Furong Ye

Carola Doerr

Hao Wang

Fellow, IEEE Thomas Bäck

Automated Configuration of Genetic Algorithms by Tuning for Anytime Performance

Keywords: Evolutionary Computation, Algorithm Configuration, Black-Box Optimization, Performance Measures

Finding the best configuration of algorithms' hyperparameters for a given optimization problem is an important task in evolutionary computation. We compare in this work the results of four different hyperparameter optimization approaches for a family of genetic algorithms on 25 diverse pseudo-Boolean optimization problems. More precisely, we compare previously obtained results from a grid search with those obtained from three automated configuration techniques: iterated racing, mixedinteger parallel efficient global optimization, and mixed-integer evolutionary strategies.

Using two different cost metrics, expected running time and the area under the empirical cumulative distribution function curve, we find that in several cases the best configurations with respect to expected running time are obtained when using the area under the empirical cumulative distribution function curve as the cost metric during the configuration process. Our results suggest that even when interested in expected running time performance, it might be preferable to use anytime performance measures for the configuration task. We also observe that tuning for expected running time is much more sensitive with respect to the budget that is allocated to the target algorithms.

I. INTRODUCTION

Mutation and crossover are two key operators of genetic algorithms (GAs), and there is a long debate about the effectiveness of the two operators and their combinations [START_REF] Mitchell | When will a genetic algorithm outperform hill climbing?[END_REF], [START_REF] Spears | Crossover or mutation?[END_REF].

Mutation can maintain the population diversity of the GA, and it allows the GA to exploit promising search areas by using small mutation rates. In the early years of evolutionary computation, evolution strategies [START_REF] Rechenberg | Evolution strategy: Nature's way of optimization[END_REF] concentrated on mutation and showed their power with adaptive mutation rates for optimization [START_REF] Bäck | A survey of evolution strategies[END_REF].

Classic GAs, on the other hand, used crossover as their main variation operator [START_REF] Bäck | Evolutionary algorithms in theory and practice: Evolution strategies, evolutionary programming, genetic algorithms[END_REF], [START_REF] De | An analysis of the behavior of a class of genetic adaptive systems[END_REF], [START_REF] Goldberg | Genetic algorithms in search, optimization and machine learning[END_REF]. Intuitively speaking, the key idea of crossover is to create offspring by recombining information from two or more "parents". As is the case for mutation operators [START_REF] Doerr | Fast genetic algorithms[END_REF], [START_REF] Ye | Interpolating local and global search by controlling the variance of standard bit mutation[END_REF], novel types of crossover keep being developed [START_REF] Thierens | Optimal mixing evolutionary algorithms[END_REF], [START_REF] Tinós | Partition crossover for continuous optimization: ePX[END_REF], [START_REF] Tinós | Partition crossover for pseudoboolean optimization[END_REF].

Several works study the synergy of mutation and crossover, both by empirical and by theoretical means, see [START_REF] De | An analysis of the behavior of a class of genetic adaptive systems[END_REF], [START_REF] Elsayed | Multi-operator based evolutionary algorithms for solving constrained optimization problems[END_REF], [START_REF] Yoon | An empirical study on the synergy of multiple crossover operators[END_REF], [START_REF] Murata | Positive and negative combination effects of crossover and mutation operators in sequencing problems[END_REF] and [START_REF] Sudholt | The benefits of population diversity in evolutionary algorithms: A survey of rigorous runtime analyses[END_REF], respectively, as well as references mentioned therein. However, most of these results focus on specific Furong Ye (E-mail: f.ye@liacs.leidenuniv.nl), Hao Wang (E-Mail:

h.wang@liacs.leidenuniv.nl), and Thomas Bäck (E-Mail: t.h.w.baeck@liacs.leidenuniv.nl) are with Leiden Institute of Advanced Computer Science, Leiden University, Leiden, Netherlands.

Carola Doerr is with Sorbonne Université, CNRS, LIP6, Paris, France. Email: Carola.Doerr@lip6.fr. algorithms and problems. Widely accepted guidelines for their deployment are sparse, leading to a situation in which users often rely on their own experience. To reduce the bias inherent to such manual decisions, a number of automated algorithm configuration techniques have been developed, to assist the user with data-driven suggestions. To deploy these techniques, one formulates the operator choice and/or their intensity as a meta-optimization problem, widely referred to as the algorithm configuration (AC) or the hyperparameter optimization (HPO) problem.

The AC problem was classically addressed by standard search heuristics such as mixed-integer evolution strategies [START_REF] Bäck | Parallel optimization of evolutionary algorithms[END_REF], [START_REF] Grefenstette | Optimization of control parameters for genetic algorithms[END_REF], [START_REF] Li | Mixed integer evolution strategies for parameter optimization[END_REF]. More specific AC tools have been developed in recent years, among them surrogate-based models (e.g., SPOT [START_REF] Bartz-Beielstein | Sequential parameter optimization[END_REF], SMAC [START_REF] Hutter | Sequential model-based optimization for general algorithm configuration[END_REF], MIP-EGO [START_REF] Van Stein | Automatic configuration of deep neural networks with parallel efficient global optimization[END_REF]), racing-based methods (Irace [START_REF] López-Ibáñez | The irace package: Iterated racing for automatic algorithm configuration[END_REF], F-race [START_REF] Birattari | F-race and iterated f-race: An overview[END_REF]) and optimization-based methods (ParamILS [START_REF] Hutter | ParamILS: An automatic algorithm configuration framework[END_REF]).

Summary of Our Contributions. In this work, we study the effects of automated algorithm configuration on a genetic algorithm (GA) framework, applied to 25 diverse pseudo-Boolean optimization problems. We compare the results of four different configuration techniques. Our main interest is in analyzing the influence of the cost metric that is used to score different configurations on the quality of the configuration suggested by the AC methods. We consider two different cost metrics: minimizing the expected running time (ERT) and maximizing the area under the empirical cumulative distribution function curve of running times (AUC). While minimizing ERT favors the average first hitting time of a single fixed-target, maximizing the AUC metric aims at optimizing anytime performance, which is measured across a whole set of (budget, target value) pairs. We show that in several cases tuning for AUC yields configurations that have smaller ERT values than those that were obtained when directly tuning for ERT.

Concretely, we build on our previous work [START_REF] Ye | Benchmarking a (µ + λ) genetic algorithm with configurable crossover probability[END_REF] in which we analyzed a configurable framework of (µ + λ) GAs that scales the relevance of mutation and crossover by means of the crossover probability p c ∈ [0, 1]. The framework creates new solution candidates by applying either mutation (with probability 1-p c ) or crossover (with probability p c ). This way, it can separate the influence of these operators from each other. While we have studied several operator choices in [START_REF] Ye | Benchmarking a (µ + λ) genetic algorithm with configurable crossover probability[END_REF] via plain grid search, we consider here only one type of crossover (uniform crossover) and one type of mutation (standard bit mutation), to keep the search space manageable and to better highlight our key findings.

We compare in this work the results from [START_REF] Ye | Benchmarking a (µ + λ) genetic algorithm with configurable crossover probability[END_REF] with those obtained from three different types of automated algorithm configuration methods, one based on iterated racing (we use Irace [START_REF] López-Ibáñez | The irace package: Iterated racing for automatic algorithm configuration[END_REF]), one surrogate-assisted technique (we use the mixed-integer parallel efficient global optimization MIP-EGO [START_REF] Van Stein | Automatic configuration of deep neural networks with parallel efficient global optimization[END_REF]), and a classic heuristic optimization method (we use the mixed-integer evolutionary strategies MIES suggested in [START_REF] Li | Mixed integer evolution strategies for parameter optimization[END_REF]). Our testbed are the 25 functions from the pseudo-Boolean optimization (PBO) test suite, suggested in [START_REF] Doerr | Benchmarking discrete optimization heuristics with IOHprofiler[END_REF], [START_REF] Ye | Benchmarking a (µ + λ) genetic algorithm with configurable crossover probability[END_REF] and available in the IOHprofiler benchmarking environment [START_REF] Doerr | IOHprofiler: A benchmarking and profiling tool for iterative optimization heuristics[END_REF]. Compared to the (1 + 1) EA and to the configurations obtained by the grid search in [START_REF] Ye | Benchmarking a (µ + λ) genetic algorithm with configurable crossover probability[END_REF], we observe that (1 + λ) mutation-only GAs perform well on ONEMAX, LEADINGONES, and most of their so-called W-model extensions (see Sec. 3.7 in [START_REF] Doerr | Benchmarking discrete optimization heuristics with IOHprofiler[END_REF]), and on Ising-Models, whereas the (µ + λ) GA benefits from using crossover and different mutation rates on the more complex optimization problems. Thanks to its ability to handle conditional configuration spaces, Irace is the only method that finds configurations of (1 + λ) mutationonly GAs for problems on which these are efficient. We also observe that, on some problems, the automatic configurators cannot obtain hyperparameter settings that are as good as those provided by a simple grid search. However, our key finding is that the configuration methods can find better configurations in terms of the ERT by using AUC as the cost metric, compared to directly using ERT instead. This advantage is particularly pronounced when the budget of the GAs is small compared to the ERT obtained by the best possible configuration. In this case, ERT cannot reflect the differences among the configurations, and the anytime performance provides more information to guide the configuration process.

Related Work. Automated algorithm configuration for improving the anytime performance of algorithms has been applied in several works, both with respect to classical CPU time (e.g., for the traveling salesman problem [START_REF] Bossek | Anytime behavior of inexact TSP solvers and perspectives for automated algorithm selection[END_REF], for a MAX-MIN ant system [START_REF] López-Ibánez | Automatically improving the anytime behaviour of optimisation algorithms[END_REF], and for mixed-integer programming ibid) and for the here-considered function evaluation budgets (see [START_REF] Aziz-Alaoui | Towards large scale automated algorithm design by integrating modular benchmarking frameworks[END_REF] for a recent example). Sensitivity of algorithm configuration approaches with respect to the cutoff time (i.e., the budget of function evaluations for which algorithms are evaluated) has been studied in [START_REF] Hall | On the impact of the performance metric on efficient algorithm configuration[END_REF], [START_REF] Hall | Analysis of the performance of algorithm configurators for search heuristics with global mutation operators[END_REF]. In particular, [START_REF] Hall | Analysis of the performance of algorithm configurators for search heuristics with global mutation operators[END_REF] provides an example in which tuning for fixed-budget performance can be preferable under small cutoff times than tuning for optimization time, even if the latter is the original objective. We are not aware of any works using anytime performance measures with the objective to identify configurations that minimize ERT values.

