Solving Any Nonlinear Problem with a MILP Model

Although the author is an Electrical Engineer he got interested in optimization problems using the GAMS software. Rapidly he understood the limitations of the nonlinear solvers, like the necessity to have an initial feasible solution and the high probability of the solver being trapped in a local optimum and since 2004 he solved a set of complex nonlinear problems using MILP models. From the solution of the optimization of AGVs (Autonomous Guided Vehicles) Network [1] resulted techniques of implementation of logical functions over variables of the model only with algebraic expressions and the division of two model variables with the aid of auxiliary binary variables. From the solution of generation of optimal error correcting codes [2,9] resulted a first sketch of the automatic relaxation of a set of constraints, after improved in the solution of university timetabling with soft (that can be relaxed) and hard (that cannot be relaxed) constraints. We claim that our solution is the optimal solution to the well know problem of maximal constraint satisfaction. Finally from the solution of the generation of additive-multiplicative magic squares [3–5] resulted in a technique of multiplication of a set of variables of a MILP model. Obviously these latter techniques can be extended to implement any nonlinear operation over a set of variables of a MILP model. This way any nonlinear problem can be solved with a MILP model. This will guarantee that the solver reach always the global optimum even for a multimodal nonlinear problem. This way we solve an open fundamental problem in optimization theory: obtaining the global optimum for any multimodal nonlinear problem. Nevertheless the MILP model that solves the nonlinear problem can be too large and the computational resources can be not enough. Finally we propose new ways to implement nonlinear solvers based on meta-heuristics and Neural Networks, since our techniques require great computational resources for moderate size problems.