I have a function with an indefinite hessian and I would like to make sure GEKKO can support quasiconvex optimization. Note that I am formulating an MINLP model and those functions are present in the objective while the constraints are linear for the parameters.
Are there any special conditions to achieve a global optimum when dealing with quasiconvex function optimization using GEKKO? Additionally, are there any performance differences with a quasiconvex function as opposed to a convex problem?
Thank you for your time.
There are at two approaches to deal with quasiconvex problems in the solvers that GEKKO uses. The solvers employ regularization by adding a diagonal term to the diagonal of the Hessian until it is positive definite. For solvers that use only the first derivative (Jacobian), a BFGS update approximates the second derivatives (Hessian). This guarantees that the Hessian approximation is positive definite. A third option that solvers employ is a trust region method to avoid divergent solutions.
For nonconvex problems, a multi-start method is often needed to find the global optimum. Otherwise, another solver such as BARON or others that are specifically designed for nonconvex problems should be used.