Partial derivatives

! XNLP example demonstrating the cost of numerical derivatives
!
! This example solves the same simple problem several times using different 
! differentiation techniques and solvers
!
! 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) 2013-2025 Fair Isaac Corporation
!       author: Zsolt Csizmadia
model mmxnlp_nlp_duals
uses "mmxnlp", "mmsystem"; 

! Scaling parameter for problem size.
parameters
	 N = 200
end-parameters

declarations	
 R   = 1..N
 x   : array(R) of mpvar
 start, finish : real;
end-declarations

!setparam("xnlp_verbose",true)

forall(i in R) do
 x(i) >= 1 - 1/i
 x(i) <= 1 + 1/i
end-do

Objective := sum(i in R) ( x(i)^3 )

! Solve with a local solver to use derivatives
setparam("XPRS_NLPSOLVER",1)

! Default solve usign analytical derivatives, expected to take no time
write("Solve problem using analytical derivatives: ") 
start := gettime
minimize(Objective)
finish := gettime
writeln(finish-start,"s")

! Use the first order solver SLP to solve the problem
setparam("xnlp_solver",XNLP_SOLVER_XSLP)
setparam("xslp_derivatives", 0)
write("Solve problem using finite differences, 1st order solver: ") 
start := gettime
minimize(Objective)
finish := gettime
writeln(finish-start,"s")

! Now resolve using second order numerical derivatives
setparam("xnlp_solver",XNLP_SOLVER_KNITRO)
! Numerical derivatives here are enforced by the means of setting a control. 
! However, the need for numerical derivatives can arise naturally from 
! black-bock user functions
setparam("xslp_derivatives", 0)
write("Solve problem using finite differences, 2nd order solver: ") 
start := gettime
minimize(Objective)
finish := gettime
writeln(finish-start,"s")

! Revert back to the second order Knitro solver, but approximate the Hessian 
! matrix instead
setparam("xnlp_solver",XNLP_SOLVER_KNITRO)
setparam("xktr_param_hessopt", 2) ! BFGS approximation
write("Solve problem using finite differences, 2nd order solver, BFGS: ") 
start := gettime
minimize(Objective)
finish := gettime
writeln(finish-start,"s")

end-model