Availability of Data and Source Code. All our data is publicly available at [START_REF] Ye | Data sets for the study "Automated Configuration of Genetic Algorithms by Tuning for Anytime Performance[END_REF] and at the public repository of IOHanalyzer, accessible via https://iohanalyzer.liacs.nl/. Users interested in an interactive evaluation of our data can find our data by selecting 'PBO' as dataset source and '2021mlga' as dataset. After loading the data from the repository, various statistics and visualizations are available for further investigation, see [START_REF] Wang | IOHanalyzer: Performance analysis for iterative optimization heuristic[END_REF] for an overview. The source code of our experiments is available at https://github.com/FurongYe/ Configuration-of-Genetic-Algorithms.

For readers' convenience, we provide in the supplementary material fixed-target charts for each of the 25 functions, similar to the one presented in Fig. 1. Paper Organization. Sec. II describes the configurable (µ+ λ) GA, the benchmark problems, and the cost metrics used to evaluate the algorithms. In Sec. III we present results for the three selected AC methods, evaluate their performance against the (1 + 1) EA, and discuss the impact of the cost metrics for the AC methods. We conclude our work and discuss directions for future research in Sec. IV.

II. PRELIMINARIES

In this section, we describe our (µ + λ) GA framework (Sec. II-A), the benchmark problems (Sec. II-B), and the performance measures used to evaluate the algorithms (Sec. II-C).

Notational conventions: Throughout this paper, we study the maximization of so-called pseudo-Boolean functions, i.e., functions of the type f : {0, 1} n → R, mapping length-n bit strings to real numbers. In particular, we always denote by n the dimension of the problem. For each positive integer k, we abbreviate

[k] = {j ∈ N | 1 ≤ j ≤ k} and [0..k] = [k] ∪ {0}.

A. A Family of (µ + λ) Genetic Algorithms

To study how mutation, crossover, and their combinations can be beneficial for GAs, we investigate a configurable genetic algorithm framework that can switch from mutationonly GAs towards crossover-only GAs by tuning the crossover probability p c . Details of the framework can be found in Alg. 1, and we refer to it as the (µ + λ) GA in the following.

The (µ + λ) GA initializes the population uniformly at random (u.a.r.) and terminates when the optimum is found or when the cutoff time (i.e., the maximal number of fitness evaluations that the algorithm may perform) is reached. For each iteration, the (µ + λ) GA creates λ offspring. Each offspring is created by applying either mutation or crossover. More precisely, crossover is applied with probability p c and mutation is used otherwise. Consequently, the (µ+λ) GA with p c = 0 refers to a mutation-only GA, and the (µ+λ) GA with p c = 1 refers to a crossover-only GA. After each iteration, the best µ points out of the combined set of µ parents and λ offspring form the next set of parents ("plus selection").

For this work, we consider uniform crossover and standard bit mutation with mutation rate p m . These two variation operators are briefly described as follows. Note that the framework can easily be extended to include other operators.

• Uniform crossover: Each bit of the offspring is copied either from the first or from second parent, chosen independently and u.a.r. for each position. • Standard bit mutation: The offspring is created by first copying the parent and then flipping distinct bits that are chosen u.a.r. The value is called the mutation strength.

It is sampled from a conditional binomial distribution Bin >0 (n, p m ), which assigns to each k ∈

[n] a probability of n k p k m (1 -p m ) n-k /(1 -(1 -p m ) n ).
Compared to the classic textbook description of standard bit mutation, this version avoids sampling copies of the parents, by allocating the probability of sampling = 0 proportionally to all positive values ∈ [n]; see [START_REF] Pinto | A simple proof for the usefulness of crossover in black-box optimization[END_REF] for a discussion of this "resampling" strategy. 5 Set P = {x (1) , x (2) , ..., x (µ) }; 6 Optimization: for t = 1, 2, 3, . . . do

B. The IOHprofiler Problem Suite PBO

The PBO problem suite suggested in [START_REF] Doerr | Benchmarking discrete optimization heuristics with IOHprofiler[END_REF], [START_REF] Ye | Benchmarking a (µ + λ) genetic algorithm with configurable crossover probability[END_REF] and available in the IOHprofiler benchmarking environment [START_REF] Doerr | IOHprofiler: A benchmarking and profiling tool for iterative optimization heuristics[END_REF] contains 25 pseudo-Boolean optimization problems. The definitions of these problems is available in the supplementary material. We provide here only a short summary: F1 is ONEMAX, F2 is LEADINGONES, F3 is a linear function with harmonic weights, F4-10 and F11-17 are so-called W-model extensions of F1 and F2, respectively,1 F18 is the Low Autocorrelation Binary Sequences (LABS) problem, F19-21 are Ising Model problems, F22 is an instance of the Maximum Independent Vertex Set (MIVS) problem, F23 is the N-Queens problem (NQP), F24 is a Concatenated Trap (CT) problem, and F25 is a randomly chosen NK Landscape instance.

Our algorithms are unbiased in the sense of Lehre and Witt [START_REF] Lehre | Black-box search by unbiased variation[END_REF], i.e., they are invariant under reordering of the bit string and under XOR of the solution candidates with arbitrary strings. They are also comparison-based, making them invariant with respect to strictly motononic transformations of the function values. We can therefore restrict our experiments to a single problem instance, and do not make use of IOHprofiler's option to transform the instances into problems with isomorphic fitness landscape and/or translated fitness values. More precisely, we tune on the first instance of each problem in the PBO suite.

C. Algorithm Performance Measures

As mentioned in the introduction, we use two main performance criteria: expected running time (ERT) and the area under the curve representing the empirical cumulative distribution function (AUC). While the ERT concerns only the hitting time for a single "final" target, AUC considers a set of targets. The formal definitions are as follows.

Definition 2.1 (ERT: Expected Running Time): Given a target value φ for a problem P , the ERT of algorithm A for hitting φ is

ERT(A, P, φ) = r i=1 min{t i (A, P, φ), B} r i=1 1{t i (A, P, φ) < ∞} , ( 1
)
where r is the number of independent runs of A, B is the cutoff time (i.e., the maximal number of solution candidates that algorithm A may evaluate in one run), t i (A, P, φ) ∈ N ∪ {∞} is the running time (for finite values, the running time is the number of function evaluations that the i-th run of A on problem P uses to hit the target φ and t i (A, P, φ) = ∞ is used if none of the solutions is better than φ), and 1(E) is the indicator function returning 1 if event E is true and 0, otherwise. If the algorithm hits the target φ in all r runs, the ERT is equal to the average hitting time (AHT). Definition 2.2 (ECDF: empirical cumulative distribution function of the running time): Given a set of targets Φ = {φ i ∈ R | i ∈ [m]} for a real-valued problem P and a set of budgets T = {t j ∈ [B] | j ∈ [z]} for an algorithm A, the ECDF value of A at budget t j is the fraction of (run, target)pairs (s, φ i ) that satisfy that in run s of algorithm A at least one of the first t j evaluated solutions has fitness at least as good as φ i .

Definition 2.3 (AUC: area under the ECDF running time curve): Given a set of targets

Φ = {φ i ∈ R | i ∈ [m]} and a set of budgets T = {t j ∈ [B] | j ∈ [z]}, the AUC ∈ [0, 1]
(normalized over B) of algorithm A on problem P is the area under the ECDF curve of the running time over multiple targets. For maximization, it reads,

AUC(A, P, Φ, T ) = r h=1 m i=1 z j=1 1{φ h (A, P, t j ) ≥ φ i } rmz ,
where r is the number of independent runs of A and φ h (A, P, t) denotes the value of the best solution that A evaluated within its first t evaluations of the run h.

III. ALGORITHM CONFIGURATION FOR BENCHMARKING

To explore promising configurations of the (µ + λ) GA, we study its associated algorithm configuration problem: Definition 3.1 (Algorithm Configuration (AC) Problem): Given a problem P , a parametrized algorithm A with parameter space Θ, and a cost metric c : Θ × P → R that is subject to minimization, the objective of the AC problem is to find a configuration θ * ∈ arg min θ∈Θ c(θ, P ).

Note that this definition is adjusted to our setting in which we tune for a single problem instance. It therefore ignores aggregation over multiple problem instances, hence the deviation from other common definitions as, for example, available in [START_REF] Eggensperger | Pitfalls and best practices in algorithm configuration[END_REF], [START_REF] Stützle | Automated Design of Metaheuristic Algorithms[END_REF].

A. Algorithm Configurators Utilized

We apply three AC methods to configure the (µ + λ) GA: Irace [START_REF] López-Ibáñez | The irace package: Iterated racing for automatic algorithm configuration[END_REF], a mixed-integer parallel efficient global optimization (MIP-EGO [START_REF] Wang | A new acquisition function for Bayesian optimization based on the moment-generating function[END_REF]), and the mixed-integer evolution strategy (MIES [START_REF] Li | Mixed integer evolution strategies for parameter optimization[END_REF]), which we briefly describe in the following paragraphs. All configurators work with a user-defined configuration budget, which is the maximal number of algorithm runs that the AC method is allowed to perform before recommending its final outcome.

Irace [START_REF] López-Ibáñez | The irace package: Iterated racing for automatic algorithm configuration[END_REF] is a so-called iterated racing method designed for hyperparameter optimization. The main steps of Irace are (1) sampling hyperparameter values from distributions that evolve over time, [START_REF] Bäck | Parallel optimization of evolutionary algorithms[END_REF] evaluating the configurations across a set of instances, (3) applying a racing method (i.e., statistical tests) to select the configurations that are not further evaluated in the subsequent rounds, and (4) updating the sampling distributions. At the end of the configuration process, one or several elite configurations are returned to the user, along with their performance observed during the configuration. The distributions from which the hyperparameters are sampled are independent of each other, unless the user specifies constraints and/or other requirements.

