domain: math shs

text:
Houssam Kasti
email: houssamkasti@gmail.com

Murad Jurdak
email: jurdak@aub.edu.lb

The effect of GeoGebra collaborative and iterative professional development on in-service secondary mathematics teachers' practices

Keywords: Technology integration, professional development, in-service secondary teachers, GeoGebra, design based research

Integrating technology in education is still not an easy task, teachers' adoption of technology in their teaching is even more problematic and the wide availability of technology made things more challenging. This research is a multiple case study that aims to study in depth the effect of a GeoGebra (a free mathematics software) intervention on the teaching of in-service mathematics teachers in secondary schools who follow the Lebanese curriculum. The type of the study is Design-Based Research that focuses on working closely with practitioners in collaborative and iterative manner in the real context to add principles to theory and practice. Results showed an increase in the extent teachers use GeoGebra in their student-centered teaching approach.

Introduction

When new technologies appear in medical or industrial fields, there is often a rush to replace obsolete tools with new ones, the staff get immediate training on their use and the adoption level is high and quick. Why does this not happen in the education field? Answering this question is not an easy task due to the multiple factors are involved in adopting technology and the rate of change in the education field, which is known to be slow.

Literature review

Research has extensively focused on the problem of technology integration in general and in mathematics in particular. First, research in many countries has shown that technology still plays a marginal role in mathematics classrooms and that access to technology resources, educational policies, and institutional support are insufficient conditions for ensuring an effective integration of technology into teachers' everyday practices (e.g., [START_REF] Cox | ICT and pedagogy -A review of the literature[END_REF][START_REF] Cuban | High access and low use of technologies in high school classrooms: Explaining an apparent paradox[END_REF][START_REF] Goos | Surveying the Technology Landscape: Teachers' Use of Technology in Secondary Mathematics Classrooms[END_REF]. Second, research studies in general focused on some aspects of the integration problem such as lack of teachers training (e.g., [START_REF] Law | Teacher learning beyond knowledge for pedagogical innovations with ICT[END_REF]Tondeur et al., 2008) or lack of theory [START_REF] Mishra | Technological pedagogical content knowledge: A framework for teacher knowledge[END_REF]. Others suggested certain solution(s) such as conducting professional development of specific characteristics, working with mentors [START_REF] Kratcoski | Teacher Technology Mentors[END_REF], working in a community-based inquiry environment [START_REF] Lavicza | Establishing a professional development network around dynamic mathematics software in England[END_REF], or working based on a theoretical framework such as TPACK, but most of these suggestions "have crashed on the hard rocks of the classroom" (Herrington, McKenney, Reeves, & Oliver, 2007, p. 9). Third, in most studies the methodology used is not sufficient for such a complicated multi-faceted problem, and this partially explains why research has had limited impact on practices [START_REF] Herrington | Design-based research and doctoral students: Guidelines for preparing a dissertation proposal[END_REF]. A key factor is that teachers should be able to actively participate in the process of technology integration [START_REF] Voogt | Under which conditions does ICT have a positive effect on teaching and learning? A Call to Action[END_REF]. To summarise, this research aims to study how a collaborative and iterative work with in-service mathematics teachers affects their level of GeoGebra integration in their teaching to answer the following research questions:

1. How does a cooperative and iterative intervention affect in-service secondary mathematics teachers' practices regarding the integration of GeoGebra in their teaching? 2. How do participants' Valsiner's three zones mediate the impact of the intervention on teachers' practices regarding the integration of GeoGebra in their teaching?

In this study we have used the Valsiner's zone theory, which states that the factors that affect teachers' use of technology can be categorized into three zones: (1) Zone of proximal development (ZPD) which includes skill, experience, and general pedagogical beliefs;

(2) Zone of free movement (ZFM) which includes access to hardware support, curriculum and assessment requirements, students (3) Zone of promoted action (ZPA) which includes pre-service education, practicum courses and professional development [START_REF] Goos | Teachers and Teaching: Theoretical perspectives and issues concerning classroom implementation[END_REF].

Methodology

Three iterations of a design based research (DBR) methodology were used in this study across two stages (Figure 1).

The first pre-intervention stage was dedicated to understanding the situation of integrating GeoGebra in the Lebanese curriculum, piloting the GeoGebra activities and testing the instruments. Six workshops were conducted over two years and a pilot study with two teachers. At the end of this stage four teachers (other than the ones in the pilot study) were selected as cases for the study. After selecting the participants, a 3 hour-workshop was conducted by the researcher with the four participants to ensure that all participants had acquired the basic features of the software (GeoGebra).

