Optimization of a Spring Example

This GAUSS optimization example demonstrates the optimization of the equilibrium of a spring system.

Code for optimization

new;
cls;

library optmt;


//Objective function
proc fct(x, k, F, ind);

    //Declare 'mm' to be a modelResults
    //struct, local to this procedure
    struct modelResults mm;

    //If the first element of the indicator vector
    //is non-zero, calculate the objective function
    if ind[1];
        //Assign the value of the objective function to the
        //'function' member of the 'modelResults' struct
        mm.function = k[1] * (sqrt(x[1]^2 + (x[2] + 1)^2) - 1)^2 + k[2] * (sqrt(x[1]^2 + (x[2] - 1)^2) - 1)^2 - (F'x);
    endif;

    //Return the modelResults structure
    retp(mm);
endp;

// Starting values
x0 = { 1, 1 };

//Extra data needed by function
k = { 100, 90 };
F = { 20, 40 };

//Declare 'out' to be an optmtResults struct
//to hold the optimization results
struct optmtResults out;

//Declare 'ctl' to be an optmtControl structure
struct optmtControl ctl;

//Fill 'ctl' with default options
ctl = optmtControlCreate();

//Print report on every 5th iteration
ctl.PrintIters = 5;

//Use Newton Algorithm
ctl.Algorithm = 3;

//Use Strong Wolfe Conditions
ctl.LineSearch = 4;

//Minimize objective function
//Pass in extra data needed by objective function
//between starting parameters and control structure
out = optmt(&fct, x0, k, F, ctl);

//Print optimization results
call optmtPrt(out);

The code prints results to the Command Window.

Results

Convergence details

The first portion of the results provide details about convergence and performance.

Return code    =    0
Function value =   -9.65623
Convergence    :    normal convergence

These results indicate that the optimization converged normally, with a return code of 0. Any return Code other than 0 would indicate some issue with the convergence. The exact meaning of the return code can be found in the optmt() documentation.

Parameter estimates

The next section of the results reports the parameter estimates and the associated gradients.

Parameters  Estimates   Gradient
---------------------------------------------------------------------
x[1,1]      0.5044      0.0000
x[2,1]      0.1219      0.0000

In this example, the gradients are all 0 for the estimates, as is expected at or near an optimum.

Computation time

The final section of the results reports the number of iterations and computation time.

Number of iterations    6
Minutes to convergence     0.00123