Efficient global optimization (EGO), also known as Bayesian optimization, is designed to solve costly-to-evaluate global optimization problems. For our configuration problem, we use an EGO-variant called mixed-integer parallel EGO (MIP-EGO [START_REF] Wang | A new acquisition function for Bayesian optimization based on the moment-generating function[END_REF]) capable of handling mixed-integer search space. EGO starts with an initial set of solution candidates {θ 1 , θ 2 , ...} and evaluating their fitness {c 1 , c 2 , ...}. From these observations, EGO learns a predictive distribution of the fitness value for each unseen configuration using stochastic models, e.g., random forests or Gaussian processes. Aiming at balancing the trade-off between the accuracy and uncertainty of this predictive distribution, EGO uses a so-called acquisition function to decide which solution candidates to sample next. Common acquisition functions are expected improvement and probability of improvement; see [START_REF] Shahriari | Taking the Human Out of the Loop: A review of Bayesian optimization[END_REF], [START_REF] Frazier | A tutorial on Bayesian optimization[END_REF] for an overview. For this work, we use the moment-generating function of improvement (MGFI) [START_REF] Wang | A new acquisition function for Bayesian optimization based on the moment-generating function[END_REF], which is defined as the weighted combination of all moments of the predictive distribution. For the weights, we used the setting recommended in [START_REF] Wang | Cooling Strategies for the Moment-Generating Function in Bayesian Global Optimization[END_REF]. To learn the predictive distribution, we choose a random forest model as it deals with the mixed-integer/categorical search space more naturally than Gaussian processes.

The mixed-integer evolution strategy (MIES) uses principles from evolution strategies for handling continuous, discrete, and nominal parameters by using self-adaptive mutation operators for all three parameter types [START_REF] Li | Mixed integer evolution strategies for parameter optimization[END_REF]. MIES starts with a randomly initialized parent population of size µ. In each iteration it generates λ offspring, by recombining two randomly selected parents and then mutating the solution resulting from their recombination, after which (µ, λ) selection is applied to the offspring population, i.e., the µ best of the λ offspring form the parent population of the next iteration.

B. Experimental Setup

Each AC method is granted a budget of 5 000 target runs, where each target run corresponds to ten independent runs of the (µ + λ) GA using the configuration that the AC method wishes to evaluate. As previously mentioned, we use ERT and AUC as performance metric, and these values are computed from the ten independent runs.

Irace requires a set of instances for the tuning process. We imitate these instances by the independent runs of the (stochastic) solvers. This is in line with previous approaches, suggested, for example, in [START_REF] Dang | Hyper-parameter tuning for the (1 + (λ, λ)) GA[END_REF]. MIP-EGO starts with 10 initial candidates by the default setting of the package [START_REF] Wang | A new acquisition function for Bayesian optimization based on the moment-generating function[END_REF]. We use a (4, 28)-MIES, following the parameter settings suggested in [START_REF] Li | Mixed integer evolution strategies for parameter optimization[END_REF].

To obtain a useful baseline against which we can compare other algorithms, we configure the (µ + λ) GA on each PBO problem separately. We focus on the problems in dimension n = 100.

Once the optimized parameter settings are obtained from the algorithm configuration, we perform 100 independent validation runs with the suggested parameter settings. This is to have a fair comparison, as the number of algorithm evaluations may differ quite drastically between the configurators and between different runs of the same configurator. All performance comparisons discussed in this paper are made with respect to these validation runs.

For evaluating GA candidates during the configuration process, the ERT values are calculated with respect to the targets listed in Tab. I and to the cutoff time of 50 000 function evaluations. This is half the budget used in [START_REF] Ye | Benchmarking a (µ + λ) genetic algorithm with configurable crossover probability[END_REF], but, according to the results presented there, our cutoff time is still larger than the number of function evaluations needed to hit the corresponding targets -except for F18, which is a very challenging problem.

For the AUC, the set of targets are 100 values, equally spaced in the interval [φ min , φ max ], where φ max is equal to the ERT targets listed in Tab. I and φ min is equal to 0 except for the following functions: φ min = -19 590 for function F22, φ min = -3 950 000 for function F23, and φ min = -1 for function F25. We evaluate the AUC at each budget, i.e., in the notation of Definition 2.3 we use T = [50 000].

The configuration space of the AC problem is Θ = {µ, λ, p c , p m }, where µ ∈ [100] is the parent population size, λ ∈ [100] is the offspring population size, p c ∈ [0, 1] is the crossover probability, and p m ∈ [0.005, 0.5] is the mutation rate. A positive crossover probability p c > 0 requires µ > 1; the configuration is considered infeasible otherwise. The results for the grid search are based on our work [START_REF] Ye | Benchmarking a (µ + λ) genetic algorithm with configurable crossover probability[END_REF], where we used µ ∈ {10, 50, 100}, λ ∈ {1, µ/2, µ}, p c ∈ {0, 0.5}, and p m = 0.01. Note that this is a considerably smaller search space, whose full enumeration requires only 18 different configurations, which is much less than the budget allocated to the automated configuration techniques. We will nevertheless observe that for some problems none of the automated configurators could find hyperparameter settings that are equally good as those provided by this small grid search.

C. Results Obtained by Automated Configuration

The (1 + 1) EA with p m = 1/n has shown competitive results in [START_REF] Doerr | Benchmarking discrete optimization heuristics with IOHprofiler[END_REF] for the PBO problems, so we use it as the baseline against which we compare the GAs obtained by the configurators. Note that this algorithm is part of the GA framework in Alg. 1 and could therefore be identified by the configuration methods.

Tab. II lists the configurations of the (µ+λ) GA obtained by the grid search and by the three AC methods. Tab. III compares the performance of these configurations, by listing the ERT and AUC values of the (1 + 1) EA and the corresponding relative deviations of the configured GAs. More precisely, the ERT and AUC values of the AC methods result from using ERT and AUC as the cost metric, respectively. The relative improvement of ERT is computed as (ERT (1+1) EA -ERT) / ERT (1+1) EA , and the relative improvement of AUC is computed as (AUC -AUC (1+1) EA ) / AUC (1+1) EA .

1) ERT Results: For the ONEMAX-based problems F1 and F4-F6, configurations of the (µ + λ) GA using crossover outperform the mutation-only GA with µ ≥ 10 [START_REF] Ye | Benchmarking a (µ + λ) genetic algorithm with configurable crossover probability[END_REF]. However, according to the values in Tab. III, the (1+1) EA outperforms the configurations with p c > 0, relatively large µ, and also relatively large λ. This observation matches our expectation because our previous study has shown that the (1 + 1) EA is efficient on ONEMAX. Meanwhile, we observe an interesting configuration with p m < 0.01, µ = λ = 2, and p c > 0 that achieves competitive ERT values against the (1 + 1) EA for F1. This configuration ties well with the results on different (µ + 1) GAs that were shown to outperform the (1 + 1) EA (and any mutation-based algorithm, in fact) in a series of recent works [START_REF] Pinto | A simple proof for the usefulness of crossover in black-box optimization[END_REF], [START_REF] Corus | Standard steady state genetic algorithms can hillclimb faster than mutation-only evolutionary algorithms[END_REF], [START_REF] Sudholt | How crossover speeds up building block assembly in genetic algorithms[END_REF].

On F4, F6, F7, F13, F15-F17, and F19-22, none of the configurations returned by the AC methods was able to outperform the (1 + 1) EA, whereas on F9-10, F14, F18, F24, all AC methods find configurations that perform much better than the (1 + 1) EA.

On LEADINGONES, a slightly better result compared to the ERT value 5 574 of the (1 + 1) EA is found by MIES.

This result is quite sensitive with respect to the mutation rate. When changing it from the MIES-suggestion of p m = 0.005 to p m = 0.01 we obtain an ERT value of 5 829. MIES also obtains an improvement on F11, which corresponds to a (2 + 2) GA with p m = 0.0245. On F14, we already observed in [START_REF] Ye | Benchmarking a (µ + λ) genetic algorithm with configurable crossover probability[END_REF] that mutation-only GAs with p m = 1/n are inferior to other GA configurations with a larger offspring population size and higher mutation rate. As expected, all three methods easily suggest configurations that outperform the (1 + 1) EA by a great margin.

On F18 and F24, all configurators unanimously suggest fairly small values for the crossover probabilities. For F25, however, GAs with p c > 0.8 show the (by far) best performance.

2) AUC results: Since we evaluate the AUC at each budget [50 000], the AUC values tend to be very close to 1, especially for the GAs that require much fewer than 50 000 evaluations to find an optimal solution.

On 13 out of the 25 problems, none of the configuration methods is able to identify a hyperparameter setting that yields better AUC values than those of the (1 + 1) EA. In some, but clearly not all of these cases, the achieved AUC values are not much worse than those of the (1 + 1) EA. The largest improvements are obtained on functions F10, F14, F17, F18, F24, and F7 (in this order). In most of these cases all four configuration techniques found improvements over the (1 + 1) EA except for F17, where Irace is the only method finding an improvement and except for F7, where all three automated configuration techniques find an improvement but not the grid search (the inverse is true on F23 but the advantage of the grid search is fairly small, not statistically significant, and the algorithm a mutation-only (10+1) GA with p m = 1/n, which is very close to the (1 + 1) EA).

We next discuss the results for the functions for which large improvements over the (1 + 1) EA were obtained. On F10, large crossover probabilities seem beneficial; all three automated configuration techniques return settings with p c > 0.98. The mutation rate seems to have less impact on the results, and the suggested settings vary from 8.5/n (Irace) to 41.3/n (MIES, which achieves the best AUC value). Interestingly, also two of the three configurations tuned for minimizing ERT have large crossover probabilities, with the exception of the one returned by MIES (p c = 0.32). The performances of all these configurations are very similar, as we can see in Fig. 1. On F14, the best improvement is obtained by a mutationonly (100+50) GA using p m = 1/n, closely followed by the Irace result, which is a (48+83) GA using crossover probability p c = 0.812 and p m ≈ 41/n. We cannot observe any clear pattern in the results, and the four suggested configurations all differ quite a bit.

For F17, as mentioned, only Irace finds a better configuration, which is also a (1+1) EA, but using a slightly larger mutation rate of p m = 1.9/n instead of 1/n.

For F18, all configurators return settings with small crossover probability p c < 0.031 and small mutation rate p m < 0.01, which indicates that local search methods may be more suitable for this problem than globally searching GAs. Interestingly, the performance of randomized local search is not very good (see [START_REF] Doerr | Benchmarking discrete optimization heuristics with IOHprofiler[END_REF] for details), which suggests that a positive probability for escaping local optima via small jumps in the search space or via crossover are needed to be efficient on this problem.

