#### DMCA

## Rigorous algorithms for nonlinear biobjective optimization (2015)

### Citations

627 | Comparison of multiobjective evolutionary algorithms: Empirical results.
- Zitzler, Deb, et al.
- 2000
(Show Context)
Citation Context ...ume that will be sufficient for our analysis. Performances of a solver of course vary from one problem to another. Several suite of benchmark problems have been defined in the literature like the ZDT =-=[136]-=-, DTLZ [22], CTP [21], LZ [72] or WFG [55]. These problems have different characteristics : convex Pareto front, constrained problems, nondifferentiable problems, etc. They have a known Pareto front, ... |

294 | On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming
- Wächter, Biegler
- 2006
(Show Context)
Citation Context ...rategy implies to use a local search procedure, an optimization method that computes by local means a locally optimal solution, for example Newton methods (for optimization) or interior point methods =-=[134]-=-. Local search implies many computations that make it maybe not reasonable to use at every sub-problem. Nevertheless, it is worth spending some efforts in finding a good quality upper bound before the... |

185 | Numerica: A Modeling Language for Global Optimization.
- Hentenryck, Michel, et al.
- 1997
(Show Context)
Citation Context ...euses (calculant les optima globaux et prenant compte de l’imprécision numérique), comme les méthodes de Branch & Bound intervalle, existent et ont été étudiées de manière étendue dans la littérature =-=[48, 61, 92, 131]-=-. Il apparaît plutôt que la raison principale pour laquelle les méthodes globales rigoureuses sont si peu étudiées dans le cas multiobjectif est dû à la nature des solutions Pareto-optimales. En effet... |

168 | V.: Performance assessment of multiobjective optimizers: an analysis and review.
- Zitzler, Thiele, et al.
- 2003
(Show Context)
Citation Context ...also not sufficient to measure the difference between two sets of nondominated solutions. Therefore, different quality measures have been developed in order to evaluate sets of nondominated solutions =-=[138]-=-. The hypervolume, for example, is the volume dominated by a set of nondominated points (images in the objective space of nondominated solutions) with respect to a reference point, usually taken as th... |

124 |
Newton’s method estimates from data at one point. In The merging of disciplines: new directions in pure, applied, and computational mathematics
- Smale
- 1985
(Show Context)
Citation Context ... methods where such jumps would prevent computing all the solutions of the original system. Several methods have been proposed to certify the connectivity between continuation steps. Smale’s α-theory =-=[125]-=- has been used in [5] to derive some maximal step size that certifies the connectivity of a predictor corrector continuation in a homotopy method for computing roots of square polynomial systems in th... |

17 |
The two phases method: An efficient procedure to solve biobjective combinatorial optimization problems.
- Ulungu, Teghem
- 1995
(Show Context)
Citation Context ...ions, it can be interesting to design search strategies, that focus on the regions of the search space not discovered by ParCont. For example, an approach similar in principle to the two phase method =-=[129]-=- used for mutliobjective linear combinatorial problems seems promising. In any case, a possible immediate collaboration of ParCont with biobjective B&B is to use the former as a local search technique... |

16 | G.: Constructive Interval Disjunction
- Trombettoni, Chabert
- 2007
(Show Context)
Citation Context ...can also be extended by considering not -facets but a decomposition of the box that have to be consistent. This principle yield to Constructive Interval Disjunction consistency, i.e. CID-consistency =-=[128]-=-. Definition 2.3.8 (CID-consistency). Consider a NCSP as (2.15) and a box x. Consider in addition a regular paving P of x, and consider for all x′ ∈ P the smallest consistent box x′′ ⊆ x′ for the NCSP... |

