A scaling algorithm for polynomial constraint satisfaction problems

Domes, Ferenc; Neumaier, Arnold

doi:10.1007/s10898-008-9317-7

Cited by 6 publications

(14 citation statements)

References 18 publications

(35 reference statements)

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“…There are several novel branching strategies within Couenne [27]. -GloptLAB [42,43,44,45] GloptLAB is a Matlab-based framework for solving quadratic constraint satisfaction problems [42]. The GloptLAB bounding and scaling strategies are particularly interesting [43,44,45].…”

Section: Literature Reviewmentioning

confidence: 99%

“…-GloptLAB [42,43,44,45] GloptLAB is a Matlab-based framework for solving quadratic constraint satisfaction problems [42]. The GloptLAB bounding and scaling strategies are particularly interesting [43,44,45]. -LindoGLOBAL [63,87] Like αBB, BARON, and Couenne, LindoGLOBAL addresses generic MINLP to global optimality with specific routines for quadratic components.…”

Section: Literature Reviewmentioning

confidence: 99%

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GloMIQO: Global mixed-integer quadratic optimizer

2012

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This paper introduces the Global Mixed-Integer Quadratic Optimizer, GloMIQO, a numerical solver addressing mixed-integer quadratically-constrained quadratic programs (MIQCQP) to ε-global optimality. The algorithmic components are presented for: reformulating user input, detecting special structure including convexity and edge-concavity, generating tight convex relaxations, partitioning the search space, bounding the variables, and finding good feasible solutions. To demonstrate the capacity of GloMIQO, we extensively tested its performance on a test suite of 399 problems of diverse size and structure. The test cases are taken from process networks applications, computational geometry problems, GLOBALLib, MINLPLib, and the Bonmin test set. We compare the performance of GloMIQO with respect to four state-of-the-art global optimization solvers: BARON 10.1.2,

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Section: Literature Reviewmentioning

confidence: 99%

Section: Literature Reviewmentioning

confidence: 99%

GloMIQO: Global mixed-integer quadratic optimizer

2012

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“…Choosing µ too small may result in improvement of small bounds but it is not recommended since the gain is not significant enough compared to other methods presented in this paper. In oder to avoid numerical problems, in this section we also assume that we have found a suitable scaling vector ω ∈ R m for the constraints and a suitable scaling vector ρ ∈ R n for the variables (for finding these we refer to Domes & Neumaier [8]). …”

Section: Bounding a Polyhedronmentioning

confidence: 99%

“…We suggest the following choice of the scaling matrices U and V and the scaling constant δ: By the scaling method presented in Domes & Neumaier [8] we find matrices U and V such that the entries of U AV are between zero and one but not too close to zero. The second part of B = [U AV, δI m ] consists of an m × m identity matrix, scaled with the constant δ.…”

Section: Example 34 In Example 32 We Hadmentioning

confidence: 99%

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Rigorous filtering using linear relaxations

Domes

Neumaier

2011

J Glob Optim

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Abstract. This paper presents rigorous filtering methods for continuous constraint satisfaction problems based on linear relaxations. Filtering or pruning stands for reducing the search space of constraint satisfaction problems. Discussed are old and new approaches for rigorously enclosing the solution set of linear systems of inequalities, as well as different methods for computing linear relaxations. This allows custom combinations of relaxation and filtering. Care is taken to ensure that all methods correctly account for rounding errors in the computations. Although most of the results apply more generally, strong emphasis is given to relaxing and filtering quadratic constraints, as implemented in the GloptLab environment, which internally exploits a quadratic structure. Demonstrative examples and tests comparing the different linear relaxation methods are also presented.

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Rigorous verification of feasibility

Domes

Neumaier

2014

J Glob Optim

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This paper considers the problem of finding and verifying feasible points for constraint satisfaction problems, including those with uncertain coefficients. The main part is devoted to the problem of finding a narrow box around an approximately feasible solution for which it can be rigorously and automatically proved that it contains a feasible solution. Some examples demonstrate difficulties when attempting verification.We review some existing methods and present a new method for verifying the existence of feasible points of constraint satisfaction problems in an automatically determined narrow box. Numerical tests within GloptLab, a solver developed by the authors, show how the different methods perform. Also discussed are the search for approximately feasible points and the use of approximately feasible points within a branch and bound scheme for constraint satisfaction.

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A scaling algorithm for polynomial constraint satisfaction problems

Cited by 6 publications

References 18 publications

GloMIQO: Global mixed-integer quadratic optimizer

GloMIQO: Global mixed-integer quadratic optimizer

Rigorous filtering using linear relaxations

Rigorous verification of feasibility

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