Spatial Branch-and-Bound for Nonconvex Separable Piecewise Linear Optimization

Nonconvex separable piecewise linear functions (PLFs) frequently appear in applications and to approximate nonlinearitites. The standard practice to formulate nonconvex PLFs is from the perspective of discrete optimization using special ordered sets and mixed-integer linear programs (MILPs). In contrast, we take the viewpoint of global continuous optimization and present a spatial branch-and-bound algorithm for optimizing a separable discontinuous PLF over a closed convex set. It offers slim and sparse linear programming relaxations, sharpness throughout the search tree, and an increased flexibility in branching decisions. The main feature of our algorithm is the generation of convex underestimators at the root node of the search tree and their quick and efficient updates at each node after branching. Convergence to the global optimum is achieved when the PLFs are lower semicontinuous. A Python implementation of our algorithm is tested on knapsack and network flow problems for both continuous and discontinuous PLFs. Our algorithm is compared with four logarithmic MILP formulations solved by Gurobi’s MILP solver as well as Gurobi’s PLF solver. We also compare our method against mixed-integer nonlinear program formulations solved by Gurobi. The numerical experiments indicate significant performance gains up to two orders of magnitude for medium- to large-sized PLFs. Finally, we also give an upper bound on the additive error from PLF approximations of nonconvex separable optimization.