) follows from Prop.…”

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confidence: 99%

“…The main workhorse of the FPTAS is step 7, that computes an approximationẑ t for the value function z t , using the functionz t as defined in (16). Note that by Prop.…”

mentioning

confidence: 99%

“…We first analyze the size of the LP (16). The variables y, w used in (16) can be substituted out. Thus, (16) has p + 3 variables: p variables in the description of A t (I t ), plus one variable for the minmax formulation of each of the convex functions…”

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confidence: 99%

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Toward Breaking the Curse of Dimensionality: An FPTAS for Stochastic Dynamic Programs with Multidimensional Actions and Scalar States

Halman¹,

Nannicini²

2019

SIAM J. Optim.

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We propose a Fully Polynomial-Time Approximation Scheme (FPTAS) for stochastic dynamic programs with multidimensional action, scalar state, convex costs and linear state transition function. The action spaces are polyhedral and described by parametric linear programs. This type of problems finds applications in the area of optimal planning under uncertainty, and can be thought of as the problem of optimally managing a single non-discrete resource over a finite time horizon. We show that under a value oracle model for the cost functions this result for one-dimensional state space is "best possible", because a similar dynamic programming model with two-dimensional state space does not admit a PTAS.The FPTAS relies on the solution of polynomial-sized linear programs to recursively compute an approximation of the value function at each stage. Our paper enlarges the class of dynamic programs that admit an FPTAS by showing, under suitable conditions, how to deal with multidimensional action spaces and with vectors of continuous random variables with bounded support. These results bring us one step closer to overcoming the curse of dimensionality of dynamic programming.

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Section: Relevance To Existing Literaturementioning

confidence: 99%

mentioning

confidence: 99%

“…The main workhorse of the FPTAS is step 7, that computes an approximationẑ t for the value function z t , using the functionz t as defined in (16). Note that by Prop.…”

mentioning

confidence: 99%

mentioning

confidence: 99%

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Toward Breaking the Curse of Dimensionality: An FPTAS for Stochastic Dynamic Programs with Multidimensional Actions and Scalar States

Halman¹,

Nannicini²

2019

SIAM J. Optim.

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“…In this paper, we attempt a generalization of the results of Halman et al (2008Halman et al ( , 2011Halman et al ( , 2009Halman et al ( , 2013 for fixed dimensions k 1 , k 2 , and k 3 . By assuming L q -convexity of the costto-go functions, we obtain additive error approximations of these functions by storing their values on subsets of their domains.…”

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confidence: 99%

Fixed-Dimensional Stochastic Dynamic Programs: An Approximation Scheme and an Inventory Application

Chen

Dawande

Janakiraman

2014

Operations Research

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We study fixed-dimensional stochastic dynamic programs in a discrete setting over a finite horizon. Under the primary assumption that the cost-to-go functions are discrete L q-convex, we propose a pseudo-polynomial time approximation scheme that solves this problem to within an arbitrary prespecified additive error of s > 0. The proposed approximation algorithm is a generalization of the explicit-enumeration algorithm and offers us full control in the trade-off between accuracy and running time. The main technique we develop for obtaining our scheme is approximation of a fixed-dimensional L q-convex function on a bounded rectangular set, using only a selected number of points in its domain. Furthermore, we prove that the approximation function preserves L q-convexity. Finally, to apply the approximate functions in a dynamic program, we bound the error propagation of the approximation. Our approximation scheme is illustrated on a well-known problem in inventory theory, the single-product problem with lost sales and lead times. We demonstrate the practical value of our scheme by implementing our approximation scheme and the explicit-enumeration algorithm on instances of this inventory problem.

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Knapsack problem with objective value gaps

Dolgui¹,

Kovalyov

Quilliot

2016

Optim Lett

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International audienceWe study a 0-1 knapsack problem, in which the objective value is forbidden to take some values. We call gaps related forbidden intervals. The problem is NP-hard and pseudo-polynomially solvable independently on the measure of gaps. If the gaps are large, then the problem is polynomially non-approximable. A non-trivial special case with respect to the approximate solution appears when the gaps are small and polynomially close to zero. For this case, two fully polynomial time approximation schemes are proposed. The results can be extended for the constrained longest path problem and other combinatorial problems

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A Computationally Efficient FPTAS for Convex Stochastic Dynamic Programs

Cited by 5 publications

References 11 publications

Toward Breaking the Curse of Dimensionality: An FPTAS for Stochastic Dynamic Programs with Multidimensional Actions and Scalar States

Toward Breaking the Curse of Dimensionality: An FPTAS for Stochastic Dynamic Programs with Multidimensional Actions and Scalar States

Fixed-Dimensional Stochastic Dynamic Programs: An Approximation Scheme and an Inventory Application

Knapsack problem with objective value gaps

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