For F24, no clear pattern can be observed in the suggested configurations, and also the crossover probabilities differ widely, from 0 for the grid search, values around 0.27 for TABLE III Absolute ERT and AUC values for the (1 + 1) EA and relative improvement of ERT and AUC for the configurations suggested by the four configuration methods, in comparison against the (1 + 1) EA values. Compared to the ERT and AUC values of the (1 + 1) EA on each problem (indicated by "EA"), the relative improvement obtained from the automated configuration are shown for each AC method, where both measures are calculated from hitting times of 100 validation runs for (1 + 1) EA and AC methods. We also indicate the statistical significance in the empirical distributions of the hitting time ( * * * * for p < 0.001, * * * for p < 0.01, * * for < 0.01, and * for p < 0.05) for each pair of the AC result and that of the (1 + 1) EA. The Mann-Whitney U test is applied with the Benjamini and Hochberg method for all 120 pairwise comparisons to control the false discovery rate. Runs are cut off at 50 000 evaluations if it does not hit the final target. The significant comparisons are colour-coded with respect to the relative improvement, where a darker colour signifies a more considerable improvement. MIP-EGO and MIES, to 0.7 for Irace. On F7, the configuration obtained by Irace (which achieves the best improvement) is a (1+λ) EA with p m ≈ 3/n, the one obtained by MIP-EGO is a (2 + 20) GA with small crossover probability p c = 0.082, and the one obtained by MIES is a (23+27) GA with crossover probability p c = 0.672. These results indicate that the fraction of configurations achieving better AUC value than the (1 + 1) EA may be fairly large.

F
On the LEADINGONES problem F2, the (2 + 1) GA with p m = 0.008 and p c = 0.005, found by MIES, yields a (small) improvement over the (1 + 1) EA, whereas the configurations found by the other methods perform worse.

All in all, we find that on several problems the suggested configurations differ widely, far more than we would have expected and this across all four parameters. Analyzing the landscape of the AC problem suggests itself as an interesting follow-up study, e.g., using similar approaches to those suggested in [START_REF] Pushak | Algorithm configuration landscapes[END_REF], [START_REF] Tanabe | Analyzing adaptive parameter landscapes in parameter adaptation methods for differential evolution[END_REF].

D. Discussions on the Configurators' Performance

We now compare the performance of the three automated AC methods. The last row of Tab. III summarizes for how many settings each method was able to find configurations that outperform the (1 + 1) EA. These numbers rather balanced between the different methods, with the notable exception of the minimizing ERT objective, for which MIES suggested 13 improvements, compared to 6-8 improvements found by the other methods. MIES also suggested the best configurations in most of the cases, but, for several of these, the improvements over the (1 + 1) EA are rather minor. This shows that barely counting such cases does not do justice to the complex behavior observed in Tab. III, from which we cannot derive a clear winning configurator. We can nevertheless make a few observations. 1) Handling conditional parameter spaces: We easily see from Tab. II that Irace is the only method that obtains mutation-only GAs, and in all of these cases it returns a (1 + λ) EA. We recall that setting µ = 1 requires to set p c = 0; the configuration is infeasible otherwise. This advantage of Irace lies in its handling of conditional parameters: Irace samples non-conditional parameters first, and samples conditional parameters only if the condition is satisfied. MIP-EGO and MIES, in contrast, sample parameter values from independent distributions and give penalties to infeasible settings. With this strategy, the two methods can avoid infeasible candidates, but the probability of sampling feasible conditional candidates may be too small. For example, MIP-EGO can find a configuration with µ = 2 and p c = 0.0065 on F16 in Tab. II, but it cannot obtain the competitive configuration of (1+λ) mutation-only GA because the probability of sampling µ = 1 and p c = 0 simultaneously is too small.

2) Impact of the cost metric: We have already observed that MIES obtains better configurations for more problems when using ERT as the cost metric. For AUC, in contrast, Irace Relative ERT values of the GAs obtained by Irace, MIES, and MIP-EGO using ERT as the cost metric compared to the ERT of the GAs obtained by the same method when using AUC as the cost metric during the configuration process. Plotted values are (ERT using ERT -ERT using AUC )/ERT using AUC , capped at -1 and 1. Positive values therefore indicate that configurator obtains better results when configuring AUC. Fig. 3. The relative deviation from the best-known ERT value of the GAs obtained during the configuration process of Irace for tuning the (µ + λ) GA for ONEMAX in dimension n = 100, with the objective to minimize the ERT for the optimum f (x) = 100. The maximal number of configurations that can be tested by Irace is set to 5 000. The figure is produced by the acviz tool, and illustrations for the details of plot representation can be found in Sec. 3 of [START_REF] Souza | Acviz: A tool for the visual analysis of the configuration of algorithms with irace[END_REF].

finds more configurations that improve over the (1 + 1) EA, which can be explained as follows. In the first few iterations, AUC is able to differentiate the performance of two poor configurations if both fail to find the final target, whereas the ERT value will be infinite and thereby incomparable in this case. Hence, using AUC as the cost metric, Irace could learn to avoid evaluating those poor configurations in the following iterations. It is worth noting that such an observation is also supported by a case study of Irace [START_REF] Pérez Cáceres | An experimental study of adaptive capping in irace[END_REF], in which the authors discovered that Irace would spend too much time on poor configurations if the mean running time is taken as the cost metric. As a solution, the adaptive capping strategy [START_REF] Hutter | ParamILS: An automatic algorithm configuration framework[END_REF] is introduced to Irace in this work. Interestingly, this discussion connotes that the AUC metric realizes a similar effect as with adaptive capping for minimizing the running time of an optimization algorithm. This behavior also indicates that the choice of the cost metric might be a factor to consider when choosing which AC technique to apply. For an algorithm that cannot hit the target in all runs, the variance of its ERT values can be high due to the uncertain success rate. Besides, ERT cannot distinguish algorithms that cannot hit the target in any runs, even though their performance may differ in terms of results for other targets. This shortcoming is mitigated when tuning for large AUC, since this performance metric also takes into account the hitting times for easier targets.

Fig. 3 plots the relative deviations from the best-known ERT value of the configurations obtained during one run of Irace when using ERT as the cost metric. We observe that many configurations show large relative deviation values, which stem from GAs that cannot hit the optimum within the given budget. These configurations do not provide much useful information, since they all look equally bad for the configurator.

E. The Choice of the Cost Metric

We now evaluate how well configurations that are obtained by tuning for the AUC cost metric perform in terms of ERT. Fig. 2 summarizes these result, by plotting the relative advantage of the configurations tuned for AUC, compared to those that were explicitly tuned for ERT. More precisely, we plot (ERT using ERT -ERT using AUC )/ERT using AUC , so that positive values indicate that tuning for AUC gave better ERT values than the configurations obtained when tuning for ERT. We see that this is the case for 13, 12, and 9 out of the 25 problems when using Irace, MIP-EGO, and MIES, respectively.

We now zoom into the results obtained by Irace. We abbreviate by "Irace-ERT" ("Irace-AUC") the configurations obtained when using ERT (AUC) as the cost metric. For the problems on which the ERT of Irace-AUC was worse than that of Irace-ERT, we plot in Fig. 4 violin plots for the running times of the 100 validation runs. We observe that F15 is the only problem where Irace-ERT significantly outperforms Irace-AUC. On F2-3, F5, and F20, we observe that the result of most runs of Irace-AUC and Irace-ERT are close, but the variances of the results of Irace-AUC are higher than for Irace-ERT. On the remaining problems, we observe high variances for the result of both Irace-ERT and Irace-AUC. Irace-ERT finds the configurations with fewer unsuccessful runs, which makes sense because the number of unsuccessful runs significantly affects the ERT value. However, AUC does not only consider the evaluations needed to hit the final target, so we observe more unsuccessful runs and competitive partial runs for Irace-AUC, i.e., in cases of F21 and F24.

Although Irace-AUC does not obtain better ERT values than Irace-ERT, it can still provide valuable insights concerning the resulting configurations and performance profiles. Fig. 6 plots the fixed-target ERT values of the GAs obtained by Irace-ERT and Irace-AUC for F21. We observe that the result of Irace-ERT outperforms the result of Irace-AUC for the final target f (x) = 260. However, for the long period when f (x) < 258, Irace-AUC performs better.

We plot in Fig. 5 the violin plots of the running times for the problems where Irace-AUC obtains better ERT values than Irace-ERT. The advantage of Irace-AUC is significant on several problems, i.e., F1, F6, F11, F16, and F22. Moreover, Fig. 5. Violin plots of first hitting times for the configurations found by Irace when tuning for ERT and AUC, respectively, for problems on which Irace-AUC outperforms Irace-ERT. Results are from the 100 independent validation runs. Targets are listed in Tab. I, and the configurations of the GAs can be found in Tab. II. For each run, values are capped at the budget 50 000 if the algorithm cannot find the target.

Irace-ERT cannot find the final targets of F7, F17, F19, and F25 within the cutoff time, whereas Irace-AUC hits the targets in some (F7, F17, and F25) or all (F19) of the runs. Fig. 7 plots the fixed-target result of different GAs on F8. Compared to Irace-ERT, we observe that Irace-AUC outperforms the other algorithm at the final target and also exhibits advantages over other algorithms in most of the optimization process. Fig. 8 plots the fixed-target result of different GAs on F7. The figure shows that Irace-AUC is the only one that hits the optimum f (x) = 100, and the best-found fitness of Irace-ERT is less than 90. We observe that none of the GAs shown in the figure hits the optimum in all runs (because the ERT values are larger than the cutoff time of 50 000 function evaluations).

The results of Irace-AUC and Irace-ERT on the PBO problems reveal the questions of how the cost metric affects the performance of Irace for different configuration tasks for future study. We study in the following the impact of the cutoff time concerning the behavior of Irace on ONEMAX and LEADINGONES.

1) Sensitivity with respect to the cutoff time: Inspired by the result in Fig. 8, we study the sensitivity of ERT and AUC with respect to the cutoff time of the GAs. To this end, we consider the set {(0.5 + 0.1t) × ERT (1 + 1) EA | t ∈ [0..15]} of 16 different cutoff times. For each of these cutoff time, for each of the two cost metrics (AUC and ERT), and for each of F1 and F2, we run Irace 20 independent times with the same configuration budget of 5 000 target runs (where each target run corresponds again to ten independent runs of the respective (µ + λ) GA configuration). Fig. 9 plots ERT values of the GAs obtained this way (as before, each ERT value is based on 100 independent validation runs). For comparison, the red line indicates the performance of the (1 + 1) EA.

On ONEMAX, we observe that Irace-ERT cannot find promising configurations when the cutoff time of the GAs is too small to hit the optimum. This is the case for cutoff times smaller than 665. However, Irace-AUC can work with small budgets that are not sufficient to hit the optimum. Even with cutoff times larger than 665, Irace-AUC still obtains better ERT values than Irace-ERT. Similarly, the result for LEADINGONES shows that Irace-ERT cannot find promising configurations with insufficient cutoff times. Still, Irace-AUC performs well across all cutoff times.

