Comparison of algorithms for multi-objective optimization of radio technical device characteristics

Objectives. The selection of a method for solving multi-objective optimization problems has many practical applications in diverse fields. The present work compares the results of applying different methods to the selected classes of problems by solution quality, time consumption, and various other criteria


INTRODUCTION
Multi-objective optimization (MOO) problems are of considerable interest in radio engineering and other fields of research [1]. Multicriteria optimization is aimed at finding an approximation of the set of Pareto-optimal solutions (POS) [2], which cannot be improved by one of the quality indices (QI) without deteriorating at least one of the other QIs.
Methods for solving MOO problems can be divided into two main classes [3,4]. The first is based on solving the problem of minimizing the scalar objective function (OF) written in the generalized form [5]: Alexander V. Smirnov Comparison of algorithms for multi-objective optimization of radio technical device characteristics where Q j (x) and Q jt are the current and target values of the jth QI; W j is the weighting factor of the jth QI.
The exponent q j = 1 provided that normalizing the QI deviation from the target value for bringing the summands in (1) to the same range of values is required. The exponent q k = 0 provided that such normalization is not required. Equality (1) covers different scalarization techniques for MOO problem. Taking all Q jt = 0 and considering the denominators equal to one, the problem of minimizing the QI weighted sum arises. Another technique is to set target values Q jt , as well as W k = 1 and W j >> 1, where j = 1, ..., M, j ≠ k. In this case, the problem of minimizing QI Q k while fixing other QI near the target values arises. Methods based on scalarization allow one POS to be found in a search cycle.
The methods grouped in the second class (MOO population-based algorithms) permit the computation of several elements of approximation of the POS set in a single search cycle using agent population. The classification and description of algorithms of this type are given in [3,4].
The sets of test functions [6] and quality criteria for approximating the POS set [3] are used to compare different methods for solving the MOO problem. Generally, the results of applying MOO populationbased algorithms are evaluated using test functions, e.g., as in [7,8]. Here, no comparison with the results of scalarization-based methods is performed. This can be justified in the case of test functions whose properties are all known; the quality of found approximations of the POS set can be evaluated objectively. In terms of practical applications, however, where nothing generally is known in advance about functions describing QI, it becomes necessary to solve the "Black Box Optimization" problem for collecting information on QI values while finding.
A determination of which algorithm would give the best approximation may be achieved only by comparing the results of different methods, including both population-based and scalarization-based approaches. Such comparison for a specific class of problems is performed in [9] to support a conclusion about the superiority of population-based algorithms in terms of the quality of obtained solutions. Nevertheless, a different result may be obtained for other types of problems. In [10], the possibility of applying populationbased algorithms from the PlatEMO open source [11] running in the MATLAB environment to solve the MOO problem of the frequency response of analog electrical filters is investigated. Here, the authors conclude that, in the case of optimization by two QIs, population-based methods provide better solutions than scalarizationbased approaches. While, in the case of optimization by three QIs, the opposite result is obtained, this conclusion is reached when not using the most effective algorithm for finding an extremum of the scalar OF in experiments. Different scalarization-based algorithms are compared in [12], albeit without considering population-based approaches.
The present work aims to compare the results of applying MOO algorithms of different classes on examples of several problems of optimizing the characteristics of radio engineering devices. Among the indicators characterizing the compared methods, the highest priority is given to obtaining the best results. In case these criteria are equal, a comparison may be performed in terms of search duration per one POS along with other indices. Problem 1. MOO of frequency response of analog filters. The QI definitions and methods for calculating them are given in [13] and other works by the author. Below is a list of QIs: • uneven gain-frequency response in passband DHp, dB; • minimum attenuation in stopband Hs, dB; • exceeding gain-frequency response of a given level in transition band DHt; • uneven delay-frequency response in passband DTd, %. Frequency response is calculated on a dimensionless frequency scale normalized to the upper bound frequency of the low pass filter (LPF) passband. Here, it is necessary to minimize DHp and DTd together while fulfilling the constraints DHt ≤ 0, Hs ≥ Hst. Problem 2. MOO of simultaneous frequency and time responses of analog filters. The QI definitions and methods for calculating them are given in [13]. In addition to the frequency domain QI mentioned above, the following time domain QIs are introduced: • maximum voltage (overshoot) of the transient process Um normalized to the steady-state value; • transient rising (front) duration Tfr; • transient-process duration Tss.
The last two QIs are calculated on a dimensionless time scale referenced to the normalized frequency scale. Here, it is necessary to jointly minimize Tss and maximize Hs under the constraints DHt ≤ 0, Um ≤ Umt, and Tfr ≤ Tfrt.
Problem 3. MOO of frequency responses of digital filters. The QI definitions and methods for calculating them are given in [5]. The QI list and formulation of optimization problem is the same as for Problem 1. The method for calculating these QIs is not given here due to the limited scope of the paper and will be published separately. It is necessary to jointly minimize DKP and maximize KPmin at a given value of the matching bandwidth DFM.
