Nonlinear Optimization Examples

Constrained Betts Function

The linearly constrained Betts function (Hock and Schittkowski, 1981) is defined as

\[ f(x) = 0.01 x_1^2 + x_2^2 - 100 \]

The boundary constraints are

\begin{eqnarray*} 2 \leq & x_1 & \leq 50 \\ -50 \leq & x_2 & \leq 50 \end{eqnarray*}

The linear constraint is

\[ 10x_1 - x_2 \geq 10 \]

The following code calls the NLPCG subroutine to solve the optimization problem. The infeasible initial point $x^0 = (-1,-1)$ is specified, and a portion of the output is shown in Figure 14.3.

The NLPCG subroutine performs conjugate gradient optimization. It requires only function and gradient calls. The F_BETTS module represents the Betts function, and since no module is defined to specify the gradient, first-order derivatives are computed by finite-difference approximations. For more information about the NLPCG subroutine, see the section NLPCG Call. For details about the constraint matrix, which is represented by the CON matrix in the preceding code, see the section Parameter Constraints.

Figure 14.3: NLPCG Solution to Betts Problem

Optimization Start
Parameter Estimates
N Parameter Estimate Gradient
Objective
Function 		Lower
Bound
Constraint 		Upper
Bound
Constraint
1 X1 6.800000 0.136000 2.000000 50.000000
2 X2 -1.000000 -2.000000 -50.000000 50.000000

Linear Constraints
1 59.00000 :   10.0000 <= + 10.0000 * X1 - 1.0000 * X2

Parameter Estimates 2
Lower Bounds 2
Upper Bounds 2
Linear Constraints 1

Optimization Start
Active Constraints 0 Objective Function -98.5376
Max Abs Gradient Element 2    

Iteration   Restarts Function
Calls 		Active
Constraints 		  Objective
Function 		Objective
Function
Change 		Max Abs
Gradient
Element 		Step
Size 		Slope of
Search
Direction
1   0 3 0   -99.54682 1.0092 0.1346 0.502 -4.018
2   1 7 1   -99.96000 0.4132 0.00272 34.985 -0.0182
3   2 9 1   -99.96000 1.851E-6 0 0.500 -74E-7

Optimization Results
Iterations 3 Function Calls 10
Gradient Calls 9 Active Constraints 1
Objective Function -99.96 Max Abs Gradient Element 0
Slope of Search Direction -7.398365E-6    

Optimization Results
Parameter Estimates
N Parameter Estimate Gradient
Objective
Function 		Active
Bound
Constraint
1 X1 2.000000 0.040000 Lower BC
2 X2 -1.24028E-10 0  

Linear Constraints Evaluated at Solution
1   10.00000 = -10.0000 + 10.0000 * X1 - 1.0000 * X2 




            

            


            
Since the initial point $(-1,-1)$ is infeasible, the subroutine first computes a feasible starting point. Convergence is achieved after three iterations, and
               the optimal point is given to be $x^* = (2,0)$ with an optimal function value of $f^* = f(x^*) = -99.96$. For more information about the printed output, see the section Printing the Optimization History.