11 |
Sam-Haroud; Interval propagation and search on directed acyclic graphs for numerical constraint solving
- Vu, Schichl, et al.
(Show Context)
Citation Context ... done by applying HC4 contractors on all constraints individually. A hull contractor on all constraints can be also obtained by applying HC4 on a DAG containing all the expressions of all constraints =-=[116, 132]-=-. Contracting methods based on box-consistency [131, 6, 38] can produce stronger narrowing than HC4, but are generally more computationally expensive. There are different possible instantiations of a ... |

10 | A local search based evolutionary multiobjective approach for fast and accurate convergence,” in Parallel Problem Solving from Nature-PPSN
- Sindhya, Deb, et al.
(Show Context)
Citation Context ...ead along the Pareto front. Thus, advanced population based metaheuristics are in general coupled with local search techniques to help pushing the solutions towards (locally) Pareto optimal solutions =-=[16, 124]-=-. Another step forward can be made by using continuation techniques as local search. Indeed, continuation can be used to efficiently recover large portions of connected (locally) Pareto optimal soluti... |

9 |
Immune optimization algorithm for constrained nonlinear multiobjective optimization problems
- Zhang
- 2007
(Show Context)
Citation Context ...ctive space, the shape of the captured Pareto front depending on the variation of the multipliers λ. 4.6.2 Following many changes of constraint activity We consider the biobjective problem (A.5) from =-=[135]-=- with 7 variables and 11 constraints, for the design of a speed reducer and presented in Appendix A (p. 147). In addition, domains of variables are considered as inequality constraints : constraints a... |

7 | Inner Regions and Interval Linearizations for Global Optimization
- Trombettoni, Araya, et al.
- 2011
(Show Context)
Citation Context ..., i.e. the variable xi whose wid(xi) is the largest ; SC3. Max Sum of Relative Smear (based on objectives and potentially active constraints). The latter criterion follows the bisection strategy from =-=[127]-=- shown to be robust for single objective problems. Given an interval matrix A containing the derivatives of the objectives and potentially 5.2 – Implementing biobjective interval Branch & Bound 125 ac... |

5 |
On finding multiple pareto-optimal solutions using classical and evolutionary generating methods.
- Shukla, Deb
- 2007
(Show Context)
Citation Context ...pplications do not require finding all global Pareto optimal solutions. Still, some globality can be achieved if this continuation by scalarization is coupled with a global search algorithm, see e.g. =-=[123, 106]-=-. The second difficulty has been tackled with some success, for example in [27, 95]. For the latter difficulty, selecting the most appropriate scalarization for a given NLMOO problem is, to our knowle... |

5 | A method for generating a well-distributed pareto set in nonlinear multiobjective optimization
- Utyuzhnikov, Fantini, et al.
(Show Context)
Citation Context ...direct more accurately the solving process. For example the Directed Search Domain (DSD) [28] is a method that is inspired by preceding work on other scalarizing methods known as Physical Programming =-=[83, 130]-=-. It proposes to orient cones along the CHIM that identify a specific region of the Pareto front. Orienting these cones allows to reach any Pareto optimal solutions (even in the case of more than two ... |

5 |
da Fonseca. Why quality assessment of multiobjective optimizers is difficult
- Zitzler, Laumanns, et al.
- 2002
(Show Context)
Citation Context ... for the solution may be used, such that the solver reaching this quality level with the least computational effort is preferred. Evaluating the quality of sets of nondominated solutions is difficult =-=[137]-=-. Dominance relations allow cases where X̂1 does not dominate X̂2 entirely. These dominance relations are also not sufficient to measure the difference between two sets of nondominated solutions. Ther... |

2 | Interval Methods for Competitive Location Problems
- Tóth
- 2008
(Show Context)
Citation Context ...ar exemple, un restaurant d’une chaîne implique de prendre en compte les points de vues contradictoires des différents acteurs, notamment le propriétaire de la chaîne et le propriétaire du restaurant =-=[126]-=-. Des contraintes peuvent également s’ajouter aux problèmes. Dans le cas du problème d’ingénierie par exemple, des contraintes de structures doivent généralement être prises en compte. De plus, trouve... |