Constrained optimization

Constrained optimization takes into consideration the inequality (G) and equality (H) constraints of the optimization problem. These constraints impose requirements on the solution space that must be met. Generally, equality constraints H can be strongly enforced in the problem. Alternatively, inequality constraints G may not be active, and therefore cannot directly be enforced. Nevertheless, there are several ways of dealing with both types of constraints in optimization problems, being penalized and augmented Lagrange methods those most used.

Penalized methods

Penalized methods weakly enforce the inequality constraint, transforming a constrained problem to an unconstrained one. This is done by including the constraint in the objective function through a penalization function. The choice of this function determines the characteristics of the objective function and whether the constraint is enforced such that the resulting solution must be located inside the initial feasible domain or not.

Augmented Lagrange methods

Augmented Lagrange methods take advantage of the Lagrangian formulation

The Lagrangian is augmented to include a penalty term with weight r for the constraints:

Finally, the method is defined as follows:

1. Select initial values for the Lagrange multipliers λ and μ and the penalty weights r.
2. For fixed λ and μ, minimize L_a(s, λ, μ) with respect to s to determine the new s for this iteration step.
3. Update λ and μ to determine λ_new=max(0,λ+rG(s)) and μ_new=μ+rH(s).
4. Repeat steps 1 to 3 until convergence.

Implementation in Kratos?