Optimization-Based Antenna Miniaturization Using Adaptively Adjusted Penalty Factors

: The continuing trend for miniaturization of electronic devices necessitates size reduction of the comprising components and circuitry. Speciﬁcally, integrated circuit-antenna modules therein require compact radiators in applications such as 5G communications, implantable and on-body devices, or internet of things (IoT). The conﬂict between the demands for compact size and electrical and ﬁeld performance can be mitigated by means of constrained numerical optimization. Evaluation of performance-related constraints requires expensive electromagnetic (EM) analysis of the system at hand; therefore, their explicit handling is inconvenient. A workaround is the penalty function approach where the primary objective (typically, antenna size) is complemented by additional terms quantifying possible constraint violations. The penalty coefﬁcients that determine contributions of these terms are normally adjusted manually, which hinders precise control over antenna performance ﬁgures and often leads to inferior results in terms of achieved miniaturization rates. This paper proposes a novel algorithm featuring an automated adjustment of the penalty factors throughout the optimization process. Our methodology is validated using three broadband antenna structures. The obtained results demonstrate that the presented adaptive adjustment permits a precise control over the constraint violations while leading to better miniaturization rates as compared to manual penalty term setup.


Introduction
Accommodation along with the integration requirements of antennas with the circuit parts has rendered miniaturization a necessity in applications such as wireless communications, internet of things, or portable and on-body devices [1,2]. As the majority of antenna performance figures (reflection, gain, bandwidth, radiation efficiency, radiation pattern) are linked to the physical size [3], a miniaturization task is far from trivial. Miniaturization techniques based on utilization of high-permittivity substrates [3,4], have been successful in reducing the antenna size at the expense of degrading the bandwidth. Several other techniques based on geometrical modifications of the antenna structure have been proposed, including the use of meandered traces [1,5], introduction of corrugations in the ground plane and the radiator [6,7], or incorporation of L-shaped slits [1].
Although the aforementioned techniques provide degrees of freedom to work out a compromise between the antenna size and the electromagnetic (EM) performance, they also contribute to the complexity of the antenna design process, primarily due to an increase in the number of geometry parameters. The unintuitive interrelations between the antenna dimensions and the performance figures virtually eliminate the possibility of finding an optimum design using traditional tuning techniques, especially when multiple objectives are to be taken into account. The only way to produce the optimum design is through simultaneous numerical optimization of all geometrical parameters. Depending on the

Problem Formulation
We will use R(x) to denote the response of the EM simulation model of the antenna structure of interest. Here, x is a vector of the geometry parameters of the structure, which are to be adjusted in the course of the optimization process. The optimization problem at hand is to minimize the antenna size A(x) while ensuring acceptable levels of performance figures pertinent to electrical and field characteristics of the structure, e.g., of the standard form.
s j (x) ≤ S j , j = 1, . . . , k Note that s j (x) are evaluated based on the simulated antenna response R(x). As mentioned before, the penalty function approach provides convenient means of handling the performance constraints, in which case the objective function can be defined as where k is the total number of constraints, c j (x) is the penalty function quantifying violation of the jth constraint (1), with β j being the corresponding penalty factor. The penalty functions only contribute to (2) if the corresponding constraints are violated, i.e., s j (x) > S j . A typical definition of the penalty function would be the one measuring a relative violation of the constraint, i.e., where γ j is the constraint violation defined as The design task is defined as a nonlinear minimization problem of the form.
where X is the parameter space, typically determined by the lower and upper bounds for antenna geometry parameters x. It should be noted that without the penalty function approach, the design task would be subject to additional constraints (1). Whereas, when using (1)-(3), it becomes an unconstrained problem, apart from the aforementioned box constraints.

Trust-Region Gradient-Based Algorithm
The standard trust-region-based algorithm [33] is employed in this work as the core optimization engine. The procedure approximates x * with a series x (i) , i = 0, 1, . . . , obtained by solving In (6), L (i) (x) is a linear (first-order Taylor) approximation of R(x (i) ). The candidate solution rendered by (6) is only accepted if U A (R(x (i+1) )) < U A (R(x (i) )). The trust region radius δ is adaptively adjusted based on a gain ratio calculated as The algorithm is terminated when the trust region radius is diminished below a certain user-defined limit δ 1 , or convergence in the argument is achieved, i.e., ||x (i+1) − x (i) || ≤ δ x .

