Role of initial starting point in convergence of pattern search with non-linear constraints

When non-linear constraints are in in question, MATLAB has a variation of pattern search that used the augmenetd Lagrangian barrier algorithm to ensure solutions are feasible. A drawback of this approach , as far as I can se is that infeasible initial points (those that violate the non-linear constraints) would hamper the internal iterations as tehy would lead to the objective function being evaluated to infinity (because of the lagrangian barriers). How (If at all) does pattern search account for these anomalies?

Answer 1

[Alan Weiss]

The nonlinear constraint algorithm is described here. I believe that the nonnegative shifts s_i keep the penalty function finite even if c(x0) is positive. And there is no term that makes an infinity for ceq(x0).

Alan Weiss

MATLAB mathematical toolbox documentation

[Vineet]

From what I read in the paper references given in the link provided, the algorithm should converge to a feasible point provided the objective function and the constraints are doubly continuous.

For the project I'm working on, the algorithm doesn't converge to a feasible point even if feasible solutions exist, for some start points but it does converge however if you specify some other start points (usually points known to be feasible).

Are such cases common? If it is , is this a peculiarity of MATLAB's implementation of the algorithm or does the start point have a role in whether the algorithm converges or not?

[Alan Weiss]

I believe that all general nonlinear solvers depend on an initial point, and can fail to find a feasible solution for some initial points. This is certainly true for MATLAB, but I think it is generally true. The caveat is for some structured nonlinear problems such as quadratic, you don't need an initial point.

Alan Weiss

MATLAB mathematical toolbox documentation

[Vineet]

Is there a natural preference for feasible starting points (as in do feasible starting points more or less guarantee feasible convergence?). Also , are you saying the anomalies arise due to the discontinuous/non-differentiable nature of the non-linear constraints?