A Rigorous Generic Branch and Bound Solver for Nonlinear Problems

Abstract

Recursive branch and bound algorithms are often used, either rigorouslyor non-rigorously, to refine and isolate solutions to global optimizationproblems or systems of equations and inequalities involving nonlinearfunctions. The presented software library, Kodiak, integrates numericand symbolic computation into a generic framework for the solution of suchproblems over hyper-rectangular variable and parameter domains. Thecorrectness of both the generic branch and bound algorithm and the self-validating enclosure methods used, namely interval arithmetic and, for polynomials and rational functions, Bernstein expansion, has beenformally verified. The algorithm has three main instantiations, forsystems of equations and inequalities, for constrained global optimization, and for the computation of equilibria and bifurcation sets for systems ofordinary differential equations. For the latter category, and to enablethe computation of bisection heuristics to reduce the branching factor, advantage is taken of the partial derivatives of the constraint functions, which are symbolically manipulated. Pavings (unions of box subsets)for a continuum of solutions to underdetermined systems mayalso be produced. The capabilities of the software tool are outlined, andcomputational examples are presented.