Extensions of a Multistart Clustering Algorithm for Constrained Global Optimization Problems

As already mentioned, GLOBALf was designed to solve bound-constrained problems. Here we add constraints handling capabilities to GLOBALm. If suitable local solvers for constrained optimization problems are available, the difficulty arises in the global phase of the algorithm, i.e. the selection of good points from which the local search is to be started. In this case we will make use of the L₁ exact penalty function:

This penalty function is exact in the sense that for sufficiently large values of the penalty weights, a local minimum of P₁ is also a local minimum of the original constrained problem. In particular, if x^* is a local minimum of the constrained problem, and ^* and u^* are the corresponding optimal Lagrange multiplier vectors, x^* is also a local minimum of P₁ if (Edgar et al., 2001):

This has the advantage that the penalty weights do not have to approach infinity as in the case of e.g. the quadratic penalty function, and consequently, a lesser distortion can be expected in the transformed objective function. If the local solver provides estimates of the Lagrange multipliers, an iterative procedure can be applied in which the values of these weights are updated with the feedback resulting from the local search.

Finally, it should be noted that, although this penalty function is non-differentiable, it is only used during the global phase, i.e. to select the candidate points from which the local solver is then started.