FICO Xpress Optimization Examples Repository
FICO Optimization Community FICO Xpress Optimization Home
Back to examples browserNext example

Introductory examples

Problem name and type, featuresDifficulty
approx Approximation: Piecewise linear approximation **
SOS-2, Special Ordered Sets, piecewise linear approximation of a nonlinear function, pwlin
burglar MIP modeling: Knapsack problem: 'Burglar' *
simple MIP model with binary variables, data input from text data file, array initialization, numerical indices, string indices, record data structure
chess LP modeling: Production planning: 'Chess' problem *
simple LP model, solution output, primal solution values, slack values, activity values, dual solution values
pricebrai All item discount pricing: Piecewise linear function ***
SOS-1, Special Ordered Sets, piecewise linear function, approximation of non-continuous function, step function, pwlin
pricebrinc Incremental pricebreaks: Piecewise linear function ***
SOS-2, Special Ordered Sets, piecewise linear function, step function

Further explanation of this example: 'Applications of optimization with Xpress-MP', Introductory examples (Chapters 1 to 5) of the book 'Applications of optimization with Xpress-MP'[download all files]

Source Files

Data Files


   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"

  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

 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:
  b: array(1..NB-1) of mpvar

 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
 writeln("Objective value: ", getobjval)
 writeln("x: ", getsol(x))

Back to examples browserNext example