Total variation bounds on the expectation of periodic functions with applications to recourse approximations

Romeijnders, Ward; Vlerk, Maarten H. van der; Haneveld, Willem K. Klein

doi:10.1007/s10107-014-0829-2

Cited by 16 publications

(82 citation statements)

References 32 publications

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“…This is confirmed by numerical experiments in [42]. A tighter error bound is derived for an alternative convex approximation, called the shifted LP-relaxation approximation; see [41]. In fact, it is shown that the error bound is the best possible in a worst-case sense.…”

Section: Solution Methods For Risk-neutral Mixed-integer Recourse Modelsmentioning

confidence: 68%

“…For both approximations, a corresponding asymptotic error bound is derived, which converges to zero as the total variations of the density functions in the model go to zero. These bounds are derived by exploiting asymptotic periodicity of the second-stage value functions in combination with the total variation bounds from [41].…”

Section: Solution Methods For Risk-neutral Mixed-integer Recourse Modelsmentioning

confidence: 99%

“…3 we use the asymptotic periodicity of mixed-integer value functions to derive convex approximations for general mixedinteger mean-CVaR recourse models. Moreover, we derive error bounds for these convex approximations using the total variation error bounds on the expectation of periodic functions from [41]. We also use these total variation bounds in Sect.…”

Section: Solution Methods For Risk-neutral Mixed-integer Recourse Modelsmentioning

confidence: 99%

“…Note that this is the approximation error for a risk-neutral recourse function. Hence, we could directly apply existing results from the risk-neutral literature [41] to obtain an upper bound. Using this idea, we take the approach of conditioning on two complementary cases.…”

Section: Error Boundsmentioning

confidence: 99%

“…In this subsection we study the behavior of the error bounds from Theorem 3 in the special case of so-called one-dimensional simple integer recourse (SIR). Similar as in the risk-neutral case [23,28,41], we can exploit the special structure of this problem to construct a convex approximation with a sharp error bound. Surprisingly, for random variables h with a non-increasing positive tail, the error bound depends on the hazard rate of the distribution of h. Contrary to the bound in Theorem 1 from Sect.…”

Section: Simple Integer Recoursementioning

confidence: 99%

See 4 more Smart Citations

Convex approximations for two-stage mixed-integer mean-risk recourse models with conditional value-at-risk

Beesten

Romeijnders

2019

Math. Program.

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In traditional two-stage mixed-integer recourse models, the expected value of the total costs is minimized. In order to address risk-averse attitudes of decision makers, we consider a weighted mean-risk objective instead. Conditional value-at-risk is used as our risk measure. Integrality conditions on decision variables make the model non-convex and hence, hard to solve. To tackle this problem, we derive convex approximation models and corresponding error bounds, that depend on the total variations of the density functions of the random right-hand side variables in the model. We show that the error bounds converge to zero if these total variations go to zero. In addition, for the special cases of totally unimodular and simple integer recourse models we derive sharper error bounds. Keywords Stochastic programming • Mean-risk models • Conditional value-at-risk • Mixed-integer recourse • Convex approximations Mathematics Subject Classification 90C11 • 90C15 • 90C59 Maarten H. van der Vlerk was actively involved in an early stage of this research, but passed away during the writing of this paper.

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Section: Solution Methods For Risk-neutral Mixed-integer Recourse Modelsmentioning

confidence: 68%

Section: Solution Methods For Risk-neutral Mixed-integer Recourse Modelsmentioning

confidence: 99%

Section: Solution Methods For Risk-neutral Mixed-integer Recourse Modelsmentioning

confidence: 99%

Section: Error Boundsmentioning

confidence: 99%

Section: Simple Integer Recoursementioning

confidence: 99%

See 3 more Smart Citations

Convex approximations for two-stage mixed-integer mean-risk recourse models with conditional value-at-risk

Beesten

Romeijnders

2019

Math. Program.

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Stochastic Mixed-Integer Programming

Haneveld

Vlerk

Romeijnders

2019

Stochastic Programming

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A loose Benders decomposition algorithm for approximating two-stage mixed-integer recourse models

Laan

Romeijnders

2020

Math. Program.

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We propose a new class of convex approximations for two-stage mixed-integer recourse models, the so-called generalized alpha-approximations. The advantage of these convex approximations over existing ones is that they are more suitable for efficient computations. Indeed, we construct a loose Benders decomposition algorithm that solves large problem instances in reasonable time. To guarantee the performance of the resulting solution, we derive corresponding error bounds that depend on the total variations of the probability density functions of the random variables in the model. The error bounds converge to zero if these total variations converge to zero. We empirically assess our solution method on several test instances, including the SIZES and SSLP instances from SIPLIB. We show that our method finds near-optimal solutions if the variability of the random parameters in the model is large. Moreover, our method outperforms existing methods in terms of computation time, especially for large problem instances.

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Total variation bounds on the expectation of periodic functions with applications to recourse approximations

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Cited by 16 publications

References 32 publications

Convex approximations for two-stage mixed-integer mean-risk recourse models with conditional value-at-risk

Convex approximations for two-stage mixed-integer mean-risk recourse models with conditional value-at-risk

Stochastic Mixed-Integer Programming

A loose Benders decomposition algorithm for approximating two-stage mixed-integer recourse models

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