  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.
