Recourse-based stochastic nonlinear programming: properties and Benders-SQP algorithms

Kulkarni, Ankur A.; Shanbhag, Uday V.

doi:10.1007/s10589-010-9316-8

Cited by 13 publications

(4 citation statements)

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“…Convergence theory for the obtained estimators is examined by Shapiro [18]. Decomposition schemes, that leverage cutting-plane methods, have also been particularly successful in addressing two-period stochastic linear [36], convex [37] and nonconvex programs [38] while a scalable matrixsplitting decomposition scheme is presented in [39] for two-period stochastic Nash games. In this paper, we consider adaptive stochastic gradient and subgradient methods for solving constrained stochastic convex optimization problems.…”

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confidence: 99%

On stochastic gradient and subgradient methods with adaptive steplength sequences

2012

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Traditionally, stochastic approximation (SA) schemes have been popular choices for solving stochastic optimization problems. However, the performance of standard SA implementations can vary significantly based on the choice of the steplength sequence, and in general, little guidance is provided about good choices. Motivated by this gap, in the first part of the paper, we present two adaptive steplength schemes for strongly convex differentiable stochastic optimization problems, equipped with convergence theory, that aim to overcome some of the reliance on user-specific parameters. Of these, the first scheme, referred to as a recursive steplength stochastic approximation (RSA) scheme, optimizes the error bounds to derive a rule that expresses the steplength at a given iteration as a simple function of the steplength at the previous iteration and certain problem parameters. The second scheme, termed as a cascading steplength stochastic approximation (CSA) scheme, maintains the steplength sequence as a piecewise-constant decreasing function with the reduction in the steplength occurring when a suitable error threshold is met.In the second part of the paper, we allow for nondifferentiable objectives but with bounded subgradients over a certain domain. In such a regime, we propose a local smoothing technique, based on random local perturbations of the objective function, that leads to a differentiable approximation of the function. Assuming a uniform distribution on the local randomness, we establish a Lipschitzian property for the gradient of the approximation and prove that the obtained Lipschitz bound grows at a modest rate with problem size. This facilitates the development of an adaptive steplength stochastic approximation framework, which now requires sampling in the product space of the original measure and the artificially introduced distribution. The resulting adaptive steplength schemes are applied to three stochastic optimization problems. In particular, we observe that both schemes perform well in practice and display markedly less reliance on user-defined parameters. I. INTRODUCTIONThe use of stochastic gradient and subgradient schemes for the solution of stochastic convex optimization problems has a long tradition, beginning with an iterative scheme, first proposed by Robbins and Monro [1], that relied primarily on noisy gradient observations. Research by Ermoliev and his coauthors [2]-[5] focused largely on quasigradient (subgradient) methods and considered a host of stochastic programming problems, amongst them being two-period recourse-based problems (see [6]). To accelerate the convergence of stochastic subgradient methods, ergodic sequences, arising from the averaging of iterates, have been employed in [7]-[10]. Often gradient computations are either costly or unavailable; in such instances, a finite-difference approximation of the gradient can be constructed as first observed by Kiefer and Wolfowitz [11]. While standard finite-difference techniques perturb one direction at a time to obtain gradient...

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On stochastic gradient and subgradient methods with adaptive steplength sequences

2012

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“…In order to handle this problem, nonlinear stochastic programming approaches and solution strategies are proposed in the related literature (see, for instance, Kulkarni and Shanbhag, 2012;Beraldi et al, 2009;Shastri and Diwekar, 2009).…”

Section: Different Demand-price Modelsmentioning

confidence: 99%

Product-line planning under uncertainty

Karakaya

Köksal

2022

Computers & Operations Research

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pagesThis study addresses the problem of multi-period mix of product-lines under a product-family, which incorporates launching decisions of new products, capacity expansion decisions and product interdependencies. The problem is modelled as a two-stage stochastic program with recourse in which price, demand, production cost and cannibalisation effect of new products are treated as uncertain parameters.The solution approach employs the Sample Average Approximation based on Monte Carlo bounding technique and multi-cut version of L-shaped method to solve approximate problems efficiently, which is tested on different cases considering VSS and EVPI performance measures. The data collected through two experimental studies is analysed using ANOVA and Random Forest methodology in order to understand which problem parameters are significant on the performance measures and to generate some rule-based inferences reflecting the relationship between significant parameters and the performance of the proposed stochastic model.

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“…In [43] and [13], the authors work with the recourse subproblems' dual that is approximated by a sequence of quadratic program subproblems, and propose a Lagrangian finite generation technique and a Newton's method to solve the problem, respectively. In [29,48], Dorn duality is employed in developing a Benders scheme. Motivated by this, we incorporate quadratic recourse in the stochastic Nash game and consider the two-stage problem (SNash rec (x −i )), for which c i (x i ) is a continuously differentiable convex cost of x i , and the recourse function Q i (x i , ω) is the optimal value of a convex quadratic program parameterized by the decision x i and scenario ω:…”

Section: Quadratic Recoursementioning

confidence: 99%

On Synchronous, Asynchronous, and Randomized Best-Response Schemes for Stochastic Nash Games

et al. 2020

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In this paper, we consider a stochastic Nash game in which each player minimizes a parameterized expectationvalued convex objective function. In deterministic regimes, proximal best-response (BR) schemes have been shown to be convergent under a suitable spectral property associated with the proximal BR map. However, a direct application of this scheme to stochastic settings requires obtaining exact solutions to stochastic optimization problems at each iteration. Instead, we propose an inexact generalization of this scheme in which an inexact solution to the BR problem is computed in an expected-value sense via a stochastic approximation (SA) scheme. On the basis of this framework, we present three inexact BR schemes: (i) First, we propose a synchronous inexact BR scheme where all players simultaneously update their strategies; (ii) Second, we extend this to a randomized setting where a subset of players is randomly chosen to update their strategies while the other players keep their strategies invariant; (iii) Third, we propose an asynchronous scheme, where each player chooses its update frequency while using outdated rival-specific data in updating its strategy. Under a suitable contractive property on the proximal BR map, we proceed to derive a.s. convergence of the iterates to the Nash equilibrium (NE) for (i) and (ii) and mean-convergence for (i)-(iii). In addition, we show that for (i)-(iii), the generated iterates converge to the unique equilibrium in mean at a linear rate with a prescribed constant rather than a sub-linear rate. Finally, we establish the overall iteration complexity of the scheme in terms of projected stochastic gradient (SG) steps for computing an −Nash equilibrium and note that in all settings, the iteration complexity is O(1/ 2(1+c)+δ ) where c = 0 in the context of (i) and represents the positive cost of randomization in (ii) and asynchronicity and delay in (iii). Notably, in the synchronous regime, we achieve a near-optimal rate from the standpoint of solving stochastic convex optimization problems by SA schemes. The schemes are further extended to settings where players solve two-stage stochastic Nash games with linear and quadratic recourse. Finally, preliminary numerics developed on a multi-portfolio investment problem and a two-stage capacity expansion game support the rate and complexity statements.

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Recourse-based stochastic nonlinear programming: properties and Benders-SQP algorithms

Cited by 13 publications

References 61 publications

On stochastic gradient and subgradient methods with adaptive steplength sequences

On stochastic gradient and subgradient methods with adaptive steplength sequences

Product-line planning under uncertainty

On Synchronous, Asynchronous, and Randomized Best-Response Schemes for Stochastic Nash Games

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