In addition, we discussed as a group the topics in the secondary mathematics Lebanese curriculum that could be better taught with the use of GeoGebra. The second stage was the intervention stage, which comprised two iterations. In this stage collaboration was one-to-one between the researcher and each of the participants. In the first iteration, the participating teachers decided which lesson they wanted to teach with GeoGebra in accordance with their school mathematics scope and sequence. They were provided with a ready-made GeoGebra activities (made by the researcher) to be implemented in their classes. In the second iteration, teachers adapted already made GeoGebra activities and/or made their own GeoGebra activities. Three visits were conducted with each participant at his/her own school and according to his/her available time. The first visit was to prepare for the first lesson. The second visit was to evaluate the first lesson and prepare for the second lesson. Analysis of data collected from the instruments was done before starting the second iteration as required by a design based research methodology. The last visit was to evaluate the second lesson and give a general overview of the whole experience.

Instruments

For the pre-intervention phase, three questionnaires were administered by the participating teachers:

(1) Demographics questionnaire, (2) Instructional Practices in GeoGebra Questionnaire IPGQ (Form 1), ( 3) Barriers (grouped in zones) in Using Technology Questionnaire BUTQ (Form 1). The purpose of these questionnaires was to measure teachers' current (before the intervention) integration practices of the GeoGebra software in their teaching and the barriers (grouped in three zones) that affect their technology integration. After conducting the first lesson, a semi-structured interview parallel form was used (IPGSI (Form 2) and BUTSI (Form 2) to measure the impact of the intervention on teachers' practices and to find out to what extent the zones could mediate that effect. In addition, another instrument was used to assess the GeoGebra activity itself, the Lesson Assessment Criteria semi-structured Interview (LACI), which is based on instrument by [START_REF] Harris | Testing a TPACK-based technology integration assessment rubric[END_REF].

The analysis was done in general for the four participants and later individually. The general analysis looked for the general impact of the intervention and for the dynamicity of change in the extent of use in each category of the practices and its subcategories. For the impact of the intervention we were interested in the change in the extent of use of GeoGebra at the end of implementation, whereas for the dynamicity we were interested in the pattern in the extent of use of GeoGebra of change happened in between the implementation stages. The dynamicity could be: (1) static: there was no change in extent of use in between the implementation stages or (2) dynamic: there was a change in extent of use in between the implementation stages.

Participants

In the sixth (last) workshop conducted by the researcher attendees were given the pre-intervention questionnaires mentioned above. Based on the answers, for the practice instrument, the values were 0 (never use GeoGebra), 1(sometimes use GeoGebra), and 2(most of the time use GeoGebra). The average of all the questions was calculated. Similarly the average for each zone was calculated in the zone questionnaire that consists of 27 questions. Based on these results, four cases were selected (Pseudonyms: Tima, Sara, Amani, and Hazem) in a way that they differ among themselves in practice level and/ or in at least one barrier level. Table 1 represents the characteristics of each participant.

GeoGebra modules

The criteria used for lesson selection are based on those identified by [START_REF] Angeli | Epistemological and methodological issues for the conceptualization, development, and assessment of ICT-TPCK: Advances in technological pedagogical content knowledge (TPCK)[END_REF] called ICT-TPCK. The GeoGebra activities were prepared by the researcher and tested on both students and teachers. The activities were designed based on the following criteria: Each activity: 1) should be student centered, 2) can be conducted by students in a computer lab or elsewhere (classroom or at home), 3) allows student to discover the concept or theorem under study, 4) includes immediate application of the concept under study, 5) does not require prior knowledge of the software.

Each teacher selected an activity according to his/her scope and sequence, so each teacher applied a different GeoGebra activity.

Results

Stage of Use of GeoGebra

Figure 2 shows that the pattern of impact was the same for using GeoGebra in lesson presentation, lesson implementation, and lesson enhancement but different for assessment. For lesson presentation, implementation, and enhancement, in general, participants started with 'sometimes use GeoGebra' and ended with 'most of the time' after the second lesson. For assessment, there was a slight breakthrough from 'never use of Geogebra in assessment' to 'sometimes use' for each of the four participants. For all the stages of using GeoGebra, in general, the change was static then dynamic. Probably more time was needed for the change to happen prior to the second implementation, which was due to teachers' need to: become more confident in using the software; be more knowledgeable; and have more free movement.