Size Reduction with Adaptive Penalty Coefficients
Formulation (1) of the objective function offers an efficient way of handling the design constraints, especially those that are expensive to evaluate (i.e., require EM analysis of the antenna). Notwithstanding, an appropriate setup of penalty coefficients is a non-trivial matter and may affect the performance of the optimization process as elaborated on in Section 1.
This work proposes a novel algorithm incorporating adaptive adjustment of penalty coefficients in the course of the optimization run. The presented procedure eliminates the need for trial-and-error objective function setup as well as leads to improved antenna miniaturization rates, as demonstrated in Section 4. In the following, we discuss the penalty factor adjustment concept (Section 3.1), the adjustment procedure (Section 3.2), as well as the complete optimization algorithm (Section 3.3).
This section recalls the formulation of EM-driven antenna miniaturization as a constrained numerical.

Adaptively Adjusted Penalty Factors. The Concept
As mentioned before, an appropriate setup of penalty coefficients is of utmost importance for the performance of the optimization process. On the one hand, having the coefficients set at too low values will result in excessive violations of the constraints. On the other hand, the values that are too high make the optimization problem numerically challenging, especially in the context of size reduction, which requires exploration of the feasible region boundary.
The objective of the technique presented in this is to automate the process of setting up the penalty coefficients so that their specific values are based on currently detected constraint violations. We use the following prerequisites, which involve the current constraint violations as well as their possible improvements over the last consecutive iterations:

•
If the parameter vector x (i+1) produced at the iteration i is feasible from the point of view of the jth constraint, the corresponding penalty coefficient β j may be reduced; • If x (i+1) is infeasible but the violation of the jth constraint was reduced to a sufficient extent w.r.t. the (i-1)th iteration, the coefficient β j remains intact; • If x (i+1) is infeasible and there was no improvement of the jth constraint violation or the improvement was insufficient, the coefficient β j should be increased.
The above can be viewed as a set of rules (applied to each and every constraint), which are simple, yet allow us to relax the 'pressure' from the penalty terms when the algorithm operates in a feasible region and increase it if the discussed indicators show the lack of improvement in terms of reducing constraint violations. Furthermore, implementing these rules facilitates exploration of the feasible region boundary, which is where the minimum-size design is likely to reside. The notion of the aforementioned sufficient constraint violation improvement will be specified and discussed at length below, along with providing a rigorous formulation of the adjustment rules. To clarify the matter, Figure 1, conceptually illustrates the possible situations and the actions performed.

Adaptively Adjusted Penalty Factors. The Procedure
In order to formalize the set of penalty factor adjustment rules considered in Section 3.1, we need to quantify the sufficient constraint violation improvement, which will be defined, for the jth constraint, as In (8), γj is the absolute violation, cf. (4), whereas 0 < M < 1 is the improvement factor. We will elaborate on the selection of the value of M later in this section. Both the violation and its improvement are considered in relation to the new parameter vector x (i+1) produced in the ith iteration and the previous point x (i) , therefore, we will use the superscript i + 1 to specify which iteration the above quantities are referring to. In particular, we have γj i+1 and Δj i+1 as the constraint violation at x (i+1) and the improvement from x (i) to x (i+1) . Similarly, the penalty coefficient for the jth constraint at iteration i will be denoted as βj i .
We are now in a position to formulate the adjustment rules in a rigorous manner. These are described using the following pseudocode: The multiplication factors mdecr and mincr determine the amount of penalty factor modifications. In our numerical experiments, they are set to mdecr = 1.25 and mincr = 5, but their values are not critical. As discussed in Section 3.1, maintaining fixed penalty coefficients under sufficient constraint violation improvement (while the design is still infeasible) allows us to maintain stability of the optimization process, i.e., the values of βj do not bounce back and forth (i.e., are immediately increased or decreased after the design crosses the

Adaptively Adjusted Penalty Factors. The Procedure
In order to formalize the set of penalty factor adjustment rules considered in Section 3.1, we need to quantify the sufficient constraint violation improvement, which will be defined, for the jth constraint, as In (8), γ j is the absolute violation, cf. (4), whereas 0 < M < 1 is the improvement factor. We will elaborate on the selection of the value of M later in this section. Both the violation and its improvement are considered in relation to the new parameter vector x (i+1) produced in the ith iteration and the previous point x (i) , therefore, we will use the superscript i + 1 to specify which iteration the above quantities are referring to. In particular, we have γ j i+1 and ∆ j i+1 as the constraint violation at x (i+1) and the improvement from Similarly, the penalty coefficient for the jth constraint at iteration i will be denoted as β j i . We are now in a position to formulate the adjustment rules in a rigorous manner. These are described using the following pseudocode:

