Mixed integer nonlinear programming without relaxation of integer constraints

Question

I have a multi-objective mixed integer nonlinear problem to solve. For this particular problem the objective functions are not defined for fractional values of the integer constrained variables.

From my initial review of the a few textbooks and papers, as well as searches on this forum, The only methods to solve mixed integer nonlinear problems requires the relaxation of the integer constants to fractional values during the search for the solution (eg: branch-and-bound, outer approximation)

Since my objective functions are undefined for fractional values of the integer constrained variables, these relaxation based methods are not appropriate. Are there any known methods (other than a brute force approach) which don't rely on relaxation?

I should mention that my objective functions are non-convex, but I would still be interested in methods for convex objectives.

Thanks in advance

Edit 1: I should have mentioned that my cost function value is only available from simulations, where the simulations are set up using continuous and discrete parameters which are my decision variables. I think this could limit me to pattern search methods, but any input from forum members or confirmation that I'm looking in the right direction would be helpful.

I've also come across Mixed Variable Programming where the decision variables can also include categorical variables (eg: the colours red, yellow and blue), rather than integer and continuous values.

Answer 1

Mixed-Integer Sequential Quadratic Programming, MISQP, fulfills your criteria (leaving aside multi-objective).

http://www.klaus-schittkowski.de/misqp.htm

Purpose: MISQP solves mixed-integer nonlinear programming problems by a modified sequential quadratic programming (SQP) method. It is not assumed that integer variables are relaxable, i.e., problem functions are evaluated only at integer points. The code is applicable also to nonconvex optimization problems.

Numerical Method: The algorithm is stabilized by a trust region method including Yuan's second order corrections. The Hessian of the Lagrangian function is approximated by BFGS updates subject to the continuous and integer variables. Successively, mixed-integer quadratic programs must be solved.

Also see the references listed there. For instance, O. Exler, K. Schittkowksi (2007): "A trust region SQP algorithm for mixed-integer nonlinear programming", Optimization Letters, Vol. 1, 269-280 http://www.klaus-schittkowski.de/misqp0_rep.htm