The NLPC Nonlinear Optimization Solver: Feasible Starting Point :: SAS/OR(R) 9.2 User's Guide: Mathematical Programming

The NLPC Nonlinear Optimization Solver

Feasible Starting Point

You can specify a starting point for the optimization. If the specified point is infeasible to linear and/or bound constraints, two schemes are used to obtain a feasible starting point (feasible to linear and bound constraints only), depending on the type of problem. They are as follows.

When only bound constraints are specified:
- If the variable , , violates a two-sided bound constraint $l_i \le x_i \le u_i$ , the variable is given a new value inside the feasible interval, as follows:
  $x_i = \{ l_i, & {\rm if \:} u_i = l_i \ l_i + \frac{1}2(u_i - l_i), & {\rm ... ...i \lt 4 \ l_i + \frac{1}{10}(u_i - l_i), & {\rm if \:} u_i - l_i \ge 4 .$
- If the variable , , violates a one-sided bound constraint $l_i \le x_i$ or $x_i \le u_i$ , the variable is given a new value near the violated bound, as follows:
  $x_i = \{ l_i + \max\{1,\frac{1}{10}l_i\}, & {\rm if \:} x_i \lt l_i \ u_i - \max\{1,\frac{1}{10}u_i\}, & {\rm if \:} x_i \gt u_i .$
When general linear constraints are specified, the scheme to find a feasible starting point involves two algorithms that 1) find a feasible point independent of the starting point or 2) find a feasible point closest to the starting point. Both algorithms are active set methods.

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