end end
The multiplication factors m decr and m incr determine the amount of penalty factor modifications. In our numerical experiments, they are set to m decr = 1.25 and m incr = 5, but their values are not critical. As discussed in Section 3.1, maintaining fixed penalty coefficients under sufficient constraint violation improvement (while the design is still infeasible) allows us to maintain stability of the optimization process, i.e., the values of β j do not bounce back and forth (i.e., are immediately increased or decreased after the design crosses the feasible region boundary). This facilitates exploration of the boundary and leads to improved size reduction ratios.
Let us now consider the improvement factor M introduced in (8). Assuming that the parameter vector x (i) resides in the infeasible region, and sufficient constraint violation improvement is observed for a few consecutive iterations, say, from i to i + k, the upper bound on the constraint violation at the iteration i + k can be calculated as As M < 1, the constraint violation is reduced at a geometric rate, and the improvement is faster when M is closer to zero. At the same time, satisfaction of the sufficient improvement condition is more demanding for lower values of M. For M = 0.5, as selected in this work, we get a balance between the rate of approaching the feasible region boundary (e.g., the constraint violation is reduced to only about six percent of its original values after only four iterations), and the difficulty of satisfying the improvement condition (reduction of only half of the current violation has to be achieved per iteration).

Optimization Framework
This section provides a summary of the operation of the trust-region gradient-based algorithm incorporating the adaptive adjustment of penalty coefficients as presented in Section 3.2. Here is a summary of the control parameters of the algorithm: • δ x , δ TR -termination thresholds (cf. Section 2.1); • m inc , m dec -increase and decrease factors for the automated adjustment of penalty coefficients (cf. Section 3.2); • M-a factor used to determine sufficient constraint violation improvement (cf. Section 3.2); • β j max , β j min -maximum and minimum values of penalty coefficients; j = 1, . . . , k; • β j 0 -initial values of the penalty coefficients; j = 1, . . . , k; The algorithm operation has been presented in Figure 2. Additional clarification is provided in the form of a flow diagram in Figure 3. In the algorithm, Steps 1 and 2 are used to initialize the optimization procedure. In Steps 3 and 4, the antenna response and its sensitivity matrix are evaluated using EM analysis. The linear approximation model L (i) (x) of the antenna responses is constructed in Step 5, along with its corresponding objective function U A (L (i) (x)) (Step 6). The candidate design is produced in Step 7 by minimizing U A (L (i) (x)). It is validated in Steps 8 and 9, where the gain ratio is calculated for the purpose of either accepting or rejecting x (i+1) and computing constraint violation improvements (Step 10). The latter is then used in Step 11 to update the penalty coefficients. The termination condition for the procedure is convergence in argument or reducing the trust region size beyond the user-defined threshold (Step 12). 2. Set minc, mdec, βj i , Sj, βj max , βj min , j = 1, …, k; 3. Evaluate antenna response R(x (i) ); 4. Evaluate antenna sensitivities J(x (i) ) using finite differentiation; 8. Evaluate antenna response R(x (i) ); 9. Compute gain ratio ρA and update TR size   Initial design Evaluate sensitivity matrix J(x (i) ) and construct linear model Construct objective function (2) Solve Equation (6) Evaluate antenna response R(x (i+1) ) EM Solver ρ > 0?

Yes
Compute gain ratio using equation (7) and update TR size

Verification Case Studies
This section provides the results of numerical experiments conducted to validate the proposed procedure for adaptive adjustment of penalty coefficients, introduced in Section 3. The verification studies involve three broadband antennas optimized for minimum size with the maximum acceptable in-band reflection considered as the only constraint. The results are compared with those obtained for manually set penalty terms, ranging from the relaxed to very tight constraint satisfaction conditions. All of the considered antenna structures have been previously described in the literature and experimentally validated therein [36][37][38]. Consequently, no experimental results are provided here.
The remaining part of this section is organized as follows. The experimental setup is described in Section 4.1, whereas the benchmark antennas are described along with the numerical results in Section 4.2, Section 4.3, and Section 4.4, respectively. Section 4.5 provides a detailed discussion.

