Constraint propagation using dominance in interval Branch &amp; Bound for nonlinear biobjective optimization

Martin, BC; Goldsztejn, Alexandre; Granvilliers, Laurent; Jermann, Christophe

doi:10.1016/j.ejor.2016.05.045

Cited by 8 publications

(18 citation statements)

References 43 publications

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“…One advantage of following the standard branch-and-bound algorithm is modularity: Other bounding techniques like [47] may also be included and their efficiency within a branch-and-bound algorithm can be assessed. In addition, the proposed framework can be readily applied to more general problems, like multi-objective semi-infinite problems [44,27,62,26].…”

Section: Resultsmentioning

confidence: 99%

“…On the other hand, the branch-and-bound approach to solving (non-robust) NLPs [51,22] has been developed to the point where it has become efficient enough to address the global optimization of important applications, e.g., in robotics, control and engineering [36,10,13,28]. Although branch-and-bound algorithms are usually strongly sensitive to the number of variables, they turn out to be useful for very nonlinear small scale problems.This approach has also been successfully extended to larger classes of problems, e.g., multi-objective nonlinear problems [21,44].…”

Section: Introductionmentioning

confidence: 99%

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A standard branch-and-bound approach for nonlinear semi-infinite problems

Marendet

Goldsztejn

Chabert

et al. 2020

European Journal of Operational Research

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This paper considers nonlinear semi-infinite problems, which contain at least one semi-infinite constraint (SIC). The standard branch-and-bound algorithm is adapted to such problems by extending usual upper and lower bounding techniques for nonlinear inequality constraints to SICs. This is achieved by defining the interval evaluation of parametrized functions and their generalized gradients, by also adapting numerical constraint programming techniques to quantified inequalities, and by introducing linear relaxations and restrictions for SICs. The overall efficiency of our algorithm is demonstrated on a standard set of benchmarks from the literature, in comparison with the best state of the art alternative.

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Section: Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A standard branch-and-bound approach for nonlinear semi-infinite problems

Marendet

Goldsztejn

Chabert

et al. 2020

European Journal of Operational Research

Self Cite

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“…Good surveys on deterministic methods in multiobjective optimization can be found in the books [22,25]. Most of the known deterministic methods for biobjective optimization are based on the branch-and-bound approach, see, for example, [10,21,35]. The upper and lower bounds for "optimal" points in the image space are often obtained by using interval methods or Lipschitz properties of the objectives.…”

Section: Related Literaturementioning

confidence: 99%

Nonconvex constrained optimization by a filtering branch and bound

2020

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A major difficulty in optimization with nonconvex constraints is to find feasible solutions. As simple examples show, the $$\alpha $$ α BB-algorithm for single-objective optimization may fail to compute feasible solutions even though this algorithm is a popular method in global optimization. In this work, we introduce a filtering approach motivated by a multiobjective reformulation of the constrained optimization problem. Moreover, the multiobjective reformulation enables to identify the trade-off between constraint satisfaction and objective value which is also reflected in the quality guarantee. Numerical tests validate that we indeed can find feasible and often optimal solutions where the classical single-objective $$\alpha $$ α BB method fails, i.e., it terminates without ever finding a feasible solution.

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“…However, to our knowledge only a few exact methods have been proposed in literature for solving NLBOO problems (e.g., [6][7][8][9][10][11][12]). They are mainly based on interval arithmetic and branch & bound, and commonly find a thin envelope in the objective space, which certainly contains the set of non-dominated vectors Y * .…”

Section: Introductionmentioning

confidence: 99%

“…Filtering (or contraction) consists in removing inconsistent values from the bounds of the box, while the upper-bounding procedures attempt to find good feasible solutions for improving the upper envelope. Some filtering methods take into account the upper envelope for filtering dominated solutions (e.g., discarding tests [8,13], dominance contractors [11]). If the box has not been discarded by the filtering process, it is split into two sub-boxes by dividing the domain of one variable and generating two child nodes in the search tree.…”

Section: Introductionmentioning

confidence: 99%

Nonlinear biobjective optimization: improvements to interval branch & bound algorithms

2019

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Interval based solvers are commonly used for solving single-objective nonlinear optimization problems. Their reliability and increasing performance make them useful when proofs of infeasibility and/or certification of solutions are a must. On the other hand, there exist only a few approaches dealing with nonlinear optimization problems, when they consider multiple objectives. In this paper, we propose a new interval branch & bound algorithm for solving nonlinear constrained biobjective optimization problems. Although the general strategy is based on other works, we propose some improvements related to the termination criteria, node selection, upperbounding and discarding boxes using the non-dominated set. Most of these techniques use and/or adapt components of IbexOpt, a state-of-the-art interval-based single-objective optimization algorithm. The code of our plugin can be found in our git repository (

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Constraint propagation using dominance in interval Branch & Bound for nonlinear biobjective optimization

Cited by 8 publications

References 43 publications

A standard branch-and-bound approach for nonlinear semi-infinite problems

A standard branch-and-bound approach for nonlinear semi-infinite problems

Nonconvex constrained optimization by a filtering branch and bound

Nonlinear biobjective optimization: improvements to interval branch & bound algorithms

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