Approximating nonlinear univariate functions via piecewise linear constraints

(!******************************************************
   Mosel Example Problems
   ======================

   file piecewise_linear.mos
   `````````````````````````
   Approximating nonlinear univariate functions using
   piecewise linear constraints.

   -- Mosel version of the Python example piecewise_linear.py --

   (c) 2020 Fair Isaac Corporation
       author: Y.Colombani, Jun. 2020
*******************************************************!)
model piecewise_linear
uses 'mmxnlp'

parameters
  N = 100                    ! Number of points of the approximation
  FREQ = 27.5                ! Frequency of sine curve
end-parameters

public declarations
  STEP = 2 * M_PI / (N - 1)  ! Width of each x segment
  x: mpvar
end-declarations

! Piecewise linear, continuous concave function formulated via piecewise
! linear segments
pw1:= pwlin([pws(0,       10*x),
             pws(1, 10 +  3*(x-1)),
             pws(2, 13 +  2*(x-2)),
             pws(3, 15 +    (x-3))])

!**** Approximate sin(FREQ * x) for x in [0, 2*pi]
! Piecewise linear, discontinuous function over N points: over the
! i-th interval, the function is equal to v[i] + s[i] * (x - b[i])
! where v, s, b are value, slope, and breakpoint.
pw2:= pwlin(union(i in 0..N-1,s=i*STEP)[pws(s,sin(s*FREQ)+FREQ*cos(s*FREQ)*(x-s))])

minimise(pw1 - pw2)

writeln("solution: x = ", x.sol)
writeln("values of piecewise linear functions:", [pw1.sol, pw2.sol])
writeln("objective function:", getobjval)

end-model