Inexact stochastic mirror descent for two-stage nonlinear stochastic programs

Guigues, Vincent

doi:10.1007/s10107-020-01490-5

Cited by 14 publications

(27 citation statements)

References 18 publications

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“…Proof of formula (2.48). We prove (2.48) adapting the proof of Lemma 2.1 in [6] to the special case of value function…”

Section: End If End For End Formentioning

confidence: 94%

“…When X t is polyhedral, formula (2.48) follows from Duality for linear programming. For a more general convex set X t , formula (2.48) directly follows from applying to value function Q k−1 t Lemma 2.1 in [3] or Proposition 3.2 in [6] which respectively provide a characterization of the subdifferential and subgradients for value functions of general convex optimization problems (whose argument is in the objective function and in linear and nonlinear coupling constraints of the corresponding optimization problem). For the interested reader and for the sake of completeness, we provide in the Appendix a proof of relation (2.48) specializing to the particular case of value function Q k−1 t the proof of Lemma 2.1 in [3].…”

Section: Computation Of the Subgradient Inmentioning

confidence: 99%

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Stochastic Dynamic Cutting Plane for multistage stochastic convex programs

Guigues¹,

Monteiro²

2019

Preprint

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We introduce StoDCuP (Stochastic Dynamic Cutting Plane), an extension of the Stochastic Dual Dynamic Programming (SDDP) algorithm to solve multistage stochastic convex optimization problems. At each iteration, the algorithm builds lower bounding affine functions not only for the cost-to-go functions, as SDDP does, but also for some or all nonlinear cost and constraint functions. We show the almost sure convergence of StoDCuP. We also introduce an inexact variant of StoDCuP where all subproblems are solved approximately (with bounded errors) and show the almost sure convergence of this variant for vanishing errors.

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“…Proof of formula (2.48). We prove (2.48) adapting the proof of Lemma 2.1 in [6] to the special case of value function…”

Section: End If End For End Formentioning

confidence: 94%

Section: Computation Of the Subgradient Inmentioning

confidence: 99%

Stochastic Dynamic Cutting Plane for multistage stochastic convex programs

Guigues¹,

Monteiro²

2019

Preprint

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“…This task can be easily achieved for value functions of linear programs, see for instance Proposition 2.1 in [13]. For nonlinear differentiable problems, the derivation of inexact cuts is given in Propositions 2.2 and 2.3 in [13] and Proposition 3.8 in [10]. However, this task is more delicate for nondifferentiable optimization problems.…”

mentioning

confidence: 99%

“…Due to (H0) value function Q is convex and if x ∈ ri(dom(Q)) then Q is subdifferentiable at x and there exists a cut (a lower bounding affine function) for Q at x which coincides with Q at x. More generally, under some assumptions, the characterization of the subdifferential of Q at x ∈ X was given in [12, Lemma 2.1] and formulas for affine lower bounding functions for Q were derived in [10,Proposition 3.2] on the basis of optimal primal-dual solutions to (1.1). When only approximate primal-dual solutions are available, we can only compute inexact cuts which are still lower bounding functions for the value function but which do not coincide with this function at the point x used to compute the cut.…”

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confidence: 99%

Inexact cuts in SDDP applied to multistage stochastic nondifferentiable problems

Guigues,

Monteiro,

Svaiter

2020

Preprint

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In [13], an Inexact variant of Stochastic Dual Dynamic Programming (SDDP) called ISDDP was introduced which uses approximate (instead of exact with SDDP) primal dual solutions of the problems solved in the forward and backward passes of the method. That variant of SDDP was studied in [13] for linear and for differentiable nonlinear Multistage Stochastic Programs (MSPs). In this paper, we extend ISDDP to nondifferentiable MSPs. We first provide formulas for inexact cuts for value functions of convex nondifferentiable optimization problems. We then combine these cuts with SDDP to describe ISDDP for nondifferentiable MSPs and analyze the convergence of the method. More precisely, for a problem with T stages, we show that for errors bounded from above by ε, the limit superior and limit inferior of sequences of upper and lower bounds on the optimal value of the problem are at most at distance 3εT to the optimal value and that for asymptotically vanishing errors ISDDP converges to an optimal policy.

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“…Duality is also a fundamental tool in the reformulation of Robust Optimization problems, see for instance [3]. Finally, derivatives of the value function of classes of optimization problems can be related to optimal dual solutions, see [5], [22] and more recently [8,10,11] for the characterization of subdifferentials, subgradients, and ε-subgradients of value functions of convex optimization problems.…”

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confidence: 99%

Duality and sensitivity analysis of multistage linear stochastic programs

Guigues¹,

Shapiro²,

Cheng³

2019

Preprint

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In this paper we investigate the dual of a Multistage Stochastic Linear Program (MSLP) to study two related questions for this class of problems. The first of these questions is the study of the optimal value of the problem as a function of the involved parameters. For this sensitivity analysis problem, we provide formulas for the derivatives of the value function with respect to the parameters and illustrate their application on an inventory problem. Since these formulas involve optimal dual solutions, we need an algorithm that computes such solutions to use them, i.e., we need to solve the dual problem.In this context, the second question we address is the study of solution methods for the dual problem. Writing Dynamic Programming equations for the dual, we can use an SDDP type method, called Dual SDDP, which solves these Dynamic Programming equations computing a sequence of nonincreasing deterministic upper bounds on the optimal value of the problem. However, applying this method will only be possible if the Relatively Complete Recourse (RCR) holds for the dual. Since the RCR assumption may fail to hold (even for simple problems), we design two variants of Dual SDDP, namely Dual SDDP with penalizations and Dual SDDP with feasibility cuts, that converge to the optimal value of the dual (and therefore primal when there is no duality gap) problem under mild assumptions. We also show that optimal dual solutions can be obtained computing dual solutions of the subproblems solved when applying Primal SDDP to the original primal MSLP.The study of this second question allows us to take a fresh look at the notoriously difficult to solve class of MSLP with interstage dependent cost coefficients. Indeed, for this class of problems, cost-to-go functions are non-convex and solution methods were so far using SDDP for a Markov chain approximation of the cost coefficients process. For these problems, we propose to apply Dual SDDP with penalizations to the cost-to-go functions of the dual which are concave. This algorithm converges to the optimal value of the problem.Finally, as a proof of concept of the tools developed, we present the results of numerical experiments computing the sensitivity of the optimal value of an inventory problem as a function of parameters of the demand process and compare Primal and Dual SDDP on the inventory and a hydro-thermal planning problems.

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Inexact stochastic mirror descent for two-stage nonlinear stochastic programs

Cited by 14 publications

References 18 publications

Stochastic Dynamic Cutting Plane for multistage stochastic convex programs

Stochastic Dynamic Cutting Plane for multistage stochastic convex programs

Inexact cuts in SDDP applied to multistage stochastic nondifferentiable problems

Duality and sensitivity analysis of multistage linear stochastic programs

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