Overall, we thus see that tuning with respect to AUC is much less sensitive with respect to the cutoff time. 2) Sensitivity with respect to the configuration budget of Irace: We also analyze the sensitivity of the results with respect to the configuration budget, i.e., the number of target runs that the configurator can perform before it suggests a configuration. We use Irace for this purpose. Fig. 10 plots the ERT values of the configurations suggested by Irace, for 8 selected problems from the PBO suite. Interestingly, the ERT values are not monotonically decreasing, as one might have expected, at least for the configurations that are explicitly tuned for small ERT. Tuning for AUC gave the best ERT values for F1, 8, 19, 20, and 21.

IV. DISCUSSION AND CONCLUSIONS

In this paper, we have analyzed the performance of a family of (µ + λ) GAs in which offspring are generated by applying either crossover (with probability p c ) or mutation (probability 1-p c ). Four different configuration methods have been applied for finding promising configurations of GAs: grid search and three automated techniques. The experimental results show that mutation-only GAs usually benefit from small parent population size. On the contrary, crossover-based GAs require sufficient population sizes. On our PBO problem set, the (1 + 1) EA outperforms the other tested GAs on ONEMAX, LEADINGONES, and some of their W-model extensions. However, crossover can be beneficial for the W-model extensions with epistasis and ruggedness, concatenated trap, and NK-landscapes.

We have also investigated the performance of the AC methods, Irace, MIP-EGO, and MIES. Irace is the only method that could find (1 + λ) mutation-only GAs, highlighting the importance of handling conditional parameter spaces appropriately.

We also observed that the cost metric used as tuning objective has a major impact on the performance of the AC methods. When using ERT, the AC methods cannot obtain useful information from configurations that cannot hit the optimum. But not only for these cases we observed that tuning for AUC gave better ERT values than when directly tuning for ERT.

Our results raise several interesting topics for future re-search. Apart from analyzing more closely the configurations obtained for each problem, and the sensitivity of the performance with respect to the different parameters, we also plan to study in what sense our observation that tuning for AUC can help finding better configurations for ERT generalizes to other algorithm families and/or problems. A bi-objective (or even multi-objective, if considering different performance measures) optimization process might be able to balance advantages of the different cost metrics. Such an approach would tie well with recent works in the context of algorithm selection [START_REF] Bossek | A multi-objective perspective on performance assessment and automated selection of single-objective optimization algorithms[END_REF], [START_REF] Jesus | Algorithm selection of anytime algorithms[END_REF], [START_REF] Jesus | A model of anytime algorithm performance for bi-objective optimization[END_REF].

Our results have also demonstrated that none of the AC methods clearly outperforms all others, suggesting to either combine them or to develop guidelines that can help users select the most suitable configuration technique for their concrete problem at hand. Finally, we have also observed that in several cases none of the techniques could find configurations that outperform or perform on par with the (1 + 1) EA, which may indicate improvement potential for these configuration methods.

Algorithm 1 : 4

 14 A Family of (µ+λ) Genetic Algorithms 1 Input: Population sizes µ, λ, crossover probability p c , mutation rate p m , cutoff time B; 2 Initialization: for i = 1, . . . , µ do 3 Sample x (i) ∈ {0, 1} n u.a.r.; Evaluate f (x (i) )

7 P ← ∅; 8 for i = 1 ,

 781 . . . , λ do 9 Sample r ∈ [0, 1] u.a.r.; 10 if r ≤ p c then 11 Select two individuals x, y ∈ P u.a.r. (with replacement);

Fig. 1 .

 1 Fig. 1. Fixed-target ERT values of the configurations suggested for F10. The suffix "-ERT/AUC" indicates which cost metric was used during the tuning.

  Fig.2. Relative ERT values of the GAs obtained by Irace, MIES, and MIP-EGO using ERT as the cost metric compared to the ERT of the GAs obtained by the same method when using AUC as the cost metric during the configuration process. Plotted values are (ERT using ERT -ERT using AUC )/ERT using AUC , capped at -1 and 1. Positive values therefore indicate that configurator obtains better results when configuring AUC.

Fig. 4 .

 4 Fig. 4. Violin plots of first hitting times for the configurations found by Irace when tuning for ERT and AUC, respectively. Only showing results for problems on which Irace-ERT outperforms Irace-AUC. Results are from the 100 independent validation runs. Targets are listed in Tab. I, and the configurations of the GAs can be found in Tab. II. For each run, values are capped at the budget 50 000 if the algorithm cannot find the target.

Fig. 6 .

 6 Fig. 6. Fixed-target ERT values of the GAs listed in Tab. II for F21.

Fig. 7 .

 7 Fig. 7. Fixed-target ERT values of the GAs listed in Tab. II for F8.

Fig. 8 .

 8 Fig. 8. Fixed-target ERT values of the GAs listed in Tab. II for F7.

Fig. 9 .

 9 Fig. 9. ERT values (y-axis) of the GAs obtained by Irace for ONEMAX and LEADINGONES in dimension n = 100, for different cutoff time B that the GAs can spend to find the optimum (x-axis). Showing results for B ∈ {(0.5 + 0.1t)ERT (1+1) EA | t ∈ [0..15]}. For comparison, the ERT values of the (1 + 1) EA are plotted by horizontal red lines. Results are for the best found configurations obtained from r = 20 independent runs of Irace, and each of the ERT values is with respect to 100 independent validation runs.

Fig. 10 .

 10 Fig. 10. ERT values (y-axis) of the GAs obtained by Irace with different configuration budgets B T (the number of configurations that Irace can test, x-axis). Results are for B T ∈ {(0.5 + 0.25t)5 000 | t ∈ [0..4]}. Each ERT value is for the 100 validation runs of the configuration suggested by Irace after a single run, i.e., one for each budget.

TABLE I

 I Target values used to compute the ERT metric on each problem. 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 Target 100 100 5 050 50 90 33 100 51 100 100 50 90 33 7 51 100 100 4.216 98 180 260 42 9 17.196 -0.297

	funcId 1	2	3	4 5 6 7 8

TABLE II

 II Configurations of the (µ + λ) GA obtained by grid search, Irace, MIP-EGO, and MIES. C indicates the cost metric used by the configurators. Results for maximizing AUC are obtained independently from those obtained for minimizing ERT.

	F	C	µ	Grid Search λ pm	pc	µ	λ	Irace pm	pc	µ	MIP-EGO λ pm	pc	µ	λ	MIES pm	pc
	1	ERT AUC	10 10	1 1	0.01 0.01	0.5 0.5	15 1	5 1	0.007 0.006	0.782 0	3 3	4 11	0.024 0.006	0.774 0.208	2 2	2 2	0.006 0.005	0.778 0.521
	2	ERT AUC	10 10	1 1	0.01 0.01	0.5 0	1 1	39 34	0.006 0.013	0 0	2 22	9 5	0.006 0.009	0.008 0.461	2 2	3 1	0.005 0.008	0.026 0.005
	3	ERT AUC	10 10	1 1	0.01 0.01	0.5 0.5	4 4	32 23	0.006 0.011	0.867 0.550	3 2	24 3	0.009 0.007	0.391 0.511	2 2	3 1	0.005 0.006	0.357 0.203
	4	ERT AUC	10 10	1 1	0.01 0.01	0.5 0.5	1 1	1 1	0.013 0.013	0 0	2 12	3 9	0.022 0.052	0.655 0.668	2 2	2 1	0.024 0.055	0.198 0.614
	5	ERT AUC	10 10	5 1	0.01 0.01	0.5 0.5	3 8	17 8	0.005 0.012	0.415 0.882	4 11	2 20	0.026 0.008	0.294 0.318	2 2	3 2	0.006 0.006	0.321 0.358
	6	ERT AUC	10 10	10 10	0.01 0.01	0 0.5	30 1	23 1	0.104 0.033	0.849 0	3 2	11 1	0.031 0.032	0.033 0.424	24 2	12 3	0.135 0.059	0.908 0.236
	7	ERT AUC	50 10	50 10	0.01 0.01	0 0	45 1	22 58	0.460 0.028	0.887 0	95 2	46 20	0.018 0.016	0.971 0.082	54 23	18 27	0.268 0.025	0.907 0.672
	8	ERT AUC	10 10	10 10	0.01 0.01	0.5 0.5	25 3	48 6	0.014 0.015	0.643 0.441	78 11	7 12	0.087 0.007	0.991 0.499	5 21	9 25	0.009 0.011	0.507 0.800
	9	ERT AUC	100 50	50 25	0.01 0.01	0.5 0.5	64 46	90 40	0.056 0.029	0.938 0.815	76 6	10 7	0.488 0.020	0.977 0.158	54 18	32 25	0.013 0.025	0.643 0.505
	10	ERT AUC	100 100	50 1	0.01 0.01	0.5 0.5	95 99	69 13	0.325 0.085	0.978 0.982	87 76	84 16	0.478 0.321	0.962 0.986	79 92	17 32	0.056 0.413	0.320 0.989
	11	ERT AUC	10 10	1 1	0.01 0.01	0 0	1 1	72 13	0.031 0.035	0 0	4 2	57 1	0.031 0.040	0.004 0.003	2 2	2 1	0.025 0.042	0.000 0.016
	12	ERT AUC	10 10	1 1	0.01 0.01	0 0	1 1	32 29	0.006 0.010	0 0	4 2	1 1	0.012 0.017	0.016 0.016	2 2	1 1	0.005 0.014	0.000 0.005
	13	ERT AUC	10 10	10 10	0.01 0.01	0 0	1 1	10 1	0.044 0.044	0 0	10 2	19 4	0.032 0.038	0.282 0.023	2 3	5 8	0.032 0.047	0.008 0.036
	14	ERT AUC	100 100	100 50	0.01 0.01	0.5 0	1 48	60 83	0.343 0.409	0 0.812	57 5	34 80	0.077 0.456	0.988 0.287	57 54	23 9	0.028 0.481	0.012 0.480
	15	ERT AUC	10 10	10 10	0.01 0.01	0 0	5 21	64 71	0.014 0.020	0.042 0.165	5 3	15 9	0.008 0.016	0.018 0.078	6 2	67 3	0.011 0.011	0.023 0.024
	16	ERT AUC	10 10	10 10	0.01 0.01	0 0	6 1	34 1	0.011 0.010	0.025 0.000	2 22	2 66	0.006 0.009	0.069 0.138	2 2	4 3	0.005 0.007	0.002 0.001
	17	ERT AUC	10 10	10 10	0.01 0.01	0.5 0	60 1	57 1	0.133 0.019	0.512 0	79 2	62 5	0.345 0.012	0.430 0.007	2 2	3 4	0.006 0.019	0.003 0.034
	18	ERT AUC	100 10	50 5	0.01 0.01	0 0	50 20	44 10	0.006 0.005	0.010 0.031	13 2	77 22	0.019 0.006	0.002 0.011	26 23	21 83	0.006 0.006	0.002 0.010
	19	ERT AUC	10 50	10 50	0.01 0.01	0.5 0	98 1	94 5	0.174 0.005	0.324 0	2 7	5 18	0.014 0.014	0.286 0.834	2 2	12 7	0.006 0.005	0.061 0.133
	20	ERT AUC	50 10	50 10	0.01 0.01	0 0	1 1	32 30	0.008 0.013	0 0	6 5	53 7	0.007 0.011	0.012 0.483	5 6	66 75	0.005 0.006	0.033 0.095
	21	ERT AUC	10 10	5 10	0.01 0.01	0 0	1 4	86 33	0.014 0.006	0 0.412	7 7	52 9	0.009 0.006	0.144 0.833	5 2	66 3	0.009 0.005	0.077 0.219
	22	ERT AUC	100 10	100 10	0.01 0.01	0 0.5	16 1	48 1	0.025 0.020	0.783 0	18 2	12 3	0.019 0.007	0.820 0.250	2 2	3 2	0.005 0.010	0.270 0.088
	23	ERT AUC	10 10	1 5	0.01 0.01	0 0	62 29	39 25	0.006 0.005	0.252 0.016	10 4	40 4	0.013 0.006	0.010 0.170	5 8	1 16	0.006 0.005	0.006 0.038
	24	ERT AUC	100 10	50 5	0.01 0.01	0 0	53 7	85 86	0.025 0.026	0.004 0.776	58 8	45 3	0.031 0.032	0.077 0.277	21 8	75 33	0.060 0.045	0.014 0.273
	25	ERT AUC	50 10	1 10	0.01 0.01	0 0	13 26	52 53	0.296 0.028	0.170 0.790	68 4	91 3	0.024 0.023	0.801 0.107	98 9	40 1	0.009 0.021	0.941 0.535

