### B&B method for discrete partial order and quasiorder optimizations

V.I. Norkin
2019 Natsional'na Akademiya Nauk Ukrainy. Dopovidi: naukovyi zhurnal
ОПОВІДІ НАЦІОНАЛЬНОЇ АКАДЕМІЇ НАУК УКРАЇНИ Partial/quasi order optimization is a research field, which studies optimization problems involving order relations. Classical examples of such problems are given by the multiobjective optimization     . Various applications of the partial/quasi order optimization are considered in  . Related works include the optimization with dominance constraints [6, 7] and set-valued optimization . In the present paper, we consider a problem of
more » ... der a problem of finding the optimal (nondominated) elements (with respect to some partial/quasi order) on a discrete feasible set of elements defined by means of some other partial/quasi orders. A similar problem setting was considered in  . In practical problems, the feasible set may contain a huge number of elements, so the enumerative search is questionable. We develop a branch and bound (B&B) method for this problem and prove its convergence. The method subdivides the original problem into a sequence of subproblems, selects subproblems containing optimal elements, and proceeds until such subproblems become trivial. Acceleration with respect to the enumerative search is achieved due to the group evaluation of The paper extends the Branch and Bound (B&B) method to find all nondominated points in a partially or quasiordered space. The B&B method is applied to the so-called constrained partial/quasi order optimization problem, where the feasible set is defined by a family of partial/quasi order constraints. The framework of the generalized B&B method is standard, it includes partition, estimation, and pruning steps, but bounds are different, they are setvalued. For bounding, the method uses a set ordering in the following sense. One set is "less or equal" than the other set if, for any element of the first set, there is a "greater or equal" element in the second one. In the B&B method, the 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 the group evaluation of elements of the original space.
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