Concerning the stage of teachers' use of GeoGebra the intervention resulted in: (a) an increase in using GeoGebra in most stages mediated positively by teachers' ZPD, and (b) an increase in teachers' appreciation of GeoGebra as a teaching tool due to the characteristics of the activities. There was interdependence between confidence and the extent of using GeoGebra in each stage. When teachers applied the activities, this led in an increase in teachers' confidence which in turn led to an increase in the extent of GeoGebra use in each of their teaching stages. There was a low impact on using GeoGebra in assessment mediated by teachers' ZFM. Three particular ZFM factors mediated negatively the impact of the intervention on assessment, these factors were: (a) Lebanese national curriculum which is so demanding with little time left for discovery, (b) Lebanese national assessment policies which assess mostly procedural knowledge, and (c) school assessment policies which are mainly set by the school administration and teachers have little impact on changing them. The characteristics of the GeoGebra activities that made impact of the intervention more effective were: (a) the effectiveness of the GeoGebra activity, (b) the ease of operating the software by students, (c) the strong alignment between the activity and the curriculum, and (d) lastly the strong fit of the activity with the instructional strategies each teacher uses.

Method of use

It is important to use GeoGebra, but what is more important is how to use it. In this category of practices the intervention had, in general, no to a slight increase in the extent of use in most subcategories and the general pattern of change was static with minimum dynamicity. For example the intervention did not affect Amani's use of GeoGebra for 'presenting a lesson' or for 'conducting an activity with the help of students'. Amani used for the first time GeoGebra for 'discovery activity done by students' or for 'students to present their work' but that change was static (never use) then dynamic. The impact of the intervention on Amani's method of use was a change in her teaching method to become more student-centered (activity done with the help of students) mediated positively by her ZFM and her ZPD. A second example is Tima, despite her ZFM factors that mediated Tima's extent of use of GeoGebra in her methods of teaching she applied for the first, time discovery activities done by students in the computer lab and/or in class. The collaboration between Tima and the researcher increased her self-confidence, skills and knowledge and that mediated positively her GeoGebra application. A third example is Sara. Before the intervention Sara was a moderate user of technology in general, and GeoGebra specifically, but the lack of a computer lab in her school and the lack of hardware in her class were the main barriers to increase technology integration. Sara used to show her students some applets using her class LCD connected to her own laptop but for the first activity she made a huge effort to take her students to the computer lab to apply discovery activities and she said:

After this experience (applying GeoGebra activity) for the first time and in a lab I will change a lot of things (in my teaching) now I have a lab for secondary. Frankly I will not use that with an LCD in the class to show students such things, there is nothing called to show (not effective)

showing them is like treating them as babies not capable of applying and concluding results, when they do it, it is different even for me I felt different. (Interview 2, November 7, 2015).

The intervention had an important effect on increasing the use of conducting discovery activities done by students in the computer lab and that change was not the same dynamicity for all teachers. The barriers teachers faced in this part of the practices were the accessibility to the computer lab and curriculum requirements (ZFM) but these barriers minimally mediated the impact of the intervention.

Place of use

Similar to method of use category there was no to slight effect of the intervention on the extent of use of GeoGebra in their classroom or at home. There was a noticeable impact on the use of GeoGebra in the computer lab since three out of the four teachers tried one or both of the GeoGebra intervention lessons for the first time in the computer lab. This was not a surprise because to use GeoGebra in class or in the computer lab is related to availability of equipment and the way of using GeoGebra. This change was not the same dynamicity for all of the teachers. Amani's change was static then dynamic, Tima's change was dynamic then static, Sara's change was dynamic, and Hazem's change was static then dynamic.

An example is the case of Sara, her first student-centered discovery activity was the activity she applied in her first lesson of the intervention. In this lesson she sensed the importance of discovery activities and how this students motivated the students and she said:

I gave them four cases with aim of acquaint to GeoGebra trace, animation, and sliders. They liked a lot so and got their attention and interest. Gave them the function act printed and they started working, one student volunteered to help me… Students enjoyed a lot the activity and attained all the required objectives. They could see things (Interview 2, November 7, 2015).

A second example is Hazem' case, the intervention did not affect the place where Hazem uses GeoGebra. He mentioned availability of a computer lab and/or the accessibility to the laptops (ZFM) to be the only barriers to more extent of using GeoGebra in his teaching. He did overcome that barrier by asking every student to bring his own device mainly tablets. Since his first interview Hazem affirmed his continuous use:

I am willing to use GeoGebra if it is related to my lesson, I consider working with GeoGebra as 'clean work' contrary to board drawing (draw, redraw…). I encourage my students to use it; I already introduced them on its features and how to use. (Interview 1, November 7, 2015)

In his second interview he said: "all students contributed [in the activity discussion], to a certain extent, according to their motivation. If they bring their own device things would be more beneficial." (Interview 2, February 11, 2016) Summing up, due to the intervention the extent of using GeoGebra for discovery by students in the computer lab increased. For all categories of the practices the accessibility and availability of hardware were the main negatively mediating factors to higher levels of practices for all participants.

