Basic Optimization Example

This GAUSS optimization example demonstrates the minimization of a quadratic function as outlined in Luenberger’s “Linear and Nonlinear Programming.”

Code for optimization

The example:

• Creates a data matrix.

• Defines an objective function that takes 4 inputs.

new;
cls;

// Make comt library available
library comt;

// Create data needed by objective procedure
Q = { 0.78 -0.02 -0.12 -0.14,
   -0.02  0.86 -0.04  0.06,
   -0.12 -0.04  0.72 -0.08,
   -0.14  0.06 -0.08  0.74 };

b = { 0.76, 0.08, 1.12, 0.68 };

// Objective procedure with 4 inputs:
//    i.      x       - The parameter vector
//    ii-iii. Q and b - Extra data needed by the objective procedure
//    ii.     ind     - The indicator vector
proc  qfct(x, Q, b, ind);
  // Declare 'mm' to be a modelResults
  // struct local to this procedure, 'qfct'
  struct modelResults mm;

  // If the first element of the indicator
  // vector is non-zero, compute function value
  // and assign it to the 'function' member
  // of the modelResults struct
  if ind[1];
      mm.function = 0.5*x'*Q*x - x'b;
  endif;

  // Return modelResults struct
  retp(mm);
endp;

// Starting parameter values
x_strt = { 1, 1, 1, 1 };

// Declare 'out' to be a comtResults struct
// to hold the results from the optimization
struct comtResults out;

// Minimize objective function
out = comt(&qfct, x_strt, Q, b);

// Print output to the screen
call comtPrt(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 = -2.17466
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 comt() documentation.

Parameter estimates

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

Parameters  Estimates    Gradient
---------------------------------------------------------------------
x[1,1]         1.5350      0.0000
x[2,1]         0.1220      0.0000
x[3,1]         1.9752      0.0000
x[4,1]         1.4130      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    16
Minutes to convergence  0.00025