Interior Penalty Method

Interior penalty method for the constrained Optimization problem

In the interior penalty function methods, a new function (𝜙 function) is constructed by augmenting a penalty term to the objective function. The penalty term is chosen such that its value will be small at points away from the constraint boundaries and will tend to infinity as the constraint boundaries are approached. Hence the value of the 𝜙 function also “blows up” as the constraint boundaries are approached. Thus, once the unconstrained minimization of 𝜙(X, rk) is started from any feasible point X1, the subsequent points generated will always lie within the feasible domain since the constraint boundaries act as barriers during the minimization process.