4.5.5 Example of a Nonlinear Optimization Task

In this example the minimum of a quadratic paraboloid with the origin (-1,0)

f(x,y) = ( x + 1)² + y² (4.49)

should be found under the condition

$$g(x,y) = x + y -1 \le 0 .$$ (4.50)

This nonlinear optimization task can be described by an experiment file as listed in Figure 4.15. Here the keyword optimizer has to be chosen. Again the variable model assigns the model file to be executed, free is the list of free variables set by the optimizer and target is a scalar target function. In addition an inequality constraint is added (inequality-constraint) which can be a list of scalar outputs of the model.

Figure 4.15: The experiment file describes an optimization with nonlinear constraints.

(optimizer 
           (model                 "donopttest.mod")
           (free                  x y)
           (target                f)
           (inequality-constraint constraint1)
           (del0                  1)
           (epsdif                0.0001)
           (log                   "donopttest.log")  
           (dump                  "donopttest.crv"))

In the model file (Figure 4.16) the algorithm for the target function, and the constraints are specified. An inequality constraint is violated if its function value is greater than zero and inactive if it is zero or less.

Figure 4.16: The model file describes the model function.

(arithmetic-model
 (inputs
  (x            bound-float   2.111    -4.    5.)
  (y            bound-float   2.       -6.    5.))
 (outputs
  (f            float)
  (constraint1  float))
 (operators
  (f            (+ (* (+ 1 x) 
                      (+ 1 x)) 
                   (* y y)))
  (constraint1  (- (+ x y)
                   1))))

This optimization process took 8 gradient and 23 evaluations for adjustment of the step size. In Figure 4.17 the chosen parameter values for the single steps and in Figure 4.18 the target function and the inequality constraint are shown. In Table 4.3 the optimum values found by the optimizer are listed. The optimized result differs from the exact result in the fifth decimal digit.

Figure 4.17: Values for the independent parameters x and x.

Figure 4.18: Values for the target function f and the constraint constraint1.

Table 4.3: Comparison of the results from the optimizer and exact solution.

	optimizer	exact
x	$2.54712 \cdot 10^{-5}$	0.
y	0.999975	1.