The general pattern of change in the practices was more from static to dynamic in the stages of use, static in the method of use and in the place of use.

Figure 1 .

 1 Figure 1. The stages of the study

Figure 2 .

 2 Figure 2. The extent of using GeoGebra by the participating teachers over the three stages: Before the intervention, after implementing the first lesson , after implementing the second lesson. 0: Never; 1: sometimes; 2: Most of the time A: Amani; T: Tima; S: Sara; H: Hazem

Table 1 . Participants domographics, practice and zones level *Not: the zone is not considered as a barrier to GeoGebra integration

 1

	Name	Age	degree Highest	experience Teaching	Practice level	ZFM	ZPA	ZPD
	Amani	50-55	BS	25 years	Low	Moderate Moderate Low
	Tima	23-26	Masters +TD	2 years	Moderate	Low	Moderate Not*
	Sara	33-40	BS	7 years	Moderate	Moderate	Low	Not
	Hazem	41-50	Masters	31 years	High	Moderate	Not	Not

Table 2 . The intervention activities conducted by participating teachers

 2 Table 2 shows type and place of activities applied by each teacher.

		Activity 1	Place	Activity 2	Place
	Amani	Sign of quadratic polynomials	In class	Derivative	In lab
	Tima	Vectors	In lab	3D	In class
	Hazem	Equation of a straight line	In class	Thales Theorem	In class
	Sara	Translation of functions	In lab	Vectors	In lab

Proceedings of CERME10

Discussion

It seems that unlike the medical or the industrial fields, the educational field is more complex in integrating technology in terms of social and psychological factors of all the stakeholders. In the medical field for example the instrument for measuring blood pressure is one tool that is used for all people, young or old, under-weigh or over-weight… To use this instrument or an updated version of it does not require social acceptance or/and making the medical staff believes of its importance. On the other hand, in the educational field there is no technology that fit all ages, abilities, and intelligence levels… Deciding to use any instrument in a certain class needs to pass many filters in order to be an integral part of the teaching-leaning process.

Recommendations

To see change in mathematics teachers' extent of using GeoGebra in particular and technology in general it seems one day workshop is not the perfect choice according to this study. Maybe with such professional development teachers' knowledge might change quickly but more has to be done in order to change their practices. How should universities prepare their pre-service teachers to be ready to use technology most of the time in their teaching? How should professional development be designed to make sure teachers' practices are changed regarding integrating technology in teaching? Maybe this study answers some of these questions but more work still needs to be done to solidify them.


text:
Iwan Gurjanow
email: gurjanow@math.uni-frankfurt.de

Matthias Ludwig
email: ludwig@math.uni-frankfurt.de

Joerg Zender
email: zender@math.uni-frankfurt.de

What influences in-service and student teachers use of MathCityMap?

Keywords: Mathematics activity, handheld devices, computer uses in education, teacher education

At the Goethe University Frankfurt am Main a new digital tool was developed to easily create mathematics trails as a mathematical outdoor activity aimed at school education, called MathCityMap. Following the articles and studies of many others, the usage of a new tool is quite an issue for teachers. Following some teacher training activities offered by our team, we offer interim results of user behavior based on data from an online survey. Our results can be useful for the implementation other digital tools.

Introduction

In their publication "Learning Outside the Classroom" the English and Welsh Department of Education and Skills strongly recommended that more lessons should take place outside of the classroom. They listed many benefits "nurture creativity, develop skills, improve attitude to learning, stimulate and improve motivation" just to name a few. (DfES, 2006) The advantage is quite obvious, going outdoors means to encounter real life objects. For mathematics education, it is possible to create authentic tasks such as: What is the height of a certain building? how many stones have been used to build that wall over there? how much water is in that pond? and so on. Tasks such as these immediately require many process competences such as problem solving, reasoning and proof, communication, connections and representations. In the early 1980s Blane and Clark proposed the idea to connect those kind of tasks to form a mathematics trail. This requires a map on which to find the tasks, a description of the tasks (both together is called the trail guide) and then you can start to walk around and solve mathematical problems. [START_REF] Blane | A Mathematics Trail Around the City of Melbourne[END_REF] In short, a mathematics trail is a set of mathematical outdoor tasks in walking distance. To solve the tasks you normally will need tools like a measuring tape and so on, which should be listed in the trail guide.

