Branch and Bound With Simplicial Partitions for Global Optimization

Abstract

Abstract. Branch and bound methods for global optimization are considered in this paper. Advantages and disadvantages of simplicial partitions for branch and bound are shown. A new general combinatorial approach for vertex triangulation of hyper-rectangular feasible regions is presented. Simplicial partitions may be used to vertex triangulate feasible regions of non rectangular shape defined by linear inequality constraints. Linear inequality constraints may be used to avoid symmetries in optimization problems.

Investigation of selection strategies in branch and bound algorithm with simplicial partitions and combination of Lipschitz bounds

“…In this paper we use simplicial branch and bound with combination of Lipschitz bounds. Advantages and disadvantages of simplicial partitions are discussed in [21]. Since a simplex is a polyhedron in n-dimensional space with the minimal number of vertices, simplicial partitions are preferable when the values of an objective function at the vertices of partitions are used to compute bounds.…”

“…The most preferable initial covering is face to face vertex triangulation-partitioning of the feasible region into finitely many n-dimensional simplices, whose vertices are also the vertices of the feasible region. We use a general (any dimensional) algorithm for combinatorial vertex triangulation of hyper-rectangle [21] into n! simplices.…”

Global Optimization Using the Branch‐and‐bound Algorithm With a Combination of Lipschitz Bounds Over Simplices

“…A branch-and-bound technique may be used for implementing the covering global optimization methods (Žilinskas 2008;Paulavičius and Žilinskas 2009) as well as combinatorial optimization algorithms (Žilinskas, A. and Žilinskas, J. 2009).…”

Depth-first simplicial partition for copositivity detection, with an application to MaxClique

“…The structure of this partitioning algorithm in [10] is similar to a branch and bound algorithm with simplicial partitions for global optimization as described in [29]. Although the strategy of selection of the next simplex to process is not emphasized in [10], a depth-first strategy may be used.…”