Numerical differentation and accuracy

! This example demonstrates the solver sensitibility to errors introduced by 
! numerical derivatives 
!
! This model should be run twice, with different settings for XNLP_SOLVER
! This example is a version of the Polygon4 example.
! 
! This example demonstrates a particular non-linear optimization concept as related 
! to Xpress NonLinear.
! The version of the example is for Xpress 7.5.
!
!  (c) 2013 Fair Isaac Corporation
 
model "Polygon 4"
 uses "mmxnlp"

 parameters
  N=5                               ! Number of vertices
  SOLVER=0                          ! 0: SLP, 1: Knitro
 end-parameters
 
 declarations
  RN = 1..N
  Area: nlctr
  rho : array(RN) of mpvar          ! Distance of vertex from the base point 
  theta : array(RN) of mpvar        ! Angle from x-axis
  D: array(RN,RN) of nlctr          ! Limit on side length
  FunctionArg: list of nlctr        ! User function arguments
  AreaFunction: userfunc            ! User function definition
 end-declarations

! Objective: sum of areas. Definition of a user function
 AreaFunction := userfuncExcel("polygonsheet.xls", "Sheet1")

! Create function arguments
! Excel functions use their first columns as input: use a list to ensure correct order
 forall (i in 1..N-1) do
  FunctionArg += [nlctr(rho(i))]
  FunctionArg += [nlctr(theta(i))]
 end-do

! Use the Excel user function in a formula for the objective
 Area := F(AreaFunction,FunctionArg)

! Bounds and start values for decision variables
 forall (i in 1..N-1) do
  rho(i) >= 0.1
  rho(i) <= 1
  setinitval(rho(i),4*i*(N + 1 - i)/((N+1)^2))
  setinitval(theta(i),M_PI*i/N)
 end-do
 
! Third side of all triangles <= 1
 forall (i in 1..N-2, j in i+1..N-1) 
  D(i,j) := rho(i)^2 + rho(j)^2 - rho(i)*rho(j)*2*cos(theta(j)-theta(i)) <= 1
 
! Vertices in increasing order 
 forall (i in 2..N-1) theta(i) >= theta(i-1) +.01
 
! Boundary conditions (last vertex above x-axis)
 theta(N-1) <= M_PI

! Uncomment to display user function info
! userfuncinfo(AreaFunction)

! Optional parameter settings
 setparam("xnlp_verbose", true)         ! Enable XNLP output log
 setparam("xnlp_solver", SOLVER)        ! Select the solver
 setparam("xslp_excelvisible", true)

! Solve the problem
 maximise(Area)

! Solution output
 writeln("Area = ", getobjval)
 forall (i in 1..N-1)
  writeln("V",i,": r=",getsol(rho(i))," theta=",getsol(theta(i)))

end-model