Although mobile devices and computers are widely used in every aspect of our daily lives (especially among pupils), they play a small role in education [START_REF] Chen | Mobile technology in educational services[END_REF]. Going on a mathematics trail could be greatly enhanced by the use of mobile devices, since they allow learning to occur in an authentic context and extend to real environments. At the Goethe University of Frankfurt am Main we started the MathCityMap Project (MCM) which combines traditional mathematics trails with the opportunities of new technologies. In 2013 the first ideas were made concrete (Ludwig, Jesberg, Weiss, 2013), but it took until 2016 to finally launch an accompanying web portal and mobile application. These have been released mainly for teachers to use in class, but are openly available to anyone who wishes to use it.

In spring 2016 we started to promote mathematics trails in combination with MCM by providing inservice teacher training and student courses at the university. Although the feedback on the training and courses was highly positive, the real usage of the MCM tools falls short of our expectations. In this article, we investigate reasons for this phenomenon we have encountered.

Theoretical background

Challenges creating a mathematics trail

Many mathematical tasks today are contextualized and appear to be realistic. But are they authentic? Following the definition [START_REF] Vos | What is 'authentic' in the teaching and learning of mathematical modelling[END_REF] has given, an object is authentic, if it is clearly not created for educational purposes. Consequently, it is not easy to find authentic tasks. The objects in the tasks of MCM can be described as real-life objects, however, the authenticity of the tasks depends on the creators. We provide assistance by offering training alongside best-practice examples.

Usually the creation process of a mathematics trail consists of designing appropriate tasks and the trail guide or trail booklet [START_REF] Cross | Developing Maths Trails[END_REF]. On the one hand, creating the tasks can be challenging for teachers as studies have shown [START_REF] Jones | Research on mathematics teachers as partners in task design[END_REF]. On the other hand, manually putting the tasks together into a trail guide which should also contain a map overview and a title page, may be time consuming.

Difficulties integrating new technologies into mathematics classes

Given the availability of new technology in schools, questions have always arisen such as, do teachers work with the new tools? how do they use them? and so on. Drijvers made a study in 2012 about the factors for successful use of new technology amongst teachers. One of the three important factors is the role of the teacher [START_REF] Drijvers | Digital technology in mathematics education: Why it works (or doesn't)[END_REF]. In Germany, a majority of teachers report to have not enough time alongside their daily tasks at school (Schneider, 2015 p. 20). Consequently, the time a new tool needs to be set up is an important issue. The MathCityMap project tries to simplify the creation process of designing tasks and trails to make it less time consuming for teachers.

In addition, [START_REF] Kuntze | Teacher' Self-Perceptions of Their Pedagogical Content Knowledge Related to Modelling -An Empirical Study with Austrian Teachers[END_REF] have shown that both pre-service and in-service teachers self-perception towards mathematical modelling is mainly negative. Especially the in-service teachers lack of knowledge about new technologies and modelling. They feel unprepared for modelling by their university education. Pre-service teachers on the other hand feel a lack of diagnostic pedagogic skills and feel unable to give good hints to the pupils. There is a difficulty to integrate modelling into classes, especially with new technologies.

GPS-based applications in mathematics education

Two examples of applications in mathematics education, that already successfully use mobile GPSdata, are [START_REF] Wijers | Exploring mathematics outside the classroom[END_REF], who developed a game which allows students to walk along the shape of geometric objects outside the school, and Sollervall and de la Iglesia, who have developed a GPS-based mobile application for embodiment of geometry [START_REF] Sollervall | Designing a didactical situation with mobile and web technologies[END_REF] The MathCityMap project

The intention of the MathCityMap (MCM) project is to automate many steps in the creation of the mathematics trail booklet/guide and to provide a collection of tasks and trails that can be freely used or just viewed to get inspiration for own tasks. Furthermore, it gives users (e.g. groups of pupils) the possibility to go on a mathematics trail more independently by using mobile devices' GPS functions to find the tasks location, by giving feedback on the users answer and by providing hints in the case that one got stuck at a particular task. The core of the MCM project can be divided into two parts, the MCM web portal and the MCM app.

MCM web portal -www.mathcitymap.eu

The web portal is a mathematics trail management system. After a short registration, the user can view public trails and tasks or create his own tasks and trails by typing in the necessary data (e.g. position, the task itself, the answer, an image of the object etc.) into a form (see Figure 1). For every mathematics trail, the mathematics trail booklet can be downloaded as PDF or accessed via the MCM App (see Figure 2). It contains all task information, a map overview and a title page.

