A structure-conveying modelling language for mathematical and stochastic programming

Colombo, Marco; Grothey, Andreas; Hogg, JD; Woodsend, Kristian; Gondzio, Jacek

doi:10.1007/s12532-009-0008-2

Cited by 21 publications

(13 citation statements)

References 20 publications

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“…Standard packages can read structured MPS files without modification. (d) The last format is based on SML [15], a structure-conveying modelling language based on the popular AMPL [17] modeling language. In addition to hooking BlockIP to it, SML was extended to deal with nonlinear separable problems (see [24] for details).…”

Section: Solver Implementation Detailsmentioning

confidence: 99%

Interior-point solver for convex separable block-angular problems

Castro

2015

Optimization Methods and Software

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Constraints matrices with block-angular structures are pervasive in Optimization. Interior-point methods have shown to be competitive for these structured problems by exploiting the linear algebra. One of these approaches solved the normal equations using sparse Cholesky factorizations for the block constraints, and a preconditioned conjugate gradient (PCG) for the linking constraints. The preconditioner is based on a power series expansion which approximates the inverse of the matrix of the linking constraints system. In this work we present an efficient solver based on this algorithm. Some of its features are: it solves linearly constrained convex separable problems (linear, quadratic or nonlinear); both Newton and second-order predictor-corrector directions can be used, either with the Cholesky+PCG scheme or with a Cholesky factorization of normal equations; the preconditioner may include any number of terms of the power series; for any number of these terms, it estimates the spectral radius of the matrix in the power series (which is instrumental for the quality of the preconditioner). The solver has been hooked to SML, a structure-conveying modelling language based on the popular AMPL modeling language. Computational results are reported for some large and/or difficult instances in the literature: (1) multicommodity flow problems; (2) minimum congestion problems; (3) statistical data protection problems using ℓ 1 and ℓ 2 distances (which are linear and quadratic problems, respectively), and the pseudo-Huber function, a nonlinear approximation to ℓ 1 which improves the preconditioner. In the largest instances, of up to 25 millions of variables and 300000 constraints, this approach is from two to three orders of magnitude faster than state-of-the-art linear and quadratic optimization solvers.

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Section: Solver Implementation Detailsmentioning

confidence: 99%

Interior-point solver for convex separable block-angular problems

Castro

2015

Optimization Methods and Software

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“…Tebboth, in his thesis [26], derives some decomposable matrix structures of a linear program (LP) when it is formulated in a specific modeling language. A similar approach of indicating the matrix structure via key words in the modeling language is taken in [7,13,22,25] among others. All proposals have in common that the user, in one way or another, needs to make available her knowledge about the decomposition to be applied.…”

Section: Related Literaturementioning

confidence: 99%

Automatic Dantzig–Wolfe reformulation of mixed integer programs

et al. 2014

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International audienceDantzig–Wolfe decomposition (or reformulation) is well-known to provide strong dual bounds for specially structured mixed integer programs (MIPs). However, the method is not implemented in any state-of-the-art MIP solver as it is considered to require structural problem knowledge and tailoring to this structure. We provide a computational proof-of-concept that the reformulation can be automated. That is, we perform a rigorous experimental study, which results in identifying a score to estimate the quality of a decomposition: after building a set of potentially good candidates, we exploit such a score to detect which decomposition might be useful for Dantzig–Wolfe reformulation of a MIP. We experiment with general instances from MIPLIB2003 and MIPLIB2010 for which a decomposition method would not be the first choice, and demonstrate that strong dual bounds can be obtained from the automatically reformulated model using column generation. Our findings support the idea that Dantzig–Wolfe reformulation may hold more promise as a general-purpose tool than previously acknowledged by the research community

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“…The necessary model transformations, including the addition of any auxiliary variables and constraints, as well as transformation into a multilevel form in the cases of non-time-consistent risk measure systems, could then in principle be automated. Ideally, such facilities should be integrated into a modeling environment tailored to the efficient expression of stochastic programming problems; see for example Colombo et al (2009) and Valente et al (2009).…”

Section: Modeling Considerations and Concluding Remarksmentioning

confidence: 99%

Multilevel Optimization Modeling for Risk-Averse Stochastic Programming

Eckstein

Eskandani

Fan

2016

INFORMS Journal on Computing

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Abstract. Coherent risk measures have become a popular tool for incorporating risk aversion into stochastic optimization models. For dynamic models in which uncertainty is resolved at more than one stage, however, using coherent risk measures within a standard single-level optimization framework becomes problematic. To avoid severe time-consistency difficulties, the current state of the art is to employ risk measures of a specific nested form, which unfortunately have some undesirable and somewhat counterintuitive modeling properties. For one thing, this technique requires increasing risk aversion as risks and reward recede in time. Further, it produces objective functions that cannot be law invariant with respect to the total incurred costs and rewards, meaning that two solutions with identical probability distributions of final wealth may be assigned different levels of risk, and the nested form of the objective function cannot be simplified. These properties deter practical acceptance of such models, and are particularly undesirable for situations with close final time horizons. This paper summarizes these issues and then presents an alternative multilevel optimization modeling approach that enforces a form of time consistency through constraints rather than by restricting the modeler's choice of objective function. This technique leads to models that are time-consistent even while using time-inconsistent risk measures, and can easily be formulated to be law invariant with respect to the final wealth if so desired. We argue that this approach should be the starting point for all multistage optimization modeling. When used with time-consistent objective functions, we show its multilevel optimization constraints become redundant and the associated models thus simplify to a more familiar single-objective form. Unfortunately, this paper also shows that its proposed approach leads to N Phard models, even in the simplest imaginable setting in which it would be needed: three-stage linear problems on a finite probability space, using the standard meansemideviation and average-value-at-risk measures. Finally, we show that for a simple but reasonably realistic test application, that the kind of models we propose, while being drawn from an N P-hard family and certainly more time consuming to solve than those obtained from the nested-objective approach, are readily solvable to global optimality using a standard commercial MILP solver. Therefore, there seems some promise of the modeling approach being useful despite its computational complexity properties.Acknowledgements: This work was supported by a grant from the Air Force Office of Scientific Research, award FA9550-11-1-0164. The authors would like to thank Andrzej Ruszczyński for his assistance and support.Note: Sections 4-7 of this report essentially duplicate material presented in the earlier reports RRR 7-2013 and (using different notation) RRR 17-2012. The remaining sections contain extensive new material and results.

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A structure-conveying modelling language for mathematical and stochastic programming

Cited by 21 publications

References 20 publications

Interior-point solver for convex separable block-angular problems

Interior-point solver for convex separable block-angular problems

Automatic Dantzig–Wolfe reformulation of mixed integer programs

Multilevel Optimization Modeling for Risk-Averse Stochastic Programming

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