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## Sensitivity Based on IPOPT

## This website is still under construction

This project provides a toolbox that takes uses NLP sensitivity theory to generate fast approximations to solutions when parameters in the problem change. sIPOPT is appended to the IPOPT code, and thus takes advantage of the fast sparse linear solvers included with the solver. The approximated solutions are generated by a linearization of the KKT conditions. More details can be found in the documentation or in the implementation paper:

- Pirnay, R. Lopez-Negrete, and L.T. Biegler,
**Sensitivity Based on IPOPT**, Submitted for publication to*Math Prog Comp*, (**Apr. 2011**)

## Installation

The ﬁrst step to install the software is to install the trunk version of IPOPT, once this is done installing sIPOPT is very simple. IPOPT’s installation instructions can be found in the IPOPT documentation. Also note that in the following we refer to $IPOPT as the main folder, where the !Ipopt/, ThirdParty/, BuildTools/, . . . , folders are located.

If you wish to use the AMPL interface, make sure that your IPOPT installation also includes it. To do this you need to download the ASL library, with the get.ASL script located in $IPOPT/ThirdParty/ASL.

Finally, we assume that you created a build folder to install IPOPT in $IPOPT/build/. In this case, to download the trunk version of IPOPT you would type:

$ svn co https://projects.coin-or.org/svn/Ipopt/trunk $IPOPT

Once IPOPT has been compiled and installed, we can proceed to build sIPOPT. To do this go to the $IPOPT/build/Ipopt/contrib/sIPOPT/ folder, and type make there.

$ cd $IPOPT/build/Ipopt/contrib/sIPOPT $ make

If no errors are shown after compilation you can proceed to install the libraries and to generate the AMPL executable. To do this type

$ make install

This should copy the generated libraries (libsipopt.*) to $IPOPT/build/lib, and the AMPL executable (ipopt_sens) to $IPOPT/build/bin/.

## Example

To illustrate the use of the toolbox we solve the following NLP

min x1^2 + x2^2 + x3^2 st 6x1 + 3x2 + 2x3 − p1 = 0 p2 x1 + x2 − x3 − 1 = 0 x1 , x2 , x3 ≥ 0,

and we perturb the parameters p1 and p2 from pa^{T} = [p1 p2] = [5 1] to pb^{T} = [4.5 1]. For this case the AMPL code is shown below.

reset ; # Suffixes for sensitivity update suffix sens_state_0, IN; suffix sens_state_1, IN; suffix sens_state_value_1, IN; suffix sens_sol_state_1, OUT; suffix sens_init_constr, IN; # Original value of parameters param et1p ; param et2p ; # Original parameter values let et1p := 5 ; let et2p := 1 ; # Define variables, with bounds and initial guess var x1 >= 0, := 0.15 ; var x2 >= 0, := 0.15 ; var x3 >= 0, := 0.00 ; # Artificial variables so IPOPT sees the parameters var et1 ; var et2 ; # objective function minimize objf: x1^2 + x2^2 + x3^2 ; # constraints subject to r1: 6*x1 + 3*x2 + 2*x3 - et1 = 0 ; r2: et2*x1 + x2 - x3 - 1 = 0 ; # Artificial constraints to pass parameters to IPOPT r3: et1 = et1p ; r4: et2 = et2p ; # Define solver and Ampl options in this case we don't want Ampl's # presolve to accidentally remove artificial variables. options solver ipopt_sens ; option presolve 0; # define an order to the parameters that will change. # In step 0, only et1 changes, and has position 1 let et1.sens_state_0 := 1 ; # in the first step/change et1 has position 1 let et1.sens_state_1 := 1 ; # Perturbed value of parameter et1 (in step 1) let et1.sens_state_value_1 := 4.5 ; # In step 0, et2 has position 1 let et2.sens_state_0 := 2 ; # in the first step/change et1 has position 2 let et2.sens_state_1 := 2 ; # Perturbed value of parameter et2 (in step 1) let et2.sens_state_value_1 := 1 ; # Artificial constraints let r3.sens_init_constr := 1 ; let r4.sens_init_constr := 1 ; # solve problem solve ; #********************************************** # Print nominal solution and bound multipliers #********************************************** display x1, x2, x3, et1, et2 ; display x1.ipopt_zU_out, x2.ipopt_zU_out, x3.ipopt_zU_out, et1.ipopt_zU_out, et2.ipopt_zU_out ; display x1.ipopt_zL_out, x2.ipopt_zL_out, x3.ipopt_zL_out, et1.ipopt_zL_out, et2.ipopt_zL_out ; # Constraint multipliers display r1, r2, r3, r4 ; #************************ # Print updated solution #************************ display x1.sens_sol_state_1, x2.sens_sol_state_1, x3.sens_sol_state_1, et1.sens_sol_state_1, et2.sens_sol_state_1 ; display x1.sens_sol_state_1_z_U, x2.sens_sol_state_1_z_U, x3.sens_sol_state_1_z_U, et1.sens_sol_state_1_z_U, et2.sens_sol_state_1_z_U ; display x1.sens_sol_state_1_z_L, x2.sens_sol_state_1_z_L, x3.sens_sol_state_1_z_L, et1.sens_sol_state_1_z_L, et2.sens_sol_state_1_z_L ; # and updated constraint multipliers display r1.sens_sol_state_1, r2.sens_sol_state_1, r3.sens_sol_state_1, r4.sens_sol_state_1 ;