MCM app for mobile devices

The MCM app allows the user to access mathematics trails created within the web portal. The trail data, such as images and map tiles, can be downloaded to the mobile device. After this procedure, it is possible to use a trail without an internet connection (see Figure 2). This design decision minimizes technical issues when using the app without mobile internet or in an area with low connectivity. Furthermore, the app offers an open street map overview for orientation purposes, feedback on the entered answers and a stepped hint system. The hint system enables pupils to solve the tasks independently and additionally has a positive impact on learning performance, learning experience and communication [START_REF] Franke-Braun | Aufgaben mit gestuften Lernhilfen -ein besonderes Aufgabenformat zur kognitiven Aktivierung der Schülerinnen und Schüler und zur Intensivierung der sachbezogenen Kommunikation[END_REF].

Research question

Following the teacher training events, we had expected more teachers to become active by creating own mathematics trails with MCM. This leads us to the research question: Why do (and don't) in-service teachers and student teachers use MathCityMap? By this question we follow Drijvers study of the usage of digital tools by teachers [START_REF] Drijvers | Digital technology in mathematics education: Why it works (or doesn't)[END_REF].

Methodology

To promote MathCityMap as a digital tool (and therefore the usage of mathematics trails in school) we have implemented three teacher trainings with 143 participants and two university student courses with 30 students during spring/summer 2016. To evaluate the trainings and gather further information for future improvements of the MCM tool, an online questionnaire was created. Additionally, we have analyzed the usage statistics.

Teacher training

The training is a half-day intensive training for in-service teachers. Since they have already studied mathematics and have a lot of teaching experience, we keep the theoretical parts on outdoor mathematics and task design rather short and prefer to go out on a prepared mathematics trail so they can experience this kind of activity. Later on, we also let them find tasks and focus more on the handling of the web portal and the app. After this course every teacher will have experienced doing mathematics outdoor with MCM, but also how to create new tasks in the web portal.

Student courses

The student courses took place at Goethe-University in Frankfurt (11 students) and the University of Potsdam (19 students) in the summer semester of 2016. The following topics formed part of the seminar: Theoretical introduction to mathematics trails, introduction to the MCM App and going on a mathematics trail with the app, aspects of outdoor task design, creating new tasks and setting them up in the MCM web portal, arranging a mathematics trail, testing the trail with a school class (grade nine), reworking the trail, testing the trail with another class (grade eight). Compared to the teacher trainings the students had to really engage themselves in mathematics trails with MCM.

Online survey

About 200 people (143 participants of the teacher trainings plus registered users of the web portal), who have agreed to receive e-mails about MCM, were invited to take part in the survey. Twenty (eight students and twelve teachers) of them completed the questionnaire.

The online survey consists of 27 items, from which twenty are closed questions or statements and seven are open text fields. The questionnaire is divided into five sections:

1. General Information (Five closed questions)

Sample item: How did you hear about MathCityMap?

2. Usage of the MCM web portal (Seven mainly closed questions) Sample item: Do you already have created a task in the web portal?

3. Statements about the MCM web portal (Seven 5-point Likert scale items: I do not agree -I agree) Sample item: The interaction between web portal and app is easy to understand.

Results

Questionnaire

Four of twelve teachers stated that they had created their own tasks. Two of them had already used the mathematics trail with a class. All eight students had created tasks and went on a mathematics trail, because it was part of the seminar.

Due to the low number of participants we report the reasons why MathCityMap was used or is going to be used and the reasons why it was not used yet in a qualitative way by forming categories. The answers were collected by an open text field, so multiple reasons could be given. The following categories are sorted by the number of mentions. If we take a look at the things teachers do require from a digital tool to be used by them, MCM is doing quite fine. MCM is easy to get and free to use. It is not time consuming to learn it and some of the teachers already have positive experiences (see Figure 4).

Usage statistics

Independent of the online survey, we also analyzed the statistics relating to the web portal and the app to describe the current state. In September 2016 74 users were registered in the web portal. Thirty of these were in-service teachers who participated in the trainings, about 20 were students who were part of the student courses. The other users were not part of the trainings or courses. In total 33 unique users (45%) created 140 tasks. About 25 mathematics trails were created by 22 unique users (30%). The app has been installed 210 times which means that there must be some people who use only the app, without being registered in the portal (e.g. pupils).

Discussion

MathCityMap as a digital tool seems to be mainly used as the mathematics trail idea is considered positively (high motivation for students and connecting mathematics to the reality). Hereafter the integration of digital tools in mathematics classes is another reason to use MCM (see Figure 3).

