Validating multi-start fmincon optimization

I am using fmincon solver for optimizing a nonlinear problem with three design variables. I have ub and lb bounds for the design varibales along with onley one inequality non-linear constraint which is implemented by @nonlcon. I use 10000 diffeerent initial guesses to ensure the coverage of design space. I have to questions: 1- is it sensible to double check the optimal solution with a hybrid surrogateopt–fmincon approach which combines global search followed up by local optimization? I already did this and i obtianed results very close to multi-start fmincon process, however, i wanted to make sure if it makes sense? 2 - in both approch most of the time KKT residual is not close to zero, how necessary is to have KKT residual ot be near zero for a non-convext and non-linear problem with possibel several local minimum. not having KKT residuals at zero implies unrelibale solutions?

[Matt J]

in both approch most of the time KKT residual is not close to zero

Close relative to what? What OptimalityTolerance have you set? What are the exitflags returned by run() ?

[Sobhan]

thanks, Close relative to what? the optimal solution from the optimization process with fmincon (10000 initial guesses used) was found to be in 1-2% difference from output solution from the optimization problem modelled with surrogateopt solver. OptimalityTolerance have you set? 'StepTolerance', 1e-8, exitflags are either 1 or 2.

[Matt J]

was found to be in 1-2% difference from output solution from the optimization problem modelled with surrogateopt solver.

I don't see what that tells you about the KKT residual.

[Sobhan]

My main Q is that: does make sense to confirm the solutions form fmincon solver with global optimization like surrogateopt?-any idea will be very appreciated.

Answer 1

[Matt J]

My main Q is that: does make sense to confirm the solutions form fmincon solver with global optimization like surrogateopt?

It can't hurt, and may add some extra confidence, but none of the solvers can be used to definitely determine whether a local optimum has been found. The exitflags are provided to you as an indication of how optimistic the solver is about the termination point being a solution.