The typical formulation of a constrained optimization problem (COP), which has the constraint form of 𝐠(𝐱)≤0 , has widely used in evolutionary computation field. However, it is not suitable for Gaussian processes (GPs) based expensive optimization. In this paper, a more general and suitable formulation of the COP, which has the constraint form of 𝐥_𝐠≤𝐠(𝐱)≤𝐮_𝐠 , is recommended for the expensive optimization. Modeling a real world COP as a typical formulation will probably introduce additional constraints and dependencies among the objective and constraints while that as a suitable one introduces none additional. In the case of typical formulation, the additional constraints and dependencies have to be handled and the handling would cost additional computational resource, especially, the additional dependencies would lead to degenerating the performance of expensive optimization technologies since most of the technologies are based on the assumption of mutual independency among the objective and constraints. Experiments show that the performance of the expensive optimization technologies in the aspect of precision on solving the problems with suitable formulation is better than that with typical one. However, we could not verify the expense of additional computational resource since there are few expensive optimization technologies dealing with dependent objective and constraints.

This paper presents a new crooked-wire antenna. The antenna design is first modeled as a constrained optimization problem (COP). Then a differential evolution solves the COP. Consequently, an antenna is designed. The main difference of the antenna design in this paper from traditional evolutionary antenna is that the objective in the COP can potentially enhance the robustness of the antenna and widen the frequency band without additional computational cost. Actually, the objective is the sum of variance of the gain, axial ratio and VSWR over the frequency band. This paper chooses NASA LADEE satellite antenna design as an example to verify the method of this paper. The performance of the evolved antenna meets the requirements. Furthermore, the VSWR of the antenna is less than 1.5 over the required frequency range, much less than the requirement 2. The axis ratio is 2.06 dB at the beam direction which is much better than that of the antenna designed by the referred literature.

The Primal-dual interior-point methods with a filter are one of hot issues in optimization with both equality and inequality constraints. Extensive attention has been paid and great progress has been made for a long time. Interior-point methods not only have polynomial complexity, but are also highly efficient in practice. In this paper, we first generalize the dual interior filter algorithm for referred paper. Then a dual interior filter algorithm based on the orthogonal design is proposed. The orthogonal design is used to evenly sample the solution space, then the points with small constraint violations are chosen as the initial points of the algorithm. The dual interior point filter algorithm based on orthogonal design (DIPFA-OD) choosing the initial points is compared with the dual interior point filter algorithm based on random initial points (DIPFA-RND). The experimental results show that DIPFA-OD has a faster convergence speed and obtains better objective values than DIPFA-RND.

In this paper, an antenna with a miniature structure and wide-band is presented. We designed a two-arm conical spiral antenna according to the structure features of the Archimedes spiral and the conical helical antenna, and proposed an exponential asymptote balun to match the impedance. Unlike traditional antenna designs which optimize antenna and matching module separately, we adopted the differential evolution (DE) algorithm to optimize both the antenna and balun simultaneously. In addition, the peak radiation direction of the antenna was added as a constraint when evolving the antenna, which is usually ignored in normal evolutionary antenna designs. Simulation results indicate that the evolved antenna can basically fulfills the requirements. And the evolved antenna with the additional constraint has smaller deviation angle between the peak radiation direction and the antenna’s axis than that without the constraint.