Computational Comparison of Convex Underestimators for Use in a Branch-and-Bound Global Optimization Framework

Abstract

Applications that require the optimization of nonlinear functions involving nonconvex terms include reactor network synthesis, separations design and synthesis, robust process control, batch process design, protein folding, and molecular structure prediction. The global optimization method α-branch-and-bound (αBB; Adjiman and Floudas, J. Glob. Optim. 9(1):23–40, 1996; Adjiman et al., Comput. Chem. Eng. 22(9):1137–1158, 1998; Adjiman et al., Comput. Chem. Eng. 22(9):1159–1179, 1998; Androulakis et al., J. Glob. Optim. 7(4):337–363, 1995; Floudas, Deterministic Global Optimization: Theory, Methods and Applications, vol. 37. Springer, New York, 2000; Maranas and Floudas, J. Glob. Optim. 4(2):135–170, 1994), guarantees the global optimum with e-convergence for any \(\mathcal{C}^{2}\)-continuous function within a finite number of iterations via fathoming nodes of a branch-and-bound tree through assignment of lower and upper bounds. Lower bounds are generated through convexification over a node’s subdomain to yield a convex nonlinear program at each node. This chapter explores the performance of the αBB method as well as number of competing methods designed to provide tight, convex underestimators, including the piecewise (Meyer and Floudas, J. Glob. Optim. 32(2):221–258, 2005), generalized (Akrotirianakis and Floudas, J. Glob. Optim. 30(4):367–390, 2004; Akrotirianakis and Floudas, J. Glob. Optim. 29(3):249–264, 2004), and nondiagonal (Skjal et al., J. Optim. Theory Appl. 154(2):462–490, 2012) αBB methods, the Brauer and Rohn+E (Skjal and Westerlund, J. Glob. Optim. 1–17, 2013) αBB methods, and the moment approach (Lasserre and Thanh, J. Glob. Optim. 56(1):1–25, 2013). Their performance is gauged through a test suite of 20 multivariate, box-constrained, nonconvex functions.