Different Approaches to Multi-Objective Sparse Array Problem with Social Network Optimization Alessandro Niccolai * , Francesco Grimaccia * , Marco Mussetta * , Riccardo E. Zich * * Politecnico di Milano, Dipartimento di Energia Via La Masa 34, 20156 Milano, Italy alessandro.niccolai@polimi.it Abstract—Multi-objective problems with two or more conflict- ing objectives are very common in every engineering fields, also for antenna optimization. Evolutionary Optimization Algorithms are important tools due to their effectiveness, flexibility and applicability especially for multi-objective problems because they can provide directly the non-dominated set. Among Evolutionary Algorithms, Social Network Optimization (SNO) shows very good optimization performance. In this paper three different approaches for solving a multi- objective problem are tested with SNO: the first one is the weighted sum method, the second is the epsilon-constrained method and the third one is the simultaneous search with a multi- objective implementation of SNO. The analysed application is the design of a sparse-array antenna. I. I NTRODUCTION Evolutionary Optimization algorithms (EAs) are important tools due to their flexibility and applicability: in fact, they can work properly on several type of common benchmark functions that can hardly managed by means of traditional techniques, like multi-modal problems [1], constrained prob- lems [2] and discontinuous problems. In many engineering problems the performance of the sys- tems can be expressed in terms of more than one parameters: when these parameters are conflicting, the problem is called multi-objective [3]. The aim of a multi-objective problem is not just the identification of a single solution, but the identification of the Pareto front. A solution belong to the Pareto front if there is not any other feasible solution that has better values for all the benchmarks [4]. In the field of antenna optimization, Evolutionary Opti- mization Algorithms have been widely applied due to their capability to face multimodal problems [5]. Among EAs, Genetic Algorithm (GA) is the most popular one: it can be easily formulated either for real-coded (like in the design of time-modulated linear arrays), binary-coded (applied for also thinned antennas and wire antenna design) and mixed integers problems (like for linear array design, thinned subarrays, and circularly polarized patch antenna) [6]. Another important EAs is Particle Swarm Optimization (PSO): this algorithm is native for real-coded problems [7] but it has been also implemented and successfully adopted for binary- coded problems [8]. In addition to these algorithms, recently other EAs has been applied to antenna optimization. One of the most interesting is Differential Evolution that shows a very good convergence rate even in very large-scale electromagnetic problems [9]. Another adopted algorithm is the Evolutionary Strategies that, due to its high exploration capability, is able to overcome the problem of local minima [10]. In this field, in [11] a new Evolutionary Optimization algorithm has been introduced. This algorithm, named Social Network Optimization (SNO), takes its inspiration from the interaction of social network members and it has been success- fully applied to several single objective optimization problems, from the design of tubular permanent magnet linear generators [12] to electromagnetic problems [13]. A very recent trend of the application of EAs in electromag- netic is the design of all the optimization environment: this can take into account the use of surrogate models, of several optimization strategies and it can give very good performance in terms of convergence rate and accuracy of the final solution. The System-by-Design approach is one of the most applied in antenna optimization [14], [15]. For what concerns multi- objective optimization problems in antenna design, they are often faced with different EAs, like PSO [16] or GA [17] In this paper the optimization of a planar sparse array is faced using Social Network Optimization. The problem has been formulated with two objectives and it has been faced with three different approaches: the first one is the solution of several scalarized problems, the second one is the epsilon constrained method in which many constrained single objective problem are solved and, finally, the last approach is the contemporaneous search of the entire non-dominated set. The paper is organized as follows: Section II contains a description of SNO and its modification for multi-objective problems. This implementation has been also preliminary tested on three standard multi-objective benchmarks in Section III. In Section IV the two approaches to multi-objective prob- lems used in this paper are described and in Section SNO has been tested on In Section V the optimization antenna problem is described, the objectives defined and the results of the optimization by means of SNO are shown and compared. Finally, in Section VI some conclusions are drawn. 978-1-7281-6929-3/20/$31.00 ©2020 IEEE