Problem 5. Another MOO problem of frequency response of matching multistep microwave transformers. In addition to QI defined for Problem 4, the relative unevenness of the delay-frequency response in matching band DTd, %, is introduced. It is necessary to minimize DKP and DTd at given values of matching bandwidth DFM and maximum power transfer coefficient KPt.
We shall now consider the algorithms used for solving the above problems.
The population-based algorithm is implemented using the PlatEMO library mentioned above. The algorithm GDE3 (The Third Evolution Step of Generalized Differential Evolution) showing the best response values according to [10] is selected from 71 algorithms presented in the library. The specified parameters are the size of population Npop and the number of calculations of the QI set Neval. These values are found experimentally for each problem. For this purpose, search iterations with increasing Npop and Neval values were performed until the found approximation of the POS set is improved.
Below are the algorithms for finding extrema of scalar OFs.
Multistart Pattern Search (MSPS), which describes the repeated start of stepwise search, is implemented by SOFTD [13] in Problems 1 and 2 and by HODF in Problem 3 [5]. Both programs are written in C++. In Problems 4 and 5, the algorithm is implemented in the MATLAB environment.
Particle Swarm Optimization (PSO) is the particle swarm algorithm [3,4] implemented in MATLAB by the particleswarm(..) function from the Global Optimization Toolbox module. According to the results presented in [12], this algorithm demonstrates its capacity to find global extrema of scalar OFs with complex landscape.
Multistart Sequential Quadratic Programming (MSSQP) is the repeated start of sequential quadratic programming algorithm implemented in MATLAB by the fmincon(..) function from Optimization Toolbox module. Unlike MSPS and PSO algorithms searching for minima of scalar OF of the type (1), the constraints on QI are not considered in MSSQP as penalty terms but are included in the Lagrange function that may be written in the following form: where λ i are Lagrange multipliers; Q j are quality indices; Q jt are their target values.
In the strict sense, the algorithm is not intended for finding global extrema of OFs having a complex landscape containing many local extrema. However, as shown in [12], it obtains a good approximation to the global minimum of OF from most starting points evenly distributed in the search space for some problems. At the same time, the search duration turns out to be much shorter than for other algorithms.
The number of starts, NT, of the scalarizationbased algorithms for finding one POS is selected in each problem so as to find a solution that could not be significantly improved by further NT increasing. Other parameters are set equal to default values.

PROBLEM SOLUTIONS
Problem 1. We shall consider experimental results for the analog LPF whose transfer function (TF) contains NP = 6 poles and NZ = 0 zeros. The lower bound frequency of the stopband on the frequency scale normalized to the upper bound frequency of the passband is Fs = 2. The results obtained using the GDE3 algorithm are shown in Fig. 1 in the form of Pareto front approximation graphs for Hst = 30 dB and Hst = 40 dB. In the first case, Neval = 1 · 10 6 , while in the second case, Neval = 0.5 · 10 6 . The search duration is 4 and 2 min, respectively. In both cases, Npop = 50.
The solutions obtained using algorithms based on scalarizing OFs are also presented in Fig. 1. Finding one solution takes an average of 2 min by MSPS algorithm, 10 s by MSSQP algorithm, and 5 min when using the PSO algorithm. The number of MSPS starts is 3000, while the number of PSO starts is 40; the number of MSSQP starts is 20. From the start results, the best one is selected. Here, it should be noted that about half of MSSQP starts gives the same best result when searching for each POS, with the remainder resulting in unacceptable solution. The results of the other two algorithms are distributed over a wide range of values in most experiments.
The comparison of QI solutions obtained by different algorithms shows that only MSSQP provides benefits for solution quality as compared to GDE3 at DHp small values. The other two algorithms based on scalarizing OFs show at best results similar to GDE3, being at a disadvantage to it at small values of DHp. In addition, they require significantly more time for finding solutions. Comparison of algorithms for multi-objective optimization of radio technical device characteristics Problem 2. We shall consider experimental results for the same LPF as in Problem 1. The results obtained using the compared algorithms are shown in Fig. 2. The constraints are set to Tfr ≤ 0.5, Um ≤ 1.1, and DHt ≤ 0 for the first series of experiments, while the constraint on Tfr is excluded in the second series.
First, it should be noted that the MSSQP algorithm cannot find admissible solutions, i.e., those satisfying all the restrictions, under the constraint on Tfr at all. In the absence of this constraint, the ability of MSSQP to find valid solutions appeared to be noticeably worse than that for GDE3 and PSO. The Pareto front turns out to be discontinuous in the presence of a constraint on Tfr. To explain the reasons for this effect, the Tfr values obtained with no constraint imposed on this QI are plotted on the graph also shown in Fig. 2. As long as Tfr is less than constraint Tfr = 0.5, approximations of Pareto fronts obtained in the presence and absence of this constraint coincide. The constraint on Tfr is not active and does not affect search results. However, if Tfr value should exceed the specified threshold, the constraint becomes active, and the transition process becomes oscillatory in order to fulfill it. In this case, the Pareto front discontinuity is due to transitions of the moment when the condition of the transient process completion [5] from one wave to another is satisfied.  6 and Neval = 1.5 · 10 6 , the duration is 6 and 9 min, respectively.
