Complexity Estimates for Fourier-Motzkin Elimination

Jing, Rui-Juan; Moreno-Maza, Marc; Talaashrafi, Delaram

doi:10.1007/978-3-030-60026-6_16

Cited by 9 publications

(5 citation statements)

References 24 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For instance, the Fourier-Motzkin-Elimination algorithm [27], which is a well-known general projection method, is known for its double exponential complexity, that is O((|E| + |V |) 2 (|E|+|V |) ) for our case. A recently developed general algorithm [28] enjoys a single exponential complexity, but with much higher exponents; namely O((|E|…”

Section: B Implementationmentioning

confidence: 99%

“…In general, finding a projection from a space with a higher dimension is difficult. Applying general-purpose projection methods [27], [28] requires a prohibitive amount of computation, especially for large-scale instances. Additionally, finding a minimal representation for this projection is important because performance in joint chance-constrained programs depends significantly on the number of constraints that are contained in the probabilistic constraints.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Exact Mixed-Integer Programming Approach for Chance-Constrained Multi-Area Reserve Sizing

Cho

Papavasiliou

2024

IEEE Trans. Power Syst.

View full text Add to dashboard Cite

An exact algorithm is developed for the chanceconstrained multi-area reserve sizing problem in the presence of transmission network constraints. The problem can be cast as a two-stage stochastic mixed integer linear program using sample approximation. Due to the complicated structure of the problem, existing methods attempt to find a feasible solution based on heuristics. Existing mixed-integer algorithms that can be applied directly to a two-stage stochastic program can only address smallscale problems that are not practical. We have found a minimal description of the projection of our problem onto the space of the first-stage variables. This enables us to directly apply more general Integer Programming techniques for mixing sets, that arise in chance-constrained problems. Combining the advantages of the minimal projection and the strengthening reformulation from IP techniques, our method can tackle real-world problems. We specifically consider a case study of the 10-zone Nordic network with 100,000 scenarios where the optimal solution can be found in approximately 5 minutes.Index Terms-Multi-area reserve sizing, chance constraints, probabilistic constraints, mixed-integer programming 2 The term joint comes from the fact that a probabilistic requirement is imposed on multiple constraints simultaneously.

show abstract

Section: B Implementationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Exact Mixed-Integer Programming Approach for Chance-Constrained Multi-Area Reserve Sizing

Cho

Papavasiliou

2024

IEEE Trans. Power Syst.

View full text Add to dashboard Cite

show abstract

“…Redundancies might already be contained in the input system, or they arise during the projection operation. While removing all redundancies is expensive, there are efficient methods for removing some redundancies of the latter type, for example Imbert's acceleration theorems [10,11,12]. 1 Remember that we use lower case letters for rows of matrices with the respective upper case letter as name.…”

Section: Fourier-motzkin Variable Eliminationmentioning

confidence: 99%

“…In contrast to the other two approaches, FM admits quantifier elimination, but it has a doubly exponential worst case complexity, even though there have been various efforts to improve its efficiency by recognizing and avoiding redundant computations (e.g. [11,12]).…”

Section: Introductionmentioning

confidence: 99%

FMplex: A Novel Method for Solving Linear Real Arithmetic Problems

Nalbach,

Promies,

Ábrahám

et al. 2023

Electron. Proc. Theor. Comput. Sci.

View full text Add to dashboard Cite

In this paper we introduce a novel quantifier elimination method for conjunctions of linear real arithmetic constraints. Our algorithm is based on the Fourier-Motzkin variable elimination procedure, but by case splitting we are able to reduce the worst-case complexity from doubly to singly exponential. The adaption of the procedure for SMT solving has strong correspondence to the simplex algorithm, therefore we name it FMplex. Besides the theoretical foundations, we provide an experimental evaluation in the context of SMT solving.

show abstract

“…) involving only m 1 and n 1 for W ∈ R m1×n1 (see [20,30,33]). Thus, |J| < +∞ regardless of the cardinalities of {W ξ } ⊆ R m1×n1 .…”

Section: Examples and Special Structuresmentioning

confidence: 99%

General Feasibility Bounds for Sample Average Approximation via Vapnik-Chervonenkis Dimension

Lam¹,

Li²

2021

Preprint

View full text Add to dashboard Cite

We investigate the feasibility of sample average approximation (SAA) for general stochastic optimization problems, including two-stage stochastic programming without the relatively complete recourse assumption. Instead of analyzing problems with specific structures, we utilize results from the Vapnik-Chervonenkis (VC) dimension and Probably Approximately Correct learning to provide a general framework that offers explicit feasibility bounds for SAA solutions under minimal structural or distributional assumption. We show that, as long as the hypothesis class formed by the feasbible region has a finite VC dimension, the infeasibility of SAA solutions decreases exponentially with computable rates and explicitly identifiable accompanying constants. We demonstrate how our bounds apply more generally and competitively compared to existing results.

show abstract

Complexity Estimates for Fourier-Motzkin Elimination

Cited by 9 publications

References 24 publications

Exact Mixed-Integer Programming Approach for Chance-Constrained Multi-Area Reserve Sizing

Exact Mixed-Integer Programming Approach for Chance-Constrained Multi-Area Reserve Sizing

FMplex: A Novel Method for Solving Linear Real Arithmetic Problems

General Feasibility Bounds for Sample Average Approximation via Vapnik-Chervonenkis Dimension

Contact Info

Product

Resources

About