# Automatic Initialization Methods for Photonic Components on a Silicon-Based Optical Switch
## 3 Optimization Method
In absence of constraints, the Newton optimization method is the most suitable one. On the contrary, for constrained optimizations, we can either apply the sequential quadratic programming method, the extended Lagrange method, or the non-linear internal point method [27]. Figure 2: Schematics of a double-ring switching node and an Mach–Zehnder interferometer (MZI)-based interleaver before (left panels) and after (right panels) the alignment procedure. (upper panels) A laser source with a wavelength that does not match the resonance of the switching node, is injected into the input port and all the optical power is collected at the through port (left). When the MRR heaters are properly biased, the resonance of the node is shifted on the laser wavelength and the optical signal is dropped from the input waveguide (right). (lower panels) A 200 GHz broaden input is coupled into the interleaver. Due to fabrication errors, the optical signal is unevenly distributed among the two arms (left). When the proper biasing configuration is applied to the interleaver heaters, the flat top transfer function is recovered at one of the two arms (right). However, in our case the target function was non-linear and non-differentiable, and so direct-search approaches were the most appropriate ones. In contrast to the optimization techniques that require the derivative of the objective target to determine a search direction, a direct-search method relies solely on the value of the target function on a set of points. Direct-search methods include the Nelder-Mead (NM), the simulated annealing, and the whole class of the evolutionary algorithms [27]. Most of these methods use the greedy criterion to make their decisions. Under the greedy criterion, a step is taken if and only if it increases the value of the target function. Although the greedy decision process converges fairly fast, it runs the risk of becoming trapped in a local minimum. The NM method [28], is the most popular among the unconstrained direct methods for local optimization problems. Its widespread use is due to its rapid initial convergence [29]. This method computes and compares the target function at the \(n+1\) vertices of a nondegenerate simplex (a \(n\)-dimensional polytype of nonzero volume that is the convex hull of its \(n+1\) vertices), where \(n\) is the number of design variables. The algorithm at each iteration removes the vertex with the worst (minimum) value of the target function and replaces it with another point with a better (higher) value. The new point is obtained by reflecting, expanding, or contracting the simplex along the line, which joins the worst vertex with the centroid of the remaining vertices. If a better point cannot be found in this way, the algorithm retains only the vertex with the best value of the target function, and the simplex is shrunk by pushing all other vertices toward that value. The algorithm runs as long as the simplex converges into a local optimum. Of course, the local optimum met depends on how the starting vertices of simplex is chosen. In the literature, we find only few examples of methods that automatically tune in resonance the high-order MRR-based filters [30]. [PERSON] et al. addressed this optimization problem by combining the NM with the coordinate descent method (algorithm that belongs to the class of the derivative-free optimization methods), to obtain an enhanced convergence efficiency. Instead, we have decided to apply another version of the NM method, namely the Globalized Bounded Nelder-Mead (GBNM) method. The GBNM accounts for variable bounds and is particularly robust against stagnation of the convergence solution, into local attraction basins of the target function [31; 32]. In fact, it combines a local optimizer, the pure NM search, a global optimizer, to random restarts of the local optimizer. To be more effective, the random restarts are triggered by a probability function, which keeps in memory the past local searches and pushes the new searches in domain regions where no solutions have yet been found.