# Optimal Design of Switchable Wearable Antenna Array for Wireless Sensor Networks
## 2 Models and Methods
### Inverse Problem Formulation
#### 2.2.4. Full Pareto-Like Optimization: P-EStra  
Eventually, a full multi-objective approach in terms of \((G_{s},\ G_{i})\) space was considered, without resorting to a scalar preference function like in previous Sections 2.2.2 and 2.2.3.  
To this end, an algorithm based on a multiobjective (1+1)-evolution strategy (P-EStra) inspired by Pareto optimality theory is used [25; 26]. The algorithm is implemented in such a way that a new design vector \(x=m+du\) (offspring) is accepted if and only if \(x\) dominates the current design vector \(m\) (parent) according to Pareto definition for \(m\geq 2\) objective functions, subject to problem constraints. This means that the offspring is accepted only if it simultaneously improves all the objectives. In turn, \(\mathrm{d}\) is the standard-deviation vector associated to \(m\), while \(u\in[0,1]\) is a normally distributed perturbation. Vector \(\mathrm{d}\) is initialized as \(d_{o}\), and the value of its elements is proportional to the feasible range of the corresponding design variable. Vector \(d\), which drives the search, is updated according to the prescribed rate of success in improving the objective functions; that is where the self-adaptation of the strategy parameter comes in: \(d\) itself undergoes a modification, which is ruled by a randomized process. In fact, given the correction rate \(q\in(0,1)\), considering the \(k\)-th iteration, \(d_{k+1}=q^{-1}\cdot d_{k}\) (or \(d_{k+1}=q\cdot d_{k}\)) is set to force a larger (or smaller) standard deviation of Gaussian distribution associated with \(x\) in the next iteration. In other words, the solution vector \(\mathrm{x}\) and the standard-deviation vector \(d\) are both subject to random mutation. In a basic (1+1) implementation, the operator of Paretian selection allows for the best individual, out of parent \(m\) and offspring \(x\), to survive to the next generation. In other words, an offspring individual is selected to survive if and only if it is better, or at least non-worse, than the parent individual against all the \(\mathrm{m}\) objective functions. This way, given an initial point, there  
Figure 9. The definition of \(d_{R}\) and \(d_{U}\) in the objective function components space.  
is a non-zero probability that the optimization trajectory eventually leads to a point belonging to the Pareto optimal front. The algorithm converges when the ratio of the largest value of d vector elements to the corresponding element of the initial standard-deviation vector \(d_{o}\) is smaller than the prescribed search tolerance.  
The basic computational cost c of P-EStra algorithm can be estimated as  
\[c\approx c_{0}\cdot n_{i}\cdot n_{p}\cdot n_{f}\]  
where \(c_{0}\) is the hardware-dependent time necessary to run a single solution of the direct problem associated to the optimization problem (in our case, the computation of directivity pattern of antenna array), \(n_{i}\) is the number of convergence iterations for a prescribed search accuracy, \(n_{p}\) is the number of evolving solutions (in our case, \(n_{p}=1\)), and \(n_{f}\) is the number of objectives.  
In principle, there is no limitation to the number of objectives the algorithm can process, and this is a potential advantage for a high-dimensionality problem. Summing up, the evolutionary algorithm starts from a guess solution, which can be either user-supplied or randomly generated, iteratively originates a search trajectory driven by the concept of non-dominated solution, and eventually converges to a Pareto-optimal solution.