Next, points of these approximations are obtained using algorithms based on scalarizing OFs. The MSPS algorithm gives the best results, since the solutions found by it cover the ranges of solutions obtained using GDE3 completely, not only equalling, but even slightly exceeding them in QI terms. In case of Hst = 30 dB, the set of MSPS solutions has the least low bound in terms of DHp parameters. In all experiments, the number of search starts is NT = 2000. The execution time per search ranges from 5 to 8.5 min.
The PSO algorithm for Problem 3 turns out to be worse than MSPS; here, the solutions found are within narrower ranges of QI values taking approximately the same time required for one search.
The MSSQP algorithm also loses to GDE3 and MSPS by the value of the lower bound of the DHp range. However, at HSt = 30 dB, the algorithm finds solutions within the DHp value range of 0.2-0.8 dB with lower DTd values compared to other algorithms. These solutions are located in the area of the search space into which other algorithms have not fallen. At the same time, the phase-frequency response of the solutions obtained using MSSQP differ from the phase-frequency response of the other algorithm solutions (Fig. 4) significantly, although the gain-frequency response is similar. At the same time, MSSQP results are close to those obtained by other methods at Hst = 40 dB.
Problem 4. We shall consider an example of the problem solution at matching the lines with the ratio of wave impedances Z w2 /Z w1 = 10, frequency matching bandwidth DFM = 1.2, and the number of transformer stages Nst = 4 and Nst = 5 (Fig. 5).
The search by GDE3 algorithm for both values of Nst is performed at Npop = 100 and Neval = 1 · 10 6 , taking approximately 2 minutes. Since the results only insignificantly deteriorate with the number of QI calculations, the search duration is decreased by 10 times. All three algorithms based on scalarizing OFs show the same results. Therefore, only the solutions obtained with MSSQP falling on Pareto front approximations found by GDE3 algorithm are shown in Fig. 5. However, the methods differ significantly in terms of search duration for a single solution. While MSPS and PSO require 40-50 s, MSSQP requires only 3-4 s. It should be noted that the rightmost points of the series obtained using MSSQP coincide with the results for Chebyshev approximations, while the points of GDE3 series located to their right are not POS.
Problem 5. The problem is solved under conditions Z w2 /Z w1 = 12, DFM = 1, and KPt = 1 for the number of stages Nst = 3, 4, and 5. Approximations of Pareto fronts obtained by the compared algorithms are shown in Fig. 6.
For GDE3 algorithm, parameters Npop = 100 and Neval = 1·10 5 are set. No further increase in these parameters has any positive effect. With increasing number of steps Nst, the search duration increases within the range from 55 to 71 s. The upper bound of DKP values is set to 0.1. At Nst = 5, the Pareto front approximation turns out to be discontinuous within the range DKP > 0.04. This is due to the DTd value varying insignificantly within this range, thus making it difficult to estimate the solution dominances.
For the MSPS algorithm, NT = 200. The average search duration per solution for three Nst values is 7, 14, and 20 s. In all cases, the found solutions are significantly worse than those obtained using GDE3; moreover, increasing the number of NT starts yields no improvement. Since the PSO algorithm gives solutions coincident with MSPS solutions at close durations, its results are not included in Fig. 6.
The MSSQP algorithm demonstrates high efficiency in solving this problem. If NT = 10, then 60-100% of starts yield the same result matching GDE3 solutions over the entire range of values. The other starts result in unacceptable solutions with constraint violations. The average search duration for three Nst values is 1.6, 5.2, and 7.5 s. At the same time, the algorithm also finds solutions with the given DKP values in the area wherein the Pareto front approximation obtained using GDE3 has turned out to be discontinuous.

CONCLUSIONS
The study demonstrates that the GDE3 MOO population-based algorithm can be used to find a solution for all of the considered problems and can therefore be recommended for use in solving of such types of MOO problems to obtain an approximation of the POS set across a wide range of QIs. Thus, it is reasonable to check the applicability of the MSSQP algorithm to a particular problem, as well as the possibility of obtaining solutions superior in quality to those obtained by GDE3. However, due to the lack of answers to questions why MSSQP algorithm is effective for some problems and unsuitable for others, as well as how it finds solutions for Problem 3 inaccessible to other algorithms, further experimental Alexander V. Smirnov Comparison of algorithms for multi-objective optimization of radio technical device characteristics verification is required. For this it will be necessary to analyze the impact of landscape features of individual QI in MOO problems on the solution finding process. Despite active investigations in recent years, including the use of intelligent technologies [14,15], in the field continues to be characterized by a lack of sufficiently general results.
The MSSQP algorithm (or, in case it cannot find suitable solutions, other algorithms based on scalarizing OFs) can be recommended for use in cases where it is necessary to find a small number of POS or to provide accurate values for part of QI, which is difficult when using population-based MOO algorithms.