A Role of Symmetries in Evaluation of Fundamental Bounds

Abstract: A problem of the erroneous duality gap caused by the presence of symmetries is solved in this paper utilizing point group theory. The optimization problems are first divided into two classes based on their predisposition to suffer from this deficiency. Then, the classical problem of Q-factor minimization is shown in an example where the erroneous duality gap is eliminated by combining solutions from orthogonal sub-spaces. Validity of this treatment is demonstrated in a series of subsequent examples of increasing complexity spanning the wide variety of optimization problems, namely minimum Q-factor, maximum antenna gain, minimum total active reflection coefficient, or maximum radiation efficiency with self-resonant constraint. They involve problems with algebraic and geometric multiplicities of the eigenmodes, and are completed by an example introducing the selective modification of modal currents falling into one of the symmetry-conformal sub-spaces. The entire treatment is accompanied with a discussion of finite numerical precision, and mesh grid imperfections and their influence on the results. Finally, the robust and unified algorithm is proposed and discussed, including advanced topics such as the uniqueness of the optimal solutions, dependence on the number of constraints, or an interpretation of the qualitative difference between the two classes of the optimization problems.