McCormick envelopes
Introduction
Optimization of a non-convex function f(x) is challenging since it may have multiple locally optimal points and it can take a significant amount of time or effort to determine if the problem has no solution or if the solution is global. Gradient based solvers are unable to certify optimality1. Different techniques are used to address this challenge depending on the characteristics of the problem. One technique used is convex envelopes2:
Given a non-convex function f(x), g(x) is a convex envelope of f(x) for X S if:
· g(x) is convex under-estimator of f(x)
· g(x)>=h(x) for all convex under-estimators h(x)
McCormick Envelopes
In particular, for bilinear (e.g., x*y, x2) Non-Linear Programming (NLP) problems3, the McCormick Envelope is a type of convex relaxation used for optimization.
In case of an NLP, an LP relaxation is derived by replacing each bilinear term with a new variable and adding four sets of constraints. In the case of an MINLP, an MILP relaxation is derived. This strategy is known as McCormick relaxation.
The LP solution gives a lower bound and any feasible solution gives an upper bound.
As noted by Scott et al4, McCormick envelopes are attractive due to their recursive nature of their definition, which affords wide applicability and easy implementation computationally. Furthermore, these relaxations are typically stronger than those resulting from convexification or linearization procedures.
Derivation of McCormick Envelopes
Example: Convex Relaxation
Good bounds are essential to focusing the feasible solution which may be obtained either by inspection or solving the optimization problem to minimize (maximize) x subject to the same constraints as the original problem.
As noted by Hazaji, state-of-the-art global optimization solvers implement bound contraction techniques in order to improve this bounding procedure. Once bound propagation is completed,
domain partitioning becomes necessary. Spatial branch and bound schemes [6, 7] are among the most effective partitioning methods in global optimization.
By splitting the domain of a given variable, the solver is able to divide the original domain into two smaller regions, further tightening the convex relaxations of each partition.
Example: Numerical
Application
Conclusion
McCormick Envelopes provide a relaxation technique for bilinear non-convex nonlinear programming problems3. Since non-convex NLP are challenging to solve and may be time or resource intensive, McCormick envelopes are attractive due to their recursive nature of their definition, which affords wide applicability and easy implementation computationally. Furthermore, these relaxations are typically stronger than those resulting from convexification or linearization procedures4.
References
- Castro, Pedro. "A Tighter Piecewise McCormick Relaxation for Bilinear Problems." (n.d.): n. pag. 3 June 2014. Web. 6 June 2015. <http://minlp.cheme.cmu.edu/2014/papers/castro.pdf
- You, F (2021). Notes for a lecture on Mixed Integer Non-Linear Programming (MINLP). Archives for SYSEN 5800 Computational Optimization (2021FA), Cornell University, Ithaca, NY.
- Dombrowski, J. (2015, June 7). Northwestern University Open Text Book on Process Optimization, McCormick Envelopes Retrieved from https://optimization.mccormick.northwestern.edu/index.php/McCormick_envelopes
- Scott, J. K. Stuber, M. D. & Barton, P. I. (2011). Generalized McCormick Relaxations. Journal of Global Optimization, Vol. 51, Issue 4, 569-606 doi: 10.1007/s10898-011-9664-7
- Hijazi, H., Perspective Envelopes for Bilinear Functions, unpublished manuscript, The Australian National University, Canberra, Australia 4841.pdf
- Androulakis, I., Maranas, C., Floudas, C.: alphabb: A global optimization method for general constrained nonconvex problems. Journal of Global Optimization 7(4), 337{363 (1995)
- Smith, E., Pantelides, C.: A symbolic reformulation/spatial branch-and-bound algorithm for the global optimisation of nonconvex fMINLPsg. Computers & Chemical Engineering 23(4), 457 { 478 (1999)