In this paper we consider a particular class of nonlinear optimization problems involving both continuous and discrete variables. The distinguishing feature of this class of nonlinear mixed optimization problems is that the structure and the number of variables of the problem depend on the values of some discrete variables. In particular we define a general algorithm model for the solution of this class of problems, that generalizes the approach recently proposed by Audet and Dennis (\cite{dennis}) and is based on the strategy of alternating a local search with respect to the continuous variables and a local search with respect to the discrete variables. We prove the global convergence of the algorithm model without specifying exactly the local continuous search, but only identifying its minimal requirements. Moreover we define a particular derivative-free algorithm for solving mixed variable programming problems where the continuous variables are linearly constrained and derivative information is not available. Finally we report numerical results obtained by the proposed derivative-free algorithm in solving a real optimal design problem. These results show the effectiveness of the approach.

## Citation

To appear in SIAM Journal on Optimization