Unifying Condition Numbers for Linear Programming

Cheung, Dennis; Cucker, Felipe; Peña, Javier

doi:10.1287/moor.28.4.609.20520

Cited by 18 publications

(21 citation statements)

References 33 publications

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“…We first focus our attention to linear programming where a number of authors have studied the related idea of condition measures [13,16,36,39,40]. LPs are considered ill-conditioned if small modifications in the problem data can have a large effect on the solution; in particular, if they lead to changes in the optimal objective value, changes in primal or dual feasibility, or changes in the structure of the final LP basis.…”

Section: Selecting the Test Setmentioning

confidence: 99%

A hybrid branch-and-bound approach for exact rational mixed-integer programming

et al. 2013

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We present an exact rational solver for mixed-integer linear programming that avoids the numerical inaccuracies inherent in the floating-point computations used by existing software. This allows the solver to be used for establishing theoretical results and in applications where correct solutions are critical due to legal and financial consequences. Our solver is a hybrid symbolic/numeric implementation of LP-based branch-and-bound, using numerically-safe methods for all binding computations in the search tree. Computing provably accurate solutions by dynamically choosing the fastest of several safe dual bounding methods depending on the structure of the instance, our exact solver is only moderately slower than an inexact floating-point branch-and-bound solver. The software is incorporated into the SCIP optimization framework, using the exact LP solver QSopt_ex and the GMP arithmetic library. Computational results are presented for a suite of test instances taken from

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Section: Selecting the Test Setmentioning

confidence: 99%

A hybrid branch-and-bound approach for exact rational mixed-integer programming

et al. 2013

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“…For example, Dontchev et al [9] entails that a lower bound on the Lipschitz modulus of the optimal set mapping is given by the reciprocal to the distance to strong metric irregularity. See also Cheung et al [8] in relation to different concepts of distance to ill-posedness. Another approach may be developed from the analysis of local stability regions in Nožička et al [25] (recovered, e.g., in Sections 5.4 and 5.5 of Bank et al [1]).…”

Section: The Exact Modulus In the Finite Casementioning

confidence: 99%

On the Lipschitz Modulus of the Argmin Mapping in Linear Semi-Infinite Optimization

2007

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This paper is devoted to quantify the Lipschitzian behavior of the optimal solutions set in linear optimization under perturbations of the objective function and the right hand side of the constraints (inequalities). In our model, the set indexing the constraints is assumed to be a compact metric space and all coefficients depend continuously on the index. The paper provides a lower bound on the Lipschitz modulus of the optimal set mapping (also called argmin mapping), which, under our assumptions, is single-valued and Lipschitz continuous near the nominal parameter. This lower bound turns out to be the exact modulus in ordinary linear programming, as well as in the semi-infinite case under some additional hypothesis which always holds for dimensions n 3. The expression for the lower bound (or exact modulus) only depends on the nominal problem's coefficients, providing an operative formula from the practical side, specially in the particular framework of ordinary linear programming, where it constitutes the sharp Lipschitz constant. In the semi-infinite case, the problem of whether or not the lower bound equals the exact modulus for n > 3 under weaker hypotheses (or none) remains as an open problem.

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“…However, these condition numbers are associated with different problems and thus have a different nature. It can be shown, nevertheless, that both are particular cases of a natural condition number for a unique, unifying, linear programming problem [5]. Note also that when d is feasible, C(d) K(d).…”

Section: Remark 2 (I) It Follows From the Definition Ofmentioning

confidence: 99%

“…Therefore, it may happen that the linear conic system (6) has no solutions even though the system (5) has. The next result shows that for small enough this is not the case.…”

Section: Identifying the Optimal Basismentioning

confidence: 99%

Solving linear programs with finite precision: II. Algorithms

Cheung

Cucker

2006

Journal of Complexity

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We describe an algorithm that first decides whether the primal-dual pair of linear programsis feasible and in case it is, computes an optimal basis and optimal solutions. Here, A ∈ R m×n , b ∈ R m , c ∈ R n are given. Our algorithm works with finite precision arithmetic. Yet, this precision is variable and is adjusted during the algorithm. Both the finest precision required and the complexity of the algorithm depend on the dimensions n and m as well as on the condition K(A, b, c) introduced in D. Cheung and F. Cucker [Solving linear programs with finite precision: I. Condition numbers and random programs, Math. Program. 99 (2004) 175-196].

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Unifying Condition Numbers for Linear Programming

Cited by 18 publications

References 33 publications

A hybrid branch-and-bound approach for exact rational mixed-integer programming

A hybrid branch-and-bound approach for exact rational mixed-integer programming

On the Lipschitz Modulus of the Argmin Mapping in Linear Semi-Infinite Optimization

Solving linear programs with finite precision: II. Algorithms

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