A nonlinear interval number programming method for uncertain optimization problems

Jiang, Chao; Han, Xu; Liu, G. R.; Liu, G. P.

doi:10.1016/j.ejor.2007.03.031

Cited by 332 publications

(170 citation statements)

References 16 publications

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“…It should be noted that A I ≤ B I does not mean B I is larger than A I as the comparison between two real numbers; rather, it denotes that B I is superior to A I . In interval optimization theory, the possibility degree is often used to describe the degree to which an interval is superior to another interval [35]. The meanings of A I ≤ b are similar to those of A I ≤ B I .…”

Section: Definition Of Possibility Degreementioning

confidence: 99%

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Optimal Scheduling of Microgrid with Multiple Distributed Resources Using Interval Optimization

et al. 2017

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Abstract:In this paper, an optimal day-ahead scheduling problem is studied for a microgrid with multiple distributed resources. For the sake of coping with the prediction uncertainties of renewable energies and loads and taking advantage of the time-of-use price for buying/selling electricity, an interval-based optimization model for maximum profits is developed. To reduce the computational complexity in solving the model, the possibility degree comparison between an interval and a real number is used to convert the interval constraints into the general ones; meanwhile, some slack variables and complementary conditions are introduced to eliminate the absolute-value operation. Unlike the stochastic optimization, the interval optimization only needs the upper-lower bounds of the uncertain variables instead of their probability distribution functions, which is beneficial to the practical application. Furthermore, the possible profit interval and the expected optimal profit can be determined by solving the optimization model. Numerical simulations are performed on a microgrid system modified from the benchmark low voltage network in the European Union project "Microgrid", and the results demonstrate the effectiveness of the proposed method.

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Section: Definition Of Possibility Degreementioning

confidence: 99%

“…Mathematically, an interval is usually used to describe an uncertain variable whose upper-lower bounds are known [34,35]. Recently, the interval optimization for the generation scheduling of power system has drawn increasing attention.…”

Section: Introductionmentioning

confidence: 99%

Optimal Scheduling of Microgrid with Multiple Distributed Resources Using Interval Optimization

et al. 2017

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“…In addition, ϕ and φ are two normalization factors, σ is the penalty factor which is usually specified as a large value and φ is a function with the following form (Jiang et al 2008):…”

Section: Nonlinear Interval Parameter Programmingmentioning

confidence: 99%

Nonlinear Interval Parameter Programming Combined with Cooperative Games: a Tool for Addressing Uncertainty in Water Allocation Using Water Diplomacy Framework

Zarghami

Safari

Szidarovszky

et al. 2015

Water Resour Manage

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This paper shows the utility of a new interval cooperative game theory as an effective water diplomacy tool to resolve competing and conflicting needs of water users from different sectors including agriculture, domestic, industry and environment. Interval parameter programming is applied in combination with cooperative game theoretic concepts such as Shapley values and the Nucleolus to provide mutually beneficial solutions for water allocation problems under uncertainty. The allocation problem consists of two steps: water resources are initially allocated to water users based on the Nash bargaining model and the achieved nonlinear interval parameter model is solved by transforming it into a problem with a deterministic weighted objective function. Water amounts and net benefits are reallocated to achieve efficient water usage through net benefit transfers. The net benefit reallocation is done by the application of different cooperative game theoretical methods. Then, the optimization problem is solved by linear interval programming and by converting it into a problem with two deterministic objective functions. The suggested model is then applied to the Zarrinehrud subbasin, within Urmia Lake basin in Northwestern Iran. Findings suggest that a reframing of the problem using cooperative strategies within the context of water diplomacy frameworkcreating flexibility in water allocation using mutual gains approach -provides better outcomes for all competing users of water.

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“…In [13], Chanas and Kuchta presented an approach to unify the solution methods proposed by Ishibuchi and Tanaka [12]. Jiang et al [14] suggested to solve the nonlinear interval number programming problem with uncertain coefficients both in nonlinear objective function and nonlinear constraints. Gabrel et al [15] introduced two different methods for solving interval linear programming problems.…”

Section: Introductionmentioning

confidence: 99%

Exactness Property of the Exact Absolute Value Penalty Function Method for Solving Convex Nondifferentiable Interval-Valued Optimization Problems

Antczak

2017

J Optim Theory Appl

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In the paper, the classical exact absolute value function method is used for solving a nondifferentiable constrained interval-valued optimization problem with both inequality and equality constraints. The property of exactness of the penalization for the exact absolute value penalty function method is analyzed under assumption that the functions constituting the considered nondifferentiable constrained optimization problem with the interval-valued objective function are convex. The conditions guaranteeing the equivalence of the sets of LU-optimal solutions for the original constrained interval-valued extremum problem and for its associated penalized optimization problem with the interval-valued exact absolute value penalty function are given.Keywords Interval-valued optimization problem · Exact absolute value penalty function method · Penalized optimization problem with the interval-valued exact absolute value penalty function · Exactness of the exact absolute value penalty function method · LU-convex function Mathematics Subject Classification 49M30 · 90C25 · 90C30Communicated by

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A nonlinear interval number programming method for uncertain optimization problems

Cited by 332 publications

References 16 publications

Optimal Scheduling of Microgrid with Multiple Distributed Resources Using Interval Optimization

Optimal Scheduling of Microgrid with Multiple Distributed Resources Using Interval Optimization

Nonlinear Interval Parameter Programming Combined with Cooperative Games: a Tool for Addressing Uncertainty in Water Allocation Using Water Diplomacy Framework

Exactness Property of the Exact Absolute Value Penalty Function Method for Solving Convex Nondifferentiable Interval-Valued Optimization Problems

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