## Thursday, December 22, 2011

c6.. y1*(y1-1) =e= 0;
c7.. y2*(y2-1) =e= 0;
c8.. y3*(y3-1) =e= 0;

This is not really a good idea. These constraints essentially say: y1,y2 and y3 can assume the values 0 or 1, i.e. they are binary variables.

If you really want to use binary variables in an NLP, form a proper MINLP model. If this idea of using non-convex constraints to model binary variables was really so hot, then we would no longer need a MIP solver and just use an NLP solver.

It looks a bit that this poster is attempting to run before (s)he can walk. It would be much better if you have some knowledge about NLP modeling before before actually working on “stochastic global optimization”. May be the advisor or teacher should play a more constructive role here.

Notes:

• in general NLP solvers don’t “understand” the functions you give them. They merely sample points, function values and derivatives as they go along. One exception is the global solver Baron: this solver really tries to work with and understand the functional form of the NL functions.
• although in general it is better to model binary decisions as binary variables in an MINLP (or MIP if the rest is linear), there are some attempts to use nonlinear programming techniques to solve integer models:
• W. Murray and K. M. Ng, “An Algorithm for Nonlinear Optimization Problems with Binary Variables,” Computational Optimization and Applications, Vol. 47, No. 2, 2008, pp. 257-288
• Roohollah Aliakbari Shandiz, Nezam Mahdavi-Amiri, “An Exact Penalty Approach for Mixed Integer Nonlinear Programming Problems”, American Journal of Operations Research, 2011, 1, 185-189