2.4 A Second Example

We now consider a different sort of example, the unconstrained problem,

$${\rm minimize} \;\;\;\;F(x_1,\cdots,x_{1000}) \equiv \sum_{i... ...0}^2 + x_1 - 1) + {\scriptstyle \frac{1}{2}}\sin(x_{1000}^2).$$ (2.17)

Once again, there are a number of ways of casting this problem in the form (2.1), but the most natural is to consider the argument of each sine function as a group -- the group function is then $g_i (\alpha ) = p_1 \sin \alpha$, $1 \leq i \leq 1000$, for various values of the parameter $p_1$. Each group but the last has two nonlinear elements, $x_{1000}^2$ and $x_i^2$ $1 \leq i \leq 999$ and a single linear element $x_1 - 1$. The last has no linear element and a single nonlinear element, $x_{1000}^2$. A single element type, $v_1^2$, of the elemental variable, $v_1$, covers all of the nonlinear elements.

Thus we see that we can consider our objective function to be made up of 1000 nontrivial groups, all of the same type, so we will have to provide our optimization procedure with function and derivative values for these at some stage. There are 1999 nonlinear elements, two from each group except the last, but all of the same type and again we shall have to provide function and derivative values for these. As there is so much structure to this problem, it would be inefficient to pass the data group-by-group and element-by-element. Clearly, one would like to specify such repetitious structures using a convenient shorthand.