Feasible Starting Point

The NLP Procedure

Two algorithms are used to obtain a feasible starting point.

When only boundary constraints are specified:
- If the parameter x_j, $1 \leq j \leq n$ , violates a two-sided boundary constraint (or an equality constraint) $l_j \leq x_j \leq u_j$ , the parameter is given a new value inside the feasible interval, as follows:
  $x_j = & l_j , & {if u_j \leq l_j\space } \ x_j = & l_j + {1 \over 2}(u_j - l_j)... ...l_j \lt 4} \ x_j = & l_j + {1 \over 10}(u_j - l_j) , & {if u_j - l_j \geq 4}$
- If the parameter x_j, j = 1, ... ,n, violates a one-sided boundary constraint $l_j \leq x_j$ or $x_j \leq u_j$ , the parameter is given a new value near the violated boundary, as follows:
  $x_j = & l_j + \max(1, {1 \over 10}l_j) , & {if l_j\space violated} \ x_j = & u_j - \max(1, {1 \over 10}u_j) , & {if u_j\space violated}$
When general linear constraints are specified, a feasible point is computed by the algorithm of Schittkowski and Stoer (1979), that may be quite far from a user-specified infeasible point.