Verification Case Studies
This section provides the results of numerical experiments conducted to validate the proposed procedure for adaptive adjustment of penalty coefficients, introduced in Section 3. The verification studies involve three broadband antennas optimized for minimum size with the maximum acceptable in-band reflection considered as the only constraint. The results are compared with those obtained for manually set penalty terms, ranging from the relaxed to very tight constraint satisfaction conditions. All of the considered antenna structures have been previously described in the literature and experimentally validated therein [36][37][38]. Consequently, no experimental results are provided here.
The remaining part of this section is organized as follows. The experimental setup is described in Section 4.1, whereas the benchmark antennas are described along with the numerical results in Section 4.2, Section 4.3, and Section 4.4, respectively. Section 4.5 provides a detailed discussion.

Experimental Setup
The objective is to optimize the considered antennas for minimum size, as determined by the substrate area A(x). The optimization process is subject to a single constraint imposed on the antenna reflection coefficient |S 11 More specifically, we have s 1 (x) ≤ −10, where s 1 (x) stands for the maximum value of |S 11 (x)| within the operating frequency range of the antennas, here, 3.1 GHz to 10.6 GHz. Correspondingly, a single penalty coefficient is used, denoted as β 1 . The penalty function c is defined based on a relative violation of the constraint, cf. (3).
The performance of the algorithm is evaluated statistically through multiple runs initiated from random starting points. This allows for reducing a possible bias associated with a particular choice of the initial design. The performance figures include the antenna size averaged over ten independent runs, and the average value of constraint violation, both evaluated at the final design yielded by the optimization algorithm.
The proposed algorithm is compared to the standard TR algorithm executed for different (fixed) values of the penalty coefficients β = 10 q , q = 2, 3, 4, 5, 6. The values set for the control parameters are β j min = 100, β j max = 10 6 , and β j 0 = 100. Other parameters are set as in Section 3.2. The termination thresholds are set to δ x = δ TR = 10 −3 . Figure 4 shows the geometry of the first benchmark structure (Antenna I) [36]. It is a broadband monopole antenna operating within the UWB band   Table 1 provides the data including the average footprint area, average constraint violation, as well as the standard deviations thereof; the latter is used to quantify the repeatability of solutions. Figure 5a illustrates the reflection coefficient of Antenna I at the initial and final designs obtained for a selected run of the proposed optimization algorithm, whereas Figure 5b shows the evolution of the penalty factor throughout the optimization process.

Antenna II
The second verification case is a broadband rectangular-slot monopole antenna (Antenna II), shown in Figure 6

Antenna II
The second verification case is a broadband rectangular-slot monopole antenna (Antenna II), shown in Figure 6 [37], operating in the UWB range. The design parameters of the antenna are x = [L g L 0 L s W s d dL d s dW s dW a b] T (all in mm). The feed line width is W 0 = 3 mm. The structure is fabricated on FR4 substrate (ε r = 4.3, h = 1.55 mm). The computational model incorporates the SMA connector, and it is simulated using the time-domain solver of CST Microwave Studio. Table 2 gathers the numerical results. Figure 7 shows the antenna responses and evolution of the penalty factor for a representative run of the proposed algorithm. = 3 mm. The structure is fabricated on FR4 substrate (εr = 4.3, h = 1.55 mm). The computational model incorporates the SMA connector, and it is simulated using the time-domain solver of CST Microwave Studio. Table 2 gathers the numerical results. Figure 7 shows the antenna responses and evolution of the penalty factor for a representative run of the proposed algorithm.    1 Average miniaturized antenna area for ten algorithm runs. 2 Standard deviation of the miniaturized antenna area for ten algorithm runs. 3 Average constraint violation for ten algorithm runs. 4 Standard deviation of the constraint violation γ, for ten algorithm runs.     1 Average miniaturized antenna area for ten algorithm runs. 2 Standard deviation of the miniaturized antenna area for ten algorithm runs. 3 Average constraint violation for ten algorithm runs. 4 Standard deviation of the constraint violation γ, for ten algorithm runs.  Figure 8 shows Antenna III, the third verification structure. It is also a UWB monopole antenna [38]. The design parameters are x = [L0 g a l1 l2 w1 o] T , (all dimensions in mm).  Figure 8 shows Antenna III, the third verification structure. It is also a UWB monopole antenna [38]. The design parameters are x = [L 0 g a l 1 l 2 w 1 o] T , (all dimensions in mm). We also have w 0 = 2o + a, and w r = 1.7 mm. The antenna is implemented on FR4 substrate (ε r = 4.3, h = 7.62 mm). The computational model incorporates the SMA connector, and it is simulated in the time-domain solver of CST Microwave Studio.  Figure 8 shows Antenna III, the third verification structure. It is also a UWB monopole antenna [38]. The design parameters are x = [L0 g a l1 l2 w1 o] T , (all dimensions in mm). We also have w0 = 2o + a, and wr = 1.7 mm. The antenna is implemented on FR4 substrate (εr = 4.3, h = 7.62 mm). The computational model incorporates the SMA connector, and it is simulated in the time-domain solver of CST Microwave Studio. Table 3 shows the numerical results, arranged the same way as for Antennas I and II. Antenna responses and evolution of the penalty coefficient for a selected algorithm run have been shown in Figure 9.   Table 3 shows the numerical results, arranged the same way as for Antennas I and II. Antenna responses and evolution of the penalty coefficient for a selected algorithm run have been shown in Figure 9.    1 Average miniaturized antenna area for ten algorithm runs. 2 Standard deviation of the miniaturized antenna area for ten algorithm runs. 3 Average constraint violation for ten algorithm runs. 4 Standard deviation of the constraint violation γ, for ten algorithm runs.

