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 EAGO-notebooks in the form of 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),
and Diff for differentiable relaxations (Khan2016, Khan2018, Khan2019). Conversion between MV, NS, 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 asssociated 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
Trignometric 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
Common Algebraic Expressions: xlogx, arh, xexpax
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 function via McCormick operators
In order to bound a function using a McCormick relaxation. You first construct structure that bounds the input variables then you construct pass these variables two a function.
In the example below, convex/concave relaxations of the function f(x) = sin(2x) + exp(x) - x
are calculated at x = 1 on the interval [-2,3].
using McCormick
# create MC object for x = 2.0 on [1.0,3.0] 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 IntvThe plotting the results we can easily generate visual the convex and concave relaxations, interval bounds, and affine bounds constructed using the subgradient at the middle of X.
This can readily be extended to multivariate functions as shown below
f(x) = max(x[1],x[2])
x = [2.0 1.0] # values of independent variable x
Intv = [Interval(-4.0,5.0), Interval(-5.0,3.0)] # define intervals to relax over
# create McCormick object
xMC = [MC{2,Diff}(x[i], Intv[i], i) for i=1:2)]
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
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 (2020) EAGO.jl: easy advanced global optimization in Julia,
Optimization Methods and Software, 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.