Branch and Bound Experiments in Convex Nonlinear Integer Programming

TLDR

Branch and bound is a well‑established tool for mixed‑integer linear programming, with its efficiency largely determined by branching variable and node selection rules. This study examines whether branch and bound can feasibly solve convex nonlinear integer programming problems. The authors implement pseudo‑costs, estimations, and heuristic rules to select branching variables, nodes, and locate integer solutions, yielding 27 distinct branch‑and‑bound strategies. Computational experiments on these strategies produce empirical results demonstrating their performance.

Abstract

The branch and bound principle has long been established as an effective computational tool for solving mixed integer linear programming problems. This paper investigates the computational feasibility of branch and bound methods in solving convex nonlinear integer programming problems. The efficiency of a branch and bound method often depends on the rules used for selecting the branching variables and branching nodes. Among others, the concepts of pseudo-costs and estimations are implemented in selecting these parameters. Since the efficiency of the algorithm also depends on how fast an upper bound on the objective minimum is attained, heuristic rules are developed to locate an integer feasible solution to provide an upper bound. The different criteria for selecting branching variables, branching nodes, and heuristics form a total of 27 branch and bound strategies. These strategies are computationally tested and the empirical results are presented.