Interval Analysis on Directed Acyclic Graphs for Global Optimization

Schichl, Hermann; Neumaier, Arnold

doi:10.1007/s10898-005-0937-x

Cited by 77 publications

(99 citation statements)

References 18 publications

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“…By Tarski's Fixed Point Theorem [28], the gfp of F is the join of all its pre-fixed points, so we can replace = with ⊆ in (12). The equivalence of (13) with (12) then follows by Prop.…”

Section: Theoremmentioning

confidence: 99%

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Feasibility-Based Bounds Tightening via Fixed Points

Belotti¹,

Cafieri²,

Lee³

et al. 2010

Combinatorial Optimization and Applications

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Abstract. The search tree size of the spatial Branch-and-Bound algorithm for Mixed-Integer Nonlinear Programming depends on many factors, one of which is the width of the variable ranges at every tree node. A range reduction technique often employed is called Feasibility Based Bounds Tightening, which is known to be practically fast, and is thus deployed at every node of the search tree. From time to time, however, this technique fails to converge to its limit point in finite time, thereby slowing the whole Branch-and-Bound search considerably. In this paper we propose a polynomial time method, based on solving a linear program, for computing the limit point of the Feasibility Based Bounds Tightening algorithm applied to linear equality and inequality constraints.

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“…By Tarski's Fixed Point Theorem [28], the gfp of F is the join of all its pre-fixed points, so we can replace = with ⊆ in (12). The equivalence of (13) with (12) then follows by Prop.…”

Section: Theoremmentioning

confidence: 99%

“…The FBBT was employed as a range reduction technique for Mixed Integer Linear Programs (MILP) in [9,10] and then for MINLPs in [11]. Within the context of GO, the FBBT was discussed in [12] and recently improved in [13] by considering its effect on common subexpressions of g(x).…”

Section: Introductionmentioning

confidence: 99%

Feasibility-Based Bounds Tightening via Fixed Points

Belotti¹,

Cafieri²,

Lee³

et al. 2010

Combinatorial Optimization and Applications

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show abstract

“…AMPL, ROSE, nauty [23], GAP [24]) held together via Unix scripts. In particular, ROSE is used to analyze an AMPL flat MP and produce its Directed Acyclic Graph (DAG) encoding [4,5], which is then fed into nauty in order to derive its symmetry group. ROSE can also obtain a convex relaxation of (4) based on the ideas given in [20,3,4].…”

Section: Expression Tree Librarymentioning

confidence: 99%

“…These tasks have been put together in higher-level reformulation solvers, e.g. writing a (linear) convex relaxation of a MINLP automatically [3,4]; writing a DAG representation of an AMPL-encoded MINLP [5,4]; writing a cdd [6] or Porta [7] representation of an AMPL-encoded LP. A list of applications of ROSE is given in Sect.…”

Section: Introductionmentioning

confidence: 99%

The Reformulation-Optimization Software Engine

Liberti¹,

Cafieri²,

Savourey³

2010

Mathematical Software – ICMS 2010

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Abstract. Most optimization software performs numerical computation, in the sense that the main interest is to find numerical values to assign to the decision variables, e.g. a solution to an optimization problem. In mathematical programming, however, a considerable amount of symbolic transformation is essential to solving difficult optimization problems, e.g. relaxation or decomposition techniques. This step is usually carried out by hand, involves human ingenuity, and often constitutes the "theoretical contribution" of some research papers. We describe a Reformulation-Optimization Software Engine (ROSE) for performing (automatic) symbolic computation on mathematical programming formulations.

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“…The price winning solver Baron by Sahinidis & Tawarmalani [41] also uses constraint propagation techniques. Initiated by the development of interval analysis on DAGs (Directed Acyclic Graphs) by Schichl & Neumaier [43], advanced constraint propagation techniques for solving numerical constraint satisfaction problems have been given in Vu et al [44,45], which today belong to the most efficient implementations of basic constraint propagation algorithms for individual operations. It is included in the global optimization software platform COCONUT Environment [3,42].…”

Section: Introductionmentioning

confidence: 99%

Constraint propagation on quadratic constraints

Domes

Neumaier

2009

Constraints

Self Cite

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Abstract. This paper considers constraint propagation methods for continuous constraint satisfaction problems consisting of linear and quadratic constraints. All methods can be applied after suitable preprocessing to arbitrary algebraic constraints.The basic new techniques consist in eliminating bilinear entries from a quadratic constraint, and solving the resulting separable quadratic constraints by means of a sequence of univariate quadratic problems. Care is taken to ensure that all methods correctly account for rounding errors in the computations.

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Interval Analysis on Directed Acyclic Graphs for Global Optimization

Cited by 77 publications

References 18 publications

Feasibility-Based Bounds Tightening via Fixed Points

Feasibility-Based Bounds Tightening via Fixed Points

The Reformulation-Optimization Software Engine

Constraint propagation on quadratic constraints

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