Discussion
The analysis of results provided in Section 4 allows for several conclusions regarding the importance of the automated adjustment of the penalty factors, as well as the performance-wise advantages of the proposed adaptive algorithm over the fixed-setup approach. To facilitate the interpretation, a graphical illustration of the data from Tables 1-3 has been provided in Figure 10. Therein, the average antenna footprint area along with the average constraint violation is shown versus the penalty coefficient for fixed-setup optimization runs. The horizontal lines represent the antenna area and constraint violation obtained for the automated adjustment procedure. The range of these lines is representative of the span in which the penalty coefficient varies throughout the optimization iterations. As it can be seen, not only the optimum value of the penalty coefficient, but also the span is problem-dependent and may not even reach the maximum set value in some cases. The following observations can be formulated:

•
Although the optimum value of penalty coefficient in the fixed-setup optimization seems to be about β = 10 4 for Antenna I, between β = 10 3 and β = 10 4 for Antenna II, and between β = 10 4 and β = 10 5 for Antenna III, considering the achievable miniaturization rates along with sufficient constraint satisfaction, the optimum value of penalty coefficient is problem-dependent. The optimum values should be identified for particular iterations of the optimization process and they are generally dependent on the status of constraint violation.

•
In both fixed and automated adjustment setups, using a penalty coefficient lower than the optimum value, results in significant constraint violation. As for the former, the violation can easily become as high as five times of the tolerance threshold or even more. Antennas I and II are representative examples of this design quality degradation. • Automated adjustment of penalty coefficients seeks to improve the final design quality by the optimum value of penalty coefficients at the level of iterations of the optimization process. This, in turn, permits a better control of constraint violations along with better achievable miniaturization rates.

•
The performance improvements are significant. For the corresponding levels of constraint violations, the procedure proposed in this work leads to antenna footprints that are smaller by 110, 44, and 13 mm 2 for Antenna I, II, and III, respectively.
• Automated adjustment of penalty coefficients seeks to improve the final design quality by the optimum value of penalty coefficients at the level of iterations of the optimization process. This, in turn, permits a better control of constraint violations along with better achievable miniaturization rates.

•
The performance improvements are significant. For the corresponding levels of constraint violations, the procedure proposed in this work leads to antenna footprints that are smaller by 110, 44, and 13 mm 2 for Antenna I, II, and III, respectively.

Conclusions
This paper proposed a novel algorithm for optimization-based antenna miniaturization using local trust region gradient search routines. Our methodology incorporates the frameworks employing a penalty function approach for handling design constraints in an implicit manner. Therein, the appropriate adjustment of the objective function, specifically the penalty terms, has a strong correlation with the reliability as well as the efficacy of the optimization process, both in terms of the achievable miniaturization rate and constraint satisfaction. The proposed methodology effectively eliminates the need for trial-and-error efforts by automated adaptive adjustment of the penalty coefficients. The latter is based on a sufficiency of constraint violation improvement between consecutive iterations throughout the entire optimization process. The optimum values of penalty coefficients are identified for particular iterations and consequently enabling the optimization process to realize a precise control over the constraint violations while leading to overall better results in terms of the achievable miniaturization rates as compared to the fixed adjustment setup.
The proposed procedure has been validated using three broadband antenna structures optimized for minimum size with constraints imposed on their maximum in-band reflection coefficient. Benchmarking the against the fixed penalty coefficient setups indicates superiority of automated adaptive adjustment setup in terms of achieving smaller size with a precise control over the constraint violations. The future work will involve a more comprehensive validation of the methodology in applications featuring multiple constraints, e.g., circularly polarized antennas, high-gain antennas, etc.