Computational Management Science

, Volume 16, Issue 4, pp 577–592

# B&B method for discrete partial order optimization

Original Paper

## Abstract

The paper extends the branch and bound (B&B) method, primarily developed for the solution of global, discrete, and vector programming problems, to finding nondominated points in a partially ordered space/set. The framework of the generalized B&B method is standard, it includes partition, estimation, and pruning steps, but bounds are different, they are set-valued, and, as a particular case, the bounds may be singletons. For bounding, the method uses a set ordering in the following sense. One set is “less or equal” than the other if for any element of the first set there is a “greater or equal” element in the second one. The defined set ordering is a partial order in the space of sets consisting of mutually nondominated elements. In the B&B method, partitioning is applied to the parts of the original space with nondominated upper bounds. Parts with small upper bounds (less than some lower bound) are pruned. Convergence of the method to the set of all nondominated points is established. The acceleration with respect to the enumerative search is achieved through group evaluation of elements of the original space. Further, the developed B&B method is extended to vector partial order optimization. In the latter case, several partial orders are defined in the decision space. Finally, we develop a B&B method for a so-called constrained partial order optimization problem, where the feasible set is defined by a family of partial orders.

## References

1. Belotti P, Soylu B, Wiecek MM (2013) A branch-and-bound algorithm for biobjective mixed-integer programs. Technical Report. http://www.optimization-online.org/DB_HTML/2013/01/3719.html. Accessed 3 Sept 2018
2. De Simone V, Marino M, Toraldo G (2009) Isotonic regression problems. In: Floudas CA, Pardalos PM (eds) Encyclopedia of optimization, 2nd edn. Springer, Berlin, pp 1774–1777
3. Deb K (2001) Multi-objective optimization using evolutionary algorithms. Willey, ChichesterGoogle Scholar
4. Deb K, Agrawal S, Pratap A, Meyarivan T (2000) A fast elitist non-dominated sorting genetic algorithm for multi-objective optimization: NSGA-II lecture notes in computer science. Springer, Berlin, pp 849–858.
5. Dentcheva D, Ruszczyński A (2003) Optimization with stochastic dominance constraints. SIAM J Optim 14:548–566
6. Ehrgott M (2005) Multicriteria optimization, 2nd edn. Springer, BerlinGoogle Scholar
7. Ehrgott M, Gandibleux X (2001) Bounds and bound sets for biobjective combinatorial optimization problems. In: Köksalan M, Zionts S (eds) Multiple criteria decision making in the new millennium. Lecture notes in economics and mathematical systems, vol 507. Springer, Berlin, pp 242–253Google Scholar
8. Ehrgott M, Gandibleux X (2007) Bound sets for biobjective combinatorial optimization problems. Comput Oper Res 34(9):2674–2694
9. Eichfelder G, Jahn J (2012) Vector optimization problems and their solution concepts. In: Ansari Q, Yao JC (eds) Recent developments in vector optimization. Vector optimization, vol 1. Springer, Berlin, Heidelberg, pp 1–27Google Scholar
10. Fattore M, Bruggemann R (eds) (2017) Partial order concepts in applied sciences. Springer, ChamGoogle Scholar
12. Khan AA, Tammer C, Zălinescu C (2015) Set-valued optimization. An introduction with applications. Springer, BerlinGoogle Scholar
13. Marques-Silva J, Argelich J, Graça A et al (2011) Boolean lexicographic optimization: algorithms & applications. Ann Math Artif Intell 62(3–4):317–343
14. Nishnianidze ZG (1984) Fixed points of monotone multivalued operators. Soobshch Akad Nauk Gruzin SSR 114(3):489–491Google Scholar
15. Norkin VI (2017) B&B solution technique for multicriteria stochastic optimization problems. In: Butenko S, Pardalos PM, Shylo V (eds) Optimization methods and applications. Springer, Berlin, pp 345–378
16. Parker DS (1989) Partial order programming. In: Proceedings of the 16th ACM SIGPLAN-SIGACT symposium on principles of programming languages. ACM, New York, pp 260–266Google Scholar
17. Przybylski P, Gandibleux X (2017) Multi-objective branch and bound. Eur J Oper Res.
18. Sawaragi S, Nakayama H, Tanino T (1985) Theory of multiobjective optimization. Academic Press, OrlandoGoogle Scholar
19. Sourd F, Spanjaard O (2008) A multiobjective branch-and-bound framework: application to the biobjective spanning tree problem. INFORMS J Comput 20(3):472–484
20. Statnikov RB, Matusov JB (2002) Multicriteria analysis in engineering. Kluwer, Dordrecht
21. Tuy H (2009) Decomposition in global optimization. In: Floudas CA, Pardalos PM (eds) Encyclopedia of optimization, 2nd edn. Springer, Berlin, pp 627–630
22. Ulungu E, Teghem J (1997) Solving multi-objective knapsack problem by a branch-and-bound procedure. In: Climaco J (ed) Multicriteria analysis. Springer, Berlin, pp 269–278Google Scholar
23. Visée M, Teghem J, Pirlot M, Ulungu E (1998) Two-phases method and branch and bound procedures to solve the bi-objective knapsack problem. J Glob Optim 12:139–155
24. Yang XQ (1992) A Hahn–Banach theorem in ordered linear spaces and its applications. Optimization 25(1):1–9

© Springer-Verlag GmbH Germany, part of Springer Nature 2019