Interval mathematics, algebraic equations and optimization

Wolfe, M. A.

doi:10.1016/s0377-0427(00)00421-0

Cited by 26 publications

(9 citation statements)

References 44 publications

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“…Introduction. Mathematical programming involving uncertain coefficients or coefficients represented by interval numbers has attracted much attention in recent years [3,7,8,9]. This is because many real-world decision making problems contain imprecise, uncertain and incomplete data.…”

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confidence: 99%

A novel approach in uncertain programming part II: a class of constrained nonlinear programming problems with interval objective functions

Hu¹,

Wang²

2006

Journal of Industrial &Amp; Management Optimization

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In this paper we present a novel approach to a class of constrained nonlinear programming problems with interval objective functions. In this approach we first introduce a new concept of optimal solutions to the nonlinear programming problem, based on a new linear order relation for interval numbers proposed in Part 1 of this series. We then propose two efficient methods for the solution of the nonlinear optimization problem with respect to the new interval order relation. Comparisons between our approach and some existing methods will be given using illustrative examples. The numerical results show the superiority of our method to the existing ones.

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confidence: 99%

A novel approach in uncertain programming part II: a class of constrained nonlinear programming problems with interval objective functions

Hu¹,

Wang²

2006

Journal of Industrial &Amp; Management Optimization

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“…As a final and more general observation, we note that DIRECT has some similarities with interval arithmetic methods of global optimization (see the recent survey by Wolfe [15], for example) in that it selects potentially optimal hyperboxes on the basis of estimated bounds on the objective function. In principle, evaluating an interval extension of f on a hyperbox will yield valid bounds on f rather than speculative ones involving unknown Lipschitz constants.…”

Section: Approach 3 -Building a Complete Route By Adding Waypointsmentioning

confidence: 92%

“…Moreover, it is not clear that interval arithmetic arithmetic is applicable to the calculation of l ji by sampling as discussed in section 2. However, for objective functions whose interval extension is available, an interesting topic for further study, therefore, would be a numerical comparison of DIRECT with some of the interval techniques outlined in [15].…”

Section: Approach 3 -Building a Complete Route By Adding Waypointsmentioning

confidence: 99%

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Bartholomew‐Biggs

Parkhurst

Wilson

2002

Computational Optimization and Applications

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In this paper we discuss a global optimization problem arising in the calculation of aircraft flight paths. Since gradient information for this problem may not be readily available, a direct-search algorithm (DIRECT), proposed by Jones et al.[11] appears to be a promising solution technique. We describe some numerical experience in which DIRECT is used in several different ways to solve a sample problem.

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“…Feasibility-based bounds tightening (FBBT), which uses interval arithmetic to place bounds on expressions by recursively overestimating each of the participating functions and operators, is the simplest and most computationally inexpensive technique (Adjiman et al, 1998a,b;Androulakis et al, 1995;Audet et al, 2000;Belotti et al, 2009;Neumaier, 1990;Sherali andTuncbilek, 1995, 1997;Wolfe, 2000). An alternative, optimization-based bounds tightening (OBBT) methodology for determining these interval estimates is to minimize and maximize the expression under the bound constraints and possibly additional linear and convex constraints from the problem.…”

Section: Feasibility-and Optimality-based Bounds Tighteningmentioning

confidence: 99%

Global optimization advances in Mixed-Integer Nonlinear Programming, MINLP, and Constrained Derivative-Free Optimization, CDFO

Boukouvala

Misener

Floudas

2016

European Journal of Operational Research

194

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This manuscript reviews recent advances in deterministic global optimization for Mixed-Integer Nonlinear Programming (MINLP), as well as Constrained DerivativeFree Optimization (CDFO). This work provides a comprehensive and detailed literature review in terms of significant theoretical contributions, algorithmic developments, software implementations and applications for both MINLP and CDFO. Both research areas have experienced rapid growth, with a common aim to solve a wide range of real-world problems. We show their individual prerequisites, formulations and applicability, but also point out possible points of interaction in problems which contain hybrid characteristics. Finally, an inclusive and complete test suite is provided for both MINLP and CDFO algorithms, which is useful for future benchmarking.

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Interval mathematics, algebraic equations and optimization

Cited by 26 publications

References 44 publications

A novel approach in uncertain programming part II: a class of constrained nonlinear programming problems with interval objective functions

A novel approach in uncertain programming part II: a class of constrained nonlinear programming problems with interval objective functions

Untitled

Global optimization advances in Mixed-Integer Nonlinear Programming, MINLP, and Constrained Derivative-Free Optimization, CDFO

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