Derivative-Free Methods for Mixed-Integer Constrained Optimization Problems

Abstract

Methods which do not use any derivative information are becoming popular among researchers, since they allow to solve many real-world engineering problems. Such problems are frequently characterized by the presence of discrete variables, which can further complicate the optimization process. In this paper, we propose derivative-free algorithms for solving continuously differentiable Mixed Integer NonLinear Programming problems with general nonlinear constraints and explicit handling of bound constraints on the problem variables. We use an exterior penalty approach to handle the general nonlinear constraints and a local search approach to take into account the presence of discrete variables. We show that the proposed algorithms globally converge to points satisfying different necessary optimality conditions. We report a computational experience and a comparison with a well-known derivative-free optimization software package, i.e., NOMAD, on a set of test problems. Furthermore, we employ the proposed methods and NOMAD to solve a real problem concerning the optimal design of an industrial electric motor. This allows to show that the method converging to the better extended stationary points obtains the best solution also from an applicative point of view.

Appendix: Technical Result

In this section, we report a technical result which is needed to prove convergence of DFL and EDFL. It is a slight modification of an analogous result reported in [20], which takes into account the presence of discrete variables.

Proposition 7.1

Let $\{\epsilon _k\}$ be a bounded sequence of positive penalty parameters. Let $\{x_k\}$ be a sequence of points such that $x_k\in \mathcal{X}\cap \mathcal Z$ for all $k$, and let $\bar{x}\in \mathcal{X}\cap \mathcal Z$ be a limit point of a subsequence $\{x_k\}_K$ for some infinite set $K\subseteq \{0,1,\ldots \}$. Suppose that for each $k\in K$ sufficiently large,

• (i) for all $d^i\in D^c\cap D(\bar{x})$, there exist vectors $y_k^i$ and scalars $\eta _k^i>0$ such that

  
• (ii)
  $$\begin{aligned} \lim _{k\rightarrow \infty ,k\in K}\epsilon _k\Vert g^+(x_k)\Vert = 0; \end{aligned}$$
• (iii)
  $$\begin{aligned} (y_k^{i})_z = (x_k)_z,\qquad \text{ for } \text{ all }\ i\in I_c. \end{aligned}$$

Then, $\bar{x}$ is a stationary point for Problem (1) with respect to the continuous variables, that is, $\bar{x}$ satisfies (2) and (4) with $\bar{\lambda }\in \mathbb {R}^m$ given by

$$\begin{aligned} \lim _{k\rightarrow \infty ,k\in K}\lambda _j(x_k;\epsilon _k) = \lim _{k\rightarrow \infty ,k\in K}\lambda _j(y_k^i;\epsilon _k) = \bar{\lambda }_j,\qquad \forall \ i\in I_c\ \text{ and }\ j=1,\ldots ,m, \end{aligned}$$

where $\lambda _j(x;\epsilon )$, $j=1,\ldots ,m$, are defined in (8).

Proof

Considering point (iii), namely that the discrete variables are held fixed in the considered subsequence, the proof is the same as that of Proposition 3.1 in [20]. $\square$