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

   file approx.mos
   ```````````````
   Function approximation with SOS2   

   SOS2s are generally used for modeling piecewise approximations
   of functions of a single variable. In this example, we aim 
   to represent f as a function of x and we consider four line
   segments between five points. The five points are (R1, F1), 
   (R2, F2), (R3, F3), (R4, F4) and (R5, F5) and associated with 
   each point i is a weight variable y(i). Binary variable b(i) is
   associated with each of the intervals (R1, R2), (R2, R3), (R3, 
   R4) and (R4, R5), so 'b(i)' takes value 1 if the value of x 
   lies between 'R(i)' and 'R(i+1)'.

   The SOS2 property of having at most two non-zero 'y(i)', and 
   if there are two non-zero then they must be adjacent, implies 
   that we are always on the piece-wise linear function.

   (c) 2008 Fair Isaac Corporation
       author: S. Heipcke, Sep. 2006
*******************************************************!)

model "Approximation"
 uses "mmxprs"

 declarations
  NB = 5
  BREAKS = 1..NB
  R,F: array(BREAKS) of real           ! Coordinates of break points
  x,f: mpvar                           ! Decision variables
  y: array(BREAKS) of mpvar            ! Weight variables
 end-declarations

 R:: [1, 2.2, 3.4, 4.8, 6.5]
 F:: [2, 3.2, 1.4, 2.5, 0.8]
 
 RefRow:= sum(i in BREAKS) R(i)*y(i) 
 x = RefRow
 f = sum(i in BREAKS) F(i)*y(i)
 
! Convexity constraint
 sum(i in BREAKS) y(i) = 1

! SOS2 definition
 RefRow is_sos2

! Alternative SOS definition (to be used for 0-valued RefRow coefficients):
! makesos2(union(i in BREAKS) {y(i)}, RefRow)

(! Alternative formulation using binaries instead of SOS2:
 declarations
  b: array(1..NB-1) of mpvar
 end-declarations

 sum(i in 1..NB-1) b(i) = 1
 forall(i in 1..NB-1) b(i) is_binary
 forall(i in BREAKS) y(i) <= if(i>1, b(i-1), 0) + if(i<NB, b(i), 0)
!)

! Bounds
 1<=x; x<=6.5

! Solve the problem
 minimize(f)
 
 writeln("Objective value: ", getobjval)
 writeln("x: ", getsol(x))
  
end-model