The lack of time, difficulties in creating appropriate tasks and the integration into the current lessons are the most mentioned reasons for why MCM had not yet been used. However, the findings also suggest that the task and trail creation processes in the web tool might be too complex at its current state (see Figure 3). All of these reasons could be interdependent. If one has difficulties in finding tasks or difficulties in integrating the mathematics trail into the lessons, it will take more time to solve these problems. Since many teachers report that they are short of time, this might lead to not using MCM [START_REF] Kuntze | Teacher' Self-Perceptions of Their Pedagogical Content Knowledge Related to Modelling -An Empirical Study with Austrian Teachers[END_REF][START_REF] Schneider | Zeit für Bildung -Eine Befragung unter Lehrerinnen und Lehrern[END_REF].

Conclusions

In our case the reasons for not using the tool (web portal and app) were mainly identified not in the tool itself, but in the mathematics trail concept (creating tasks, implementing the trail in classes).

The teacher training events and student courses need adjustments so that they pay more attention to the following identified difficulties:

1. Higher focus on task designguidance, best practice examples, blueprint tasks which can be easily adopted to the participants' school surroundings.

2. Creating a teaching conceptconcrete example of how to integrate mathematics trails in combination with MCM into mathematic lessons for a particular topic.

In addition, 'doing mathematics outdoors' could be integrated into the school curriculum to increase its significance. On the technical side, further research is needed on how to improve the usability of the MCM web portal to make the creation process more intuitive.

Figure 1 :

 1 Figure 1: The MCM web portal form for tasks

Figure 2 :

 2 Figure 2: Screenshots of the MCM App To describe the pedagogic functionality of MCM, we use the model by Drijvers, Boon and Van Reeuwijk (2010). It divides digital technologies into three groups of didactical functionalities: (a) do mathematics, (b) practice skills, (c) develop concepts. MCM offers mathematical tasks at real life objects where the user mainly can practice his skills.

4.

  Feedback on MCM (Four mainly open questions) Sample item: Which are the reasons for you to use MathCityMap? 5. General use of digital tools in mathematics classes (Four closed and open questions)Sample item: What are your requirements for using a digital tool in mathematics classes?

Figure 3 :

 3 Figure 3: results of the survey about the reasons for using / not using MCM (13 / 8 persons in total, open text answers)

Figure

  Figure 4: results of the survey about the requirements to use a digital tool in class (17 persons in total, preset answers, several selections possible)

Proceedings of CERME10


Cheikh Toure

Anne Auger

Dimo Brockhoff

Nikolaus Hansen

On Bi-Objective convex-quadratic problems

Keywords: Bi-objective optimization, Pareto set, Convex front, convexquadratic problems

In this paper we analyze theoretical properties of bi-objective convex-quadratic problems. We give a complete description of their Pareto set and prove the convexity of their Pareto front. We show that the Pareto set is a line segment when both Hessian matrices are proportional. We then propose a novel set of convex-quadratic test problems, describe their theoretical properties and the algorithm abilities required by those test problems. This includes in particular testing the sensitivity with respect to separability, ill-conditioned problems, rotational invariance, and whether the Pareto set is aligned with the coordinate axis.

Introduction

Convex-quadratic functions are among the simplest yet very useful test functions in optimization. Given a positive definite matrix Q of R n×n , a convex quadratic function is defined as

f (x) = 1 2 (x -x * ) Q(x -x * )
where x * is the unique optimum of the function. The Hessian of f coincides with the matrix Q. The level-sets of f defined as {x ∈ R n : (x -x * ) Q(x -x * ) = c, c ≥ 0} are hyper-ellipsoids whose main axes are the eigenvectors of the matrix Q with length proportional to the inverse of the eigenvalues of Q.

By changing the eigenvalues and eigenvectors of Q, one can model different essential difficulties in numerical optimization: if the eigenvectors are not aligned with the coordinate axes (if the matrix Q is not diagonal), then the associated function is non-separable: it cannot be efficiently optimized by coordinate-wise search. In practice, difficult optimization problems are non-separable. Having a large condition number for Q, that is a large ratio between the largest and smallest eigenvalue of Q models ill-conditioned problems where the characteristic scale along different directions is very different. Ill-conditioning is very frequent in real-world problems. They arise naturally as one often optimizes quantities that have different natures and different intrinsic scales (some variables can be akin to time, others to weights, ...) such that a unit change along each variable can have a completely different impact on the function optimized. More generally, the eigenspectrum of Q entirely characterizes the scale among the different axes of the hyper-ellipsoidal level sets and parametrizes the difficulty of the function: from the arguably easiest function, the sphere function f (x) = n i=1 x 2 i , to very difficult ill-conditioned functions where condition numbers of Q of up to 10 10 have been observed in real-world problems, for example in [START_REF] Collange | Multidisciplinary Optimisation in the Design of Future Space Launchers[END_REF].

