Some real-life optimization problems, apart from dependence on the combination of state variables, also show dependence on the complexity of the model describing the problem. Changing model complexity implies changing the number of decision space dimensions. A new method called Particle Swarm Optimization for Variable Number of Dimensions is developed here. The well-known particle swarm optimization procedure is modified to handle spaces with a variable number of dimensions within a single run. Some well-known benchmark problems are modified to depend on the number of dimensions. Novel performance metrics are defined in the article to evaluate convergence properties of the method. Some recommendations for setting the optimization are made according to results of the method on the proposed benchmark test suite. The method is compared with conventional swarm strategies able to solve problems with variable number of dimensions.

A memetic framework for optimal inverse design is proposed by combining a local gradient-based procedure and a robust global scheme. The procedure is based on method-of-moment (MoM) matrices and does not demand full inversion of a system matrix. Fundamental bounds are evaluated for all optimized metrics in the same manner, providing natural stopping criteria and quality measures for realized devices. Compared to density-based topology optimization, the proposed routine does not require filtering or thresholding. Compared to commonly used heuristics, the technique is significantly faster, still preserving a high level of versatility and robustness. This is a two-part article in which the first part is devoted to the theoretical background and properties, and the second part applies the method to examples of varying complexity.

The decoupling of modern printed circuit boards introduces a very complex task. Powerful stochastic optimizers are usually used to determine values and positions of decoupling capacitors on the board. The number of capacitors used has to be determined a priori by the user which brings problems with convergence of the optimization process or can lead to a waste of resources when the noises are to be attenuated to a certain level. In this paper, an approach based on the combination of time-domain contour integral method and optimization with variable number of dimensions is introduced. The optimizer works with models having variable dimensions and searches for the optimal one. The approach is tested on two example power circuit boards with various noise attenuation limits and constraints on capacitor positions and values.

Optimal inverse design, including topology optimization and evaluation of fundamental bounds on performance, which was introduced in Part 1, is applied to various antenna design problems. A memetic scheme for topology optimization combines local and global techniques to accelerate convergence and maintain robustness. Method-of-moments (MoMs) matrices are used to evaluate objective functions and allow determination of fundamental bounds on performance. By applying the Shermann–Morrison–Woodbury identity, the repetitively performed structural update is inversion-free yet full-wave. The technique can easily be combined with additional features often required in practice, e.g., only a part of the structure is controllable, or evaluation of an objective function is required in a subdomain only. The memetic framework supports multifrequency and multiport optimization and offers many other advantages, such as an actual shape being known at every moment of the optimization. The performance of the method is assessed, including its convergence and computational cost.

The article proposes a multi-objective optimization method called Generalized Differential Evolution (GDE3) for a Variable Number of Dimensions (VND). The well-known generalized differential evolution is adapted to handle problems where the number of decision space variables is not a priori known. The performance of the method is assessed on a set of benchmark problems based on standard multi-objective test cases modified to depend on the number of decision space variables. Experimental results include a study of the setting of controlling parameters and a comparison with the state-of-the-art Variable-Length GDE3 (VLGDE3) algorithm. Finally, the novel approach is compared to the standard approach in a linear antenna array design problem.

In this paper, linearly and circularly polarized single-layer Substrate Integrated Waveguide (SIW) horn antennas with nearly equal half power beamwidths in principal planes for the 24 GHz ISM band are designed. The uniqueness of the solution lies in the application of a pixelated structure to the design of these SIW horn antennas. For the design, a novel heuristic algorithm called Binary Ink Stamp Optimization is proposed. At first, the ability of the proposed algorithm is proved over fifteen benchmark functions and compared with the results of six state-of-the-art optimization algorithms. Then, the novel algorithm is exploited for the design of the pixelated part of the antennas. The antennas are manufactured using low-cost 3D printing technology. The experimental results prove that the linearly polarized antenna has a reflection coefficient below -18 dB in the whole 24 GHz ISM band. At the center frequency of this band, the antenna gain is 14.2 dBi, and the level of side lobes is below -15 dB in E and H-planes. In addition, E and H-plane half power beamwidths are 26.5° and 25°, respectively. The circularly polarized antenna radiates a right-hand circularly polarized (RHCP) wave, the reflection coefficient, and axial ratio are below -12 dB and 3 dB in the whole 24 GHz ISM band, respectively. At the center frequency of this band, the antenna gain is 8.9 dBi and the level of side lobes is below -7.4 dB in both principal planes. Further, the half power beamwidths in principal planes are 30.6° and 33.4°, respectively.