The W-model transformations were suggested in[START_REF] Weise | Difficult features of combinatorial optimization problems and the tunable w-model benchmark problem for simulating them[END_REF],[START_REF] Weise | Selecting a diverse set of benchmark instances from a tunable model problem for black-box discrete optimization algorithms[END_REF] and later extended in[START_REF] Doerr | Benchmarking discrete optimization heuristics with IOHprofiler[END_REF]. In a nutshell, these transformations introduce dummy variables that do not impact the quality of a solution ("dummy" transformation), perturb the fitness values ("ruggedness"), reduce the string size by partitioning the bit string and applying a majority vote within each substring ("neutrality"), or map substrings to different ones ("epistasis").

ACKNOWLEDGMENTS

We thank Manuel López-Ibáñez for pointing us to the acviz tool [13]. We thank Thomas Weise for pointing out a mistake in the previous definition of the Ising model problems F19-21.

Our work was supported by the Chinese scholarship council (CSC No. 201706310143), by the Paris Ile-de-France Region, by ANR-11-LABX-0056-LMH (LabEx LMH), and by COST Action CA15140.

Furong Ye, is a PostDoc at the Leiden Institute of Advanced Computer Science (LIACS), after finishing his PhD study at LIACS. His PhD topic is "Benchmarking discrete optimization heuristics: From building a sound experimental environment to algorithm configuration". He is part of the core development team of IOHprofiler, with a focus on the IOHexperimenter. His research interests are the empirical analysis of algorithm performance and (dynamic) algorithm configuration.

FIGURE CAPTIONS AND TABLE CAPTIONS

Fig. 1. Fixed-target ERT values of the configurations suggested for F10. The suffix "-ERT/AUC" indicates which cost metric was used during the tuning. Fig. 2. Relative ERT values of the GAs obtained by Irace, MIES, and MIP-EGO using ERT as the cost metric compared to the ERT of the GAs obtained by the same method when using AUC as the cost metric during the configuration process. Plotted values are (ERT using ERT -ERT using AUC )/ERT using AUC , capped at -1 and 1. Positive values therefore indicate that configurator obtains better results when configuring AUC. Fig. 3. The relative deviation from the best-known ERT value of the GAs obtained during the configuration process of Irace for tuning the (µ + λ) GA for ONEMAX in dimension n = 100, with the objective to minimize the ERT for the optimum f (x) = 100. The maximal number of configurations that can be tested by Irace is set to 5 000. The figure is produced by the acviz tool, and illustrations for the details of plot representation can be found in Sec. 3 of [START_REF] Souza | Acviz: A tool for the visual analysis of the configuration of algorithms with irace[END_REF].

Fig. 4. Violin plots of first hitting times for the configurations found by Irace when tuning for ERT and AUC, respectively. Only showing results for problems on which Irace-ERT outperforms Irace-AUC. Results are from the 100 independent validation runs. Targets are listed in Tab. I, and the configurations of the GAs can be found in Tab. II. For each run, values are capped at the budget 50 000 if the algorithm cannot find the target. Fig. 5. Violin plots of first hitting times for the configurations found by Irace when tuning for ERT and AUC, respectively, for problems on which Irace-AUC outperforms Irace-ERT. Results are from the 100 independent validation runs. Targets are listed in Tab. I, and the configurations of the GAs can be found in Tab. II. For each run, values are capped at the budget 50 000 if the algorithm cannot find the target. EA on each problem (indicated by "EA"), the relative improvement obtained from the automated configuration are shown for each AC method, where both measures are calculated from hitting times of 100 validation runs for (1 + 1) EA and AC methods. We also indicate the statistical significance in the empirical distributions of the hitting time ( * * * * for p < 0.001, * * * for p < 0.01, * * for < 0.01, and * for p < 0.05) for each pair of the AC result and that of the (1 + 1) EA. The Mann-Whitney U test is applied with the Benjamini and Hochberg method for all 120 pairwise comparisons to control the false discovery rate. Runs are cut off at 50 000 evaluations if it does not hit the final target. The significant comparisons are colour-coded with respect to the relative improvement, where a darker colour signifies a more considerable improvement.


domain: spi phys

text:
Angélica Sicard
email: angelica.sicard@onera.fr

Vincent Maget
email: vincent.maget@onera.fr

Didier Lazaro
email: didier.lazaro@onera.fr

Nicolas Balcon
email: nicolas.balcon@cnes.fr

Robert Ecoffet
email: robert.ecoffet@cnes.fr

GREEN upper envelope model for energetic electrons

Keywords: Electron, Model, Radiation Belt

  L'archive ouverte pluridisciplinaire

I. INTRODUCTION

HE well-known NASA AP8 and AE8 models [START_REF] Sawyer | AP-8 trapped proton environment for solar maximum and solar minimum[END_REF] [START_REF] Vette | The AE-8 Trapped Electron Model Environment[END_REF] have been the standards for a long time and have rendered huge service in satellite design. However, these models are old and need revisiting. Therefore, the AE9/AP9 (IRENE) project have existed for some time and v1.50 of this model has been available and stable since 2017 [START_REF] Ginet | AE9, AP9 and SPM: New models for Specifying the Trapped Energetic Particles and Space Plasma Environment[END_REF]. The AE9 model covers electrons from 40 keV to several MeV and the AP9 model covers protons from 100 keV to several hundred of MeV. At low energy, the IRENE model corresponds to SPM model for electrons from 1 keV to 40 keV and for protons from 1 keV to 100 keV. In parallel, ONERA have developed another global model: GREEN (Global Radiation Earth Environment) [START_REF] Sicard | GREEN: the new Global Radiation Earth ENvironment model (beta version)[END_REF] [START_REF] Sicard | New Model for the Plasma Electrons Fluxes (Part of GREEN Model)[END_REF] . This model covers a large region of space, from Low Earth Orbit (LEO) altitudes to Geostationary Earth Orbit (GEO) and above, for a wide range of energies from plasma (0.1 keV) to relativistic particles (several MeV for electrons and several hundred of meV for protons). The GREEN model provides averaged values of electron and proton fluxes as a function of the year in the solar cycle. We made the choice to focus on the dependence of the fluxes to the solar cycle and not to develop a model with confidence levels. Users can therefore get access to several different models (GREEN, AE9/AP9,…) depending on their needs. In this paper, we present the current ONERA effort to improve global models of radiation belts and in particular to upper case ones.

It is well known that the intensity of solar cycles is variable and the effects on energetic protons and electrons may vary from one cycle to the next. The purpose of this study is therefore to develop an "Upper Envelope" in the GREEN model which takes into account the variation from one solar cycle to another and which gives the maximum flux for each year of the solar cycle. Only the case for the electron population has been studied so far. The same type of "Upper Envelope" was defined in IGE-2006 model [START_REF] Sicard-Piet | A new international geostationary electron model: IGE-2006, from 1 keV to 5.2 MeV[END_REF] . This "upper case" was based on three solar cycles of LANL-GEO data. The upper case in IGE-2006 model is described in part II. The development of the upper envelope in the GREEN model is then detailed, using NOAA-POES data [START_REF] Evans | Polar Orbiting Environmental Satellite Space Environment Monitor -2: Instrument Descriptions and Archive Data Documentation[END_REF] (part III) and Salammbô simulations [START_REF] Beutier | A three-dimensional analysis of the electron radiation belt by the Salammbô code[END_REF] [9] (part IV). Finally, an example of results from the GREEN model (Mean and Upper Envelope) and comparisons with AE8 and AE9 models are shown in the last section of the paper, before the conclusions.

