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McCormick.jl

A Forward McCormick Operator Library

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McCormick.jl is a component package in the EAGO ecosystem and is reexported by EAGO.jl. It contains a library of forward McCormick operators (both nonsmooth and differentiable). Documentation for this is included in the EAGO.jl package and additional usage examples are included in EAGO-notebooks as Jupyter Notebooks.

McCormick Operator Variants

Each McCormick object is associated with a parameter T <: RelaxTag which is either NS for nonsmooth relaxations (Mitsos2009, Scott2011), MV for multivariate relaxations (Tsoukalas2014, Najman2017), or Diff for differentiable relaxations (Khan2016, Khan2018, Khan2019). Conversion between NS, MV, and Diff relax tags are not currently supported. Convex and concave envelopes are used to compute relaxations of univariate functions.

Supported Operators

In addition to supporting the implicit relaxation routines of Stuber 2015, this package supports the computation of convex/concave relaxations (and associated subgradients) for expressions containing the following operations:

Common Algebraic Expressions: inv, log, log2, log10, exp, exp2, exp10, sqrt, +, -, ^, min, max, /, *, abs, step, sign, deg2rad, rad2deg, abs2, cbrt, fma, xlogx, arh, xexpax

Trigonometric Functions: sin, cos, tan, asin, acos, atan, sec, csc, cot, asec, acsc, acot, sind, cosd, tand, asind, acosd, atand, secd, cscd, cotd, asecd, acscd, acotd, sinpi, cospi

Hyperbolic Functions: sinh, cosh, tanh, asinh, acosh, atanh, sech, csch, coth, acsch, acoth

Special Functions: erf, erfc, erfcinv, erfc

Activation Functions: relu, leaky_relu, param_relu, sigmoid, bisigmoid, softsign, softplus, maxtanh, pentanh, gelu, elu, selu, swish1

Bound Specification Functions: positive, negative, lower_bnd, upper_bnd, bnd

Other Functions: one, zero, intersect, real, dist, eps, <, <=, ==

Differentiable relaxations (Diff <: RelaxTag) are supported for the functions given in Khan2016, Khan2018, Khan2019. However, differentiable relaxations for other nonsmooth terms listed above have yet to be developed and as such have been omitted.

Bounding a Univariate Function

In order to bound a function using a McCormick relaxation, you first construct a McCormick object (x::MC) that bounds the input variables, and then you pass these variables to the desired function.

In the example below, convex/concave relaxations of the function

$$f(x) = x (x - 5) \sin(x)$$

are calculated at $x = 2$ on the interval $X = [1, 4]$.

using McCormick

# Create MC object for x = 2.0 on [1.0, 4.0] for relaxing
# a function f(x) on the interval Intv

f(x) = x*(x - 5.0)*sin(x)

x = 2.0                          # Value of independent variable x
Intv = Interval(1.0, 4.0)        # Define interval to relax over
                                 # Note that McCormick.jl reexports IntervalArithmetic.jl
                                 # and StaticArrays. So no using statement for these is
                                 # necessary.
# Create McCormick object
xMC = MC{1,NS}(x, Intv, 1)

fMC = f(xMC)             # Relax the function

cv = fMC.cv              # Convex relaxation
cc = fMC.cc              # Concave relaxation
cvgrad = fMC.cv_grad     # Subgradient/gradient of convex relaxation
ccgrad = fMC.cc_grad     # Subgradient/gradient of concave relaxation
Iv = fMC.Intv            # Retrieve interval bounds of f(x) on Intv

Plotting the results, we can easily visualize the convex and concave relaxations, interval bounds, and affine bounds constructed using the subgradient at the middle of $X$.

Bounding a Multivariate Function

This can readily be extended to multivariate functions, for example:

$$ \begin{aligned} & f(x, y) = \big(4 - 2.1 x^{2} + \frac{x^{4}}{6} \big) x^{2} + x y + (-4 + 4 y^{2}) y^{2}\\ & X = [-2, 0]\\ & Y = [-0.5, 0.5] \end{aligned} $$

using McCormick

# Define function
f(x, y) = (4.0 - 2.1*x^2 + (x^4)/6.0)*x^2 + x*y + (-4.0 + 4.0*y^2)*y^2

# Define intervals for independent variables
n = 30
X = Interval{Float64}(-2, 0)
Y = Interval{Float64}(-0.5, 0.5)
xrange = range(X.lo, stop=X.hi, length=n)
yrange = range(Y.lo, stop=Y.hi, length=n)

# Calculate differentiable McCormick relaxation
for (i,x) in enumerate(xrange)
    for (j,y) in enumerate(yrange)
        z = f(x, y)                 # Calculate function values
        xMC = MC{1,Diff}(x, X, 1)   # Differentiable relaxation for x
        yMC = MC{1,Diff}(y, Y, 2)   # Differentiable relaxation for y
        fMC = f(xMC, yMC)           # Relax the function
        cv = fMC.cv                 # Convex relaxation
        cc = fMC.cc                 # Concave relaxation
    end
end

Citing McCormick.jl

McCormick.jl is a component of the EAGO.jl ecosystem. Please cite the following paper when using McCormick.jl:

M. E. Wilhelm & M. D. Stuber (2022) EAGO.jl: easy advanced global optimization in Julia,
Optimization Methods and Software, 37:2, 425-450, DOI: 10.1080/10556788.2020.1786566

Unit Testing Note

While McCormick.jl generally supports Julia 1.1+, some functions may return an error for Julia versions less than 1.3. In particular, cbrt will result in a StackOverflow when called. McCormick is unit tested using Julia versions 1.3 and beyond.

References

  • 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.
  • Najman, J, Bongartz, D., and Mitsos A (2019). "Relaxations of thermodynamic property and costing models in process engineering." Computers & Chemical Engineering, 130, 106571.
  • 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.