EAGO provides a library of McCormick relaxations in native Julia code. The EAGO optimizer supports relaxing functions using nonsmooth McCormick relaxations (Mitsos2009, Scott2011), smooth McCormick relaxations (Khan2016, Khan2018, Khan2019), and multi-variant McCormick relaxations (Tsoukalas2014; a variant of subgradient-based interval refinement (Najman2017)). For functions with arbitrarily differentiable relaxations, the differentiable constant μ can be modified by adjusting a constant value in the package. Additionally, validated and nonvalidated interval bounds are supported via IntervalArithmetic.jl which is reexported. The basic McCormick operator and reverse McCormick operator (Wechsung2015) libraries are included in two dependent subpackages which can loaded and used independently:

NaN Numerics

When a relaxation is computed at an undefined point or over an unbounded domain, the resulting relaxation is defined as "not a number" (NaN) rather than throwing an error. This allows algorithms to check for these cases without resorting to try-catch statements. Moreover, when the interval domain is extensive enough to cause a domain violation, an x::MC structure is returned that satisfies isnan(x) === true.

  • Khan KA, Watson HAJ, Barton PI (2017). Differentiable McCormick relaxations. Journal of Global Optimization, 67(4): 687-729.
  • Khan KA, Wilhelm ME, Stuber MD, Cao H, Watson HAJ, Barton PI (2018). Corrections to: Differentiable McCormick relaxations. Journal of Global Optimization, 70(3): 705-706.
  • Khan KA (2019). Whitney differentiability of optimal-value functions for bound-constrained convex programming problems. Optimization, 68(2-3): 691-711
  • Mitsos A, Chachuat B, and Barton PI. (2009). McCormick-based relaxations of algorithms. SIAM Journal on Optimization, 20(2): 573–601.
  • Najman J, Bongratz D, Tsoukalas A, and Mitsos A (2017). Erratum to: Multivariate McCormick relaxations. Journal of Global Optimization, 68: 219-225.
  • Scott JK, Stuber MD, and Barton PI. (2011). Generalized McCormick relaxations. Journal of Global Optimization, 51(4): 569–606.
  • Stuber MD, Scott JK, Barton PI (2015). Convex and concave relaxations of implicit functions. Optim. Methods Softw., 30(3): 424–460
  • Tsoukalas A and Mitsos A (2014). Multivariate McCormick Relaxations. Journal of Global Optimization, 59: 633–662.
  • Wechsung A, Scott JK, Watson HAJ, and Barton PI. (2015). Reverse propagation of McCormick relaxations. Journal of Global Optimization, 63(1): 1-36.