II. UPPER ENVELOPE IN IGE-2006

An upper envelope or "upper case" is defined in the ESA standard (ECSS-E-ST-10-04) IGE-2006 model [START_REF] Sicard-Piet | A new international geostationary electron model: IGE-2006, from 1 keV to 5.2 MeV[END_REF] . The outputs of the IGE-2006 model are the mean flux as well as the minimum and maximum flux observed from the variation of intensity from one solar cycle to another. The reader is advised to refer to Sicard-Piet et al., 2006 [6] to see in detail how this envelope was calculated. IGE-2006 is based on three solar cycles of LANL satellites data in geostationary orbit. These data allow calculating a mean flux, depending on the solar cycle, between 1keV and 5MeV. Each year of a solar cycle is represented on a scale from year -6 to year +4 with 0 being the year of the solar minimum. Therefore, for the three solar cycles of LANL data, each year of the solar cycle (between -6 and +4) is represented by three values of electron flux: the mean flux, the upper case and the lower case. In order to obtain the upper case, for each year of the solar cycle and for each energy channel, the ratio between the maximum flux and the mean flux over the three cycles is calculated. This ratio depends on the year of the solar cycle, but only the maximum ratio over all equivalent years of solar cycles will be considered in order to have an upper envelope that does not depend on the year of the solar cycle. Fig. 1 shows this maximum ratio as a function of energy (blue diamonds). Two linear curves (one for energies from 1 keV to 1.5 MeV and the other one for energies greater than 1.5 MeV) representing an envelope bounding the data points in order to be conservative, are also shown in black in the figure. Considering only the maximum ratio over the 11 years of a solar cycle (and not a ratio depending on the year of the solar cycle) allows having a constant upper envelope whatever the year of the cycle is, and therefore allows taking into account the fact that in the space era the variability has been within a year of the 11-year nominal average.

III. DEFINITION OF THE UPPER ENVELOPE IN THE GREEN

MODEL USING NOAA-POES DATA The idea here is to use data from the NOAA-POES satellites [START_REF] Evans | Polar Orbiting Environmental Satellite Space Environment Monitor -2: Instrument Descriptions and Archive Data Documentation[END_REF] to study the dynamics of electron fluxes from one solar cycle to the next. Coherence exists between LEO and equatorial electron measurements at high energy. Kanekal et al. [START_REF] Kanekal | Multisatellite measurements of relativistic electrons: global coherence[END_REF] showed it for the first time for MeV electrons using SAMPEX data at low altitude. This property has been used previously to develop the ONERA SLOT model [START_REF] Sicard-Piet | A new ONERA-CNES Slot Electron Model[END_REF] and to deduce a radiation belt index from LEO measurements [START_REF] Lazaro | Radiation Belt Activity and Solar Proton Event Alarm[END_REF] . Therefore, assuming that the dynamics of electron flux is the same at low altitude as anywhere else on the field line (assuming that an equilibrium is obtained much faster along the field line than the timescale considered by the model), it is possible to calculate the ratio between max flux and mean flux from the NOAA-POES data. NOAA-POES data have many advantages: (1) as they are obtained along LEO orbits, satellites have a long lifetime (for example NOAA-POES-15 launched in May 1998 is still operational); (2) NOAA-POES constellation has been flying with quasi-similar SEM/SEM2 detectors [START_REF] Raben | TIROS/NOAA satellite space environment monitor data archive documentation: 1995 update[END_REF] since the 80's, so long term data can be exploited and cross-calibration is then easy to handle; (3) they scan the entire field lines covering the radiation belts region, from L*=1.08 to L* values greater than 8 (GEO orbit is between L*=5.6 and L*=6.6); (4) even if the number of channels is small, detectors have well-spread energy intervals: E>30keV (E1 channel), E>100keV (E2 channel), E>300keV (E3 channel) and E>1MeV. For the last channel, it has been shown that one SEM2 proton channel (unidirectional P>6.9MeV/90°) is contaminated by electron and so allow to measure E> 1 MeV electron [START_REF] Boscher | High-Energy Electrons in the Inner Zone[END_REF] . Therefore, the E>1MeV on SEM has been combined with the P>6.9MeV/90°o n SEM2 to have long time coverage of E>1MeV electrons fluxes. Obviously, this channel has been cleaned to keep only electron data according to the Data Analysis Procedure from COSPAR-PRBEM [15] : the removal of the inner proton belt at L*<2 and the proton flares at high L* values.

It is important to keep in mind that fluxes from NOAA-POES are unidirectional fluxes and that at LEO the assumption of isotropic flux is not valid. However, the absolute value of the fluxes will not be used here but only their dynamics as a function of time. The question of anisotropy at LEO is out of the scope of the paper and will be address in further development pf the GREEN model. The NOAA-POES data used in this study cover all years from 1979 to 2019, encompassing measurements from NOAA-POES-06, NOAA-POES-10, NOAA-POES-12 and NOAA- A decrease in low energy fluxes at low L* over the last solar cycle is quite normal: as the solar cycle was weak no strong magnetic storms happened to push the particles at low L* values. So on average the fluxes are lower. Moreover there is no decrease at high L* value for the same reasons as above: no magnetic storm pushed the particles at low L* value. Finally, at high energy (>1MeV), there were increases in electron fluxes linked to high-speed streams coming from coronal holes (always to medium-high L) [START_REF] Horne | Realistic worst case for a severe space weather event driven by a fast solar wind stream[END_REF] .

Note: Data in the > 1MeV energy range on older NOAA-POES with SEM detector (from 1979 to 1998) are not of very good quality (poorly calibrated). However, it was possible to use them anyway by recalibrating the flux from those obtained on the SEM2 instrument aboard NOAA-POES-15. On the other hand, it is important to keep in mind that the absolute value of the fluxes will not be used here but only their dynamics as a function of time.

The next step is to classify this dataset according to the years of the solar cycle. Table 1 represents the correspondence between the year and the year of the solar cycle as we used it to calculate the yearly averages of the electron fluxes as a function of the year of the solar cycle. It is important to note than some cycles only last 10 years while another last 13 years. For this one, as the minimum was longer than for the other solar cycle we choose to put three years as the minimum (2007, 2008 and 2009). Fig. 3 represents the yearly averages of the electron flux as a function of the year of the solar cycle for two values of L * (L*=2.5 and L*=4.5) and for the 4 energy channels (>30keV,>100keV, >300keV and >1MeV). The curves in solid lines correspond to the average fluxes over the entire period (1979-2019) and the curves in dotted lines correspond to the maximum fluxes for each year of the solar cycle over the entire time-period. The third step is to calculate the ratio between the maximum flux and the average flux for each energy and each L* value. In order to eliminate any discrepancies induced by the varying solar cycle length and to margin the upper envelope estimation, only the maximum ratio for an energy and a L* value will be considered for calculating this upper envelope.

Fig. 4 represents the ratio between the maximum flux and the mean flux resulting from the analysis of the NOAA-POES data as a function of L * for 4 energy ranges (>30keV,>100keV,>300keV,>1MeV). The same ratio obtained in the case of the IGE-2006 model is also plotted in dotted lines. At the bottom a zoom of the same graph is represented. The ratio is clearly dependent on the energy and L *. It is the strongest in the region of the slot, that is to say between L * = 2 and L * = 4. This ratio is a relative variation and the high ratio in the slot region is due to the absence of particles in this region over some time periods.

The ratio reaches a maximum of 5.3 for electrons with energies > 300keV at L * = 2.4. We also observe on Fig. 4 that the ratios obtained from the NOAA-POES data between L*=5.7 and L*=7 are consistent with the ones obtained in the case of the IGE-2006 model using LANL-GEO data: the ratios from the IGE-2006 model and from NOAA-POES data are between 1.4 and 1.9. In the >100 keV and > 300 keV cases the ratios calculated from NOAA-POES data are really close to those from the IGE-2006 model. For >30 keV and >1 MeV cases the ratios are slightly different but remain on the same order of magnitude. We have no information from NOAA-POES data above 1MeV thus no comparison can be made with IGE-2006 for energies above 1 MeV. • Ec is the kinetic energy of the particle The distribution functions calculated with Salammbô were then converted into omnidirectional flux for energies equivalent to those measured by NOAA-POES and for two additional energies in order to define the upper envelope at very high energies: >30keV, >100keV, >300keV, >1MeV, >3MeV and >5MeV.The omnidirectional fluxes resulting from Salammbô at the equator are shown in Fig. 5. The parameters F10.7 and Kp are also represented on this figure. We can clearly see the short-term dynamics associated with geomagnetic storms, but this is outside the scope of this study. Only solar cycle scale dynamics are taken into account in the GREEN model. This long-term dynamics is also visible in the electron fluxes plotted on Fig. 5. Note: It is important to note that the Salammbô fluxes studied here are obtained at the equator while the fluxes from NOAA-POES satellites are measured in LEO. It is well known that on a same magnetic field line electron flux is much stronger near equator than at high latitude. This is why the magnitude of electron flux from Salammbô (equator) is much stronger than electron flux from NOAA-POES data (high latitude). However, in the development of GREEN and as mentioned previously, we have always assumed that even if the absolute values of electron flux are different between equator and high latitude, the long-term flux dynamics in LEO is the same as in GEO, in terms of relative variations in amplitude. As we can see on the figure, Salammbô gives some strange results beyond L*=6 with very high fluxes at high energy (>3MeV and >5MeV) maybe due to the boundary condition which is not perfect at these high energies. Thus, we decided to use Salammbô results only up to L*=6 in the rest of the study. For L*>6 upper envelope calculated with IGE-2006 (ECSS-04c European standard for GEO) will be used. Fluxes from Salammbô were studied in the same way as those from NOAA-POES, described previously. Fig. 6 represents yearly averages of >30keV, >100keV, >300keV, >1MeV, >3MeV and >5MeV electron flux from the Salammbô simulation as a function of the year for 6 values of L * (2.0, 3.0, 4.0, 5.0, 6.0 and 7.0). The solar cycle variation is also plotted with F10.7 in arbitrary unit. The orders of magnitude of fluxes are higher than those from NOAA-POES because fluxes at the equator (Salammbô) are higher than fluxes at LEO (NOAA-POES).

