A parameterization approach for mixed-integer optimization

Mixed-integer problems represent a wide class of industrial optimization ap- plications. We talk about "mixed-integer" optimization problem when the design variables are discrete and continuous type together. In this particular case, the (integer) number of design variables of the problem represents an unknown for the problem itself. For all the optimization problems, the selec- tion of the number of design variable represents a basic assumption, and the final solution of the problem is largely influenced by this choice. By the way, an early definition of the number of variables may represent a strong limi- tation in the future development of the optimizing system. The number of blades of a turbo-machine, the number of hulls of a ship, or also the number of connections in a network are three simple examples of such a problem. On the other hand, the definition of an optimization problem classically requires an explicit definition of the type and number of involved design variables. In this report, a strategy for tackling and solve a problem with an unpre- scribed number of design variables is presented. The number of the design variables and their values will be defined implicitly, by adopting a rearrange- ment of the data structure, eliminating the direct definition of the design variable number and their values: as a consequence of this strategy, the number of design variables is undefined, or at least limited between two ex- treme values, and it will represents a part of the solution of the optimization problem itself.