Scientific Computing

This module demonstrates the penalty function method for constrained optimization. The penalty function method computes an approximate solution to a constrained optimization problem by successive unconstrained optimization of a weighted combination of the original objective function and a function that penalizes violation of the constraints. By varying the penalty parameter, the method converges iteratively to an approximate solution of the original constrained optimization problem.

The user begins by choosing either a preset example or typing in a
desired objective function *f*(*x*,
*y*)*g*(*x*,
*y*)*ρ* for
each iteration is chosen using the slider. To approach feasibility,
*ρ* should successively increase, weighting constraint
satisfaction more heavily. After the value for *ρ* has been
selected, the user clicks *Done* and moves on to the next step, in
which the unconstrained minimum of the current penalty function is
computed using the previous solution as starting point. The whole
process can be repeated until it converges to the approximate
constrained solution. Values for successive iterates are recorded in
the table below.

**Reference:** Michael T. Heath, *Scientific Computing,
An Introductory Survey*, 2nd edition, McGraw-Hill, New York,
2002. See Section 6.7.2, especially Example 6.16.

**Developers:** Jessica Schoen and Michael Heath