As for NOAA-POES data Fig. 7 represents the yearly averages of the electron flux as a function of the year of the solar cycle for 2 values of L * (L*=2.5 and L*=4.5) and for the 6 energy ranges (>30keV,>100keV, >300keV, >1MeV, >3MeV and >5MeV). The curves in solid lines correspond to the average fluxes over the entire period (1979-2019) and the curves in dotted lines correspond to the maximum fluxes for each year of the solar cycle over the entire time period. Fig. 7 shows clearly that the dynamics of electron fluxes are higher inside the slot region (L*=2.5) than outside (L*=4.5). It is really difficult to compare the dynamics of electron fluxes from NOAA-POES data and from Salammbô simulations on the previous plots. That is why, in the interest of the paper, only the ratio between maximum flux and mean flux for each energy and each L* studied in this paper will be compared thereafter. Fig. 8 represents the ratio between the maximum flux and the average flux resulting from the analysis of Salammbô simulations (dashed lines), NOAA-POES data (full lines) and IGE-2006 results (in dotted lines) as a function of L * for 6 energy ranges (>30keV,>100keV,>300keV,>1MeV, >3MeV and >5MeV). Overall, the ratios obtained by the different analysis are consistent: ratios between 4 and 6 at low L* (L*<3.5) then a decrease up to ratios between 1 and 2 (according to NOAA-POES and Salammbô) and finally an increase of the ratio beyond L * = 5, which is greater at high energy (according to Salammbô and IGE-2006). Keep in mind that we will not used Salammbô results above L*=6 as mentioned above but the results still seem to show that the ratio increases with energy at high L* values. Fig. 9 shows these ratios for each of the first four energies (there is no data >3 MeV and > 5 MeV from NPOES data). For electrons greater than 300keV (green curves) the coherence between ratios deduced from NPOES data and those deduced from Salammbô is very good. The maximum of the ratios are at the same location (L*~2.6) and are very close (5.2 for NOAA-POES and 5.8 for Salammbô). However, notable differences between the results from Salammbô and those from NOAA-POES should be mentioned, especially when considering the position of L * for the maximum of the ratio Max flux /Mean flux for the three other energies. For example, at 30 keV, the maximum ratio is at L * = 2.4 according to Salammbô while it is at L * = 3.4 according to NOAA-POES. According to the NOAA-POES data, the greatest variation of the ratio is in the Slot region, while according to Salammbô, it is actually at a slightly lower L* (L*~2.5). While the profile of ratios deduced from Salammbô results is coherent between the four energies, it is really difficult to understand why this coherence does not exist for the ratio Max flux/Mean flux deduced from NOAA-POES data with a profile at 300 keV different than the three other energies.

Obviously, neither the data nor the model is perfect and small differences in fluxes can lead to differences in the ratio studied here. Moreover, it is important to keep in mind that Salammbô simulations are at equator while NOAA-POES data are at high latitudes. It is really difficult to know which model or data is correct for each L* and each energy. Understanding in detail the differences between Salammbô simulations and NOAA-POES data is beyond the scope of this study.

Within the framework of our study, to define the "upper envelope" of GREEN model which takes into account the variations from one solar cycle to another, it is imperative to be conservative. Therefore, the ratio that will be integrated into GREEN model will be, for each energy and each L* value, the largest ratio between the one obtained from Salammbô, the one resulting from the analysis of the NOAA-POES data and the one resulting from the IGE-2006 model. Moreover, let's remember that the upper envelope keeps the logic used in the IGE-2006 model, i.e. that the maximum ratio from the yearly max to yearly mean only are applied. Thus, Fig. 10 represents the ratio between the maximum flux and the mean flux used to construct GREEN "Upper Envelope" model. For L*>6 ratios for upper envelope provided by IGE-2006 were used.

V. UPPER ENVELOPE IN GREEN AND EXAMPLES OF RESULTS

Finally, in order to obtain the upper envelope of electron flux in GREEN, the ratios defined previously have been multiplied by the mean electron fluxes provided by GREEN. In more detail, for each point of a given orbit and each energy asked by the user the mean flux is first calculated with the GREEN model. Then according to the magnetic coordinates and the energy, the ratio is interpolated is the (L*,energy) grid plotted on Fig. 10. The mean flux is then multiplied by this ratio in order to obtain the upper flux for each point of the orbit and each energy. Therefore, we can now calculate the average electron flux on different types of orbits (MEO, GEO, EOR,…) from GREEN Mean and GREEN "Upper envelope". Fig. 11 represents the mean spectra of electrons along a GPS-like orbit in 2003 (in solar maximum) at the top, along a GEO for 11 years in the middle and along an equatorial EOR (from LEO to GEO; altitude minimum: 214km and altitude maximum : 53000km) orbit at the bottom, from GREEN Mean (in blue) and GREEN "Upper Envelop" (in red). The ratio between the averaged upper flux and the average mean flux is also represented, in grey. Fig. 11 Average electron spectra for a MEO orbit (GPS like orbit) in 2003 (solar maximum) from average GREEN-e (in blue) and GREEN "Upper Envelope" (in red). The ratio between the two spectra is also represented, in gray. Results from AE8 MIN (in light green), AE8 MAX (in dark green) and AE9 Mean (in orange) are also plotted.

Results from AE8 MIN (in dashed light green), AE8 MAX (in dashed dark green) and AE9 Mean (in dashed orange) are also plotted. It can be observed that the ratio between the mean flux and the "Upper envelope" flux increases with energy and can reach a factor of 2 at 5 MeV for GPS-like orbit, a factor of 4.2 at 5MeV for GEO and a factor of 1.9 at 1.5MeV for equatorial EOR orbit. If we compare GREEN "Upper Envelope" results with AE8 and AE9, Fig. 11 shows that electron fluxes provided by the AE8/9 models are slightly lower than GREEN results in the case of GPS-like orbit, slightly higher than GREEN results in the case of GEO and really close in the case of equatorial EOR orbit.

VI. CONCLUSION

GREEN is a mean model of electron and proton flux depending on the year of the solar cycle. In this paper, we have described a new version of GREEN that provides mean flux but also an "upper envelope". This upper envelope takes into account the variation of electron flux from one solar cycle to another in the same way the IGE-2006 model has been done for GEO orbit. To define this upper envelope NOAA-POES data between 1979 and 2019 have been analyzed and ratios between mean yearly flux and maximum yearly flux have been defined. In parallel similar ratios have been calculated using Salammbô simulations between 1990 and 2019 and IGE-2006 results. In order to be conservative the highest ratio was selected for each energy and each L* value. Then the mean fluxes in GREEN were multiplied by these ratios to obtain the GREEN upper envelope. The upper envelope is defined only for electron between 30keV and 5MeV flux. Comparison of results from GREEN "Upper Envelope" and other models like AE8 and AE9 shows that for some orbits fluxes from GREEN "Upper Envelope" can be slightly higher than those models while it can be slightly lower for other orbits. The "Upper Envelope" is part of the GREEN-V4 model and will be available in the OMERE Tool [20] .

Fig. 1 .

 1 Fig. 1. Maximum (over the 11 years of a solar cycle) of the ratio of the max flux (over the three solar cycles of data) to the mean flux as a function of energy (blue diamonds). The black curves represent a fit of this max ratio taking a small margin of error.

Fig. 2

 2 Fig. 2 Yearly averages of >30keV, >100keV, >300keV and >1MeV electron flux from NOAA-POES data as a function of the year for 6 values of L * (2.0, 3.0, 4.0, 5.0, 6.0 and 7.0).

  satellites successively. Fig.2represents the annual averages of >30keV, >100keV, >300keV and > 1MeV electron flux as a function of the year for 6 values of L* (2.0, 3.0, 4.0, 5.0, 6.0 and 7.0). The solar cycle variation is also plotted with F10.7 in arbitrary unit. Several observations can be made on these graphs: (1), the last solar cycle induces much smaller fluxes than the previous ones for low energies (>30keV, <100keV and >300keV) and at low L* values; (2) the dynamics of electron fluxes as a function of time increases when the energy increases and decreases as L * increases. (3) at high L* values (L*>6) and high energy (>1MeV) the maximum flux during the declining phase of the solar cycle due to magnetic storms coming from high-speed streams (coronal holes).

Fig. 3

 3 Fig.3Yearly averages of the electron flux from NOAA-POES as a function of the year of the solar cycle for 2 values of L * (2.5 and 4.5) and for the 4 energy ranges (>30keV,>100keV,>300keV and >1MeV). The curves in solid lines correspond to the average flux over the entire period (1979-2019) and the curves in dotted lines correspond to the maximum flux.

Fig. 4

 4 Fig. 4 Top: Ratio between maximum flux and mean flux resulting from the analysis of NOAA-POES data (1979-2019) as a function of L * for 4 energy ranges (>30keV,>100keV,>300keV,>1MeV). The same ratio obtained in the case of the IGE-2006 model is also drawn in a dotted line. Bottom: zoom of the same graph.

Fig. 5

 5 Fig. 5 Omnidirectional electron flux from Salammbô-3D model at the equator. F10.7 and Kp are also plotted.

Fig. 6

 6 Fig. 6 Yearly averages of >30keV, > 100keV, >300keV, >1MeV, >3MeV and >5MeV electron flux from Salammbô simulation as a function of the year for 6 values of L * (2.0, 3.0, 4.0, 5.0, 6.0 and 7.0).

Fig. 7

 7 Fig. 7 Yearly averages of the electron flux from Salammbô simulations as a function of the year of the solar cycle for 2 values of L* (2.5 and 4.5) and for the 6 energy ranges (>30keV,>100keV,>300keV, >1MeV, >3MeV and >5MeV). The curves in solid lines correspond to the average flux over the period 1979-2019 and the curves in dotted lines correspond to the maximum flux.

Fig. 8

 8 Fig. 8 Ratio between maximum flux and mean flux resulting from the analysis of Salammbô simulations between 1979 and 2019 (dashed lines), NOAA-POES data from 1979 to 2019 (full lines) and IGE-2006 results (in dotted lines) as a function of L * for 6 energy ranges (> 30keV,> 100keV,> 300keV,> 1MeV, > 3MeV and > 5MeV).

Fig. 9

 9 Fig. 9 Ratio between maximum flux and mean flux resulting from the analysis of Salammbô simulations between 1990 and 2019 (dashed lines), NOAA-POES data from 1979 to 2019 (full lines) and IGE-2006 results (in dotted lines) as a function of L * for 4 energy ranges (>30keV,>100keV,>300keV,>1MeV).

Fig. 10

 10 Fig. 10 Ratio between maximum flux and mean flux used to construct GREEN Upper Envelope as a function of L * for 6 energy ranges (>30keV,>100keV,>300keV,>1MeV, >3MeV and >5MeV).

Table 1

 1 Correspondence between year and year of the solar cycle

					Year
		-6				2001
		-5		1981	1991	2002	2014
		-4		1982	1992	2003	2015
		-3		1983	1993	2004	2016
	Year of the	-2		1984	1994	2005	2017
	solar cycle	-1		1985	1995	2006	2018
		0		1986	1996	2007,2008,2009	2019
		1		1987	1997	2010
		2		1988	1998	2011
		3	1979	1989	1999	2012
		4	1980	1990	2000	2013

ACKNOWLEDGMENT

We thank the NOAA's National Geophysical Data Center (NGDS) for providing NOAA POES data.

* This work was supported by CNES, the French Space Agency (4500065608-DIA94).