Convex-quadratic functions have been central to the design of several important classes of optimization algorithms for single-objective optimization. Newton or quasi-Newton methods use or learn a second order approximation of the objective function optimized [START_REF] Nocedal | Numerical optimization[END_REF]. This second order approximation is done by convex-quadratic functions (assuming that the function is twice continuously differentiable and convex). Introduced more recently, the class of derivative-freeoptimization (DFO) trust-region based algorithms builds a second-order approximation of the objective function by interpolation [START_REF] Powell | The NEWUOA software for unconstrained optimization without derivatives[END_REF]. In the evolutionary computation (EC) context, convex-quadratic functions have also played a central role for the design of algorithms like CMA-ES: they have been intensively used for designing the algorithm and the performance of the method has been carefully quantified on different eigenspectra of the matrix Q for different condition numbers [START_REF] Hansen | Completely derandomized self-adaptation in evolution strategies[END_REF].

Given that a multiobjective problem is "simply" the simultaneous optimization of single-objective problems, the typical difficulties of each objective function are the same as the typical difficulties of single-objective problems. In particular non-separability and ill-conditioning are important difficulties that the single functions have. Therefore, combining convex-quadratic problems seems natural for testing and designing multiobjective algorithms. This has already been done in the past for instance for the design of multiobjective versions of CMA-ES [START_REF] Igel | Covariance matrix adaptation for multi-objective optimization[END_REF] or as a subset of the biobjective BBOB test function suite [START_REF] Brockhoff | Benchmarking Numerical Multiobjective Optimizers Revisited[END_REF][START_REF] Tušar | COCO: the bi-objective black box optimization benchmarking (bbob-biobj) test suite[END_REF].

Yet, while the difficulties encoded and parametrized within a convex-quadratic problem are well-understood for single-objective optimization, the situation is different for multiobjective optimization, starting from bi-objective optimization. Simple properties like convexity of the Pareto front associated to bi-objective convex-quadratic problems as well as properties of the Pareto set have not been systematically investigated. Additionally, convex-quadratic bi-objective test problems used in the literature do not capture all important properties one could be testing with convex-quadratic problems. There is more degree of freedom than for single objective optimization that is not exploited: we can combine two functions having the same Hessian matrix, place the optima on the functions both on one axis of the search space, ... and this will affect how the Pareto set and Pareto front look like.

This paper aims at filling the gaps from the literature on multiobjective optimization with respect to convex-quadratic problems. More precisely the objectives are twofold: clarify theoretical Pareto properties of bi-objective problems where each function is convex-quadratic and define sets of bi-objective convexquadratic problems that allow to test different (well-understood) difficulties of bi-objective problems. The paper is organized as follows: in Section 2 we present theoretical properties of convex-quadratic problems and discuss new test functions in Section 3.

Theoretical Properties of Bi-Objective

Convex-Quadratic Problems

Preliminaries

We consider bi-objective problems (f 1 , f 2 ) defined on the search space R n . The Pareto set of (f 1 , f 2 ) is defined as the set of all non-dominated (or efficient) solutions {x ∈ R n | ∃y ∈ R n such that f 1 (y) ≤ f 1 (x) and f 2 (y) ≤ f 2 (x) and at least one inequality is strict}. The image of the Pareto set (in the objective space R 2 ) is called the Pareto front of (f 1 , f 2 ). We first remark that the Pareto set remains unchanged if we compose the objective functions with a strictly increasing function. More precisely the following lemma holds.

Lemma 1 (Invariance of the Pareto set to strictly increasing transformations of the objectives). Given a bi-objective problem x → (f 1 (x), f 2 (x)) and g 1 :

Im(f 1 ) -→ R, g 2 : Im(f 2 ) -→ R two strictly increasing functions, then (f 1 , f 2 ) and (g 1 • f 1 , g 2 • f 2 ) have the same Pareto set. Proof. If x is not in the Pareto set of (g 1 • f 1 , g 2 • f 2 ), then their exists y such that g 1 • f 1 (y) ≤ g 1 • f 1 (x) and g 2 • f 2 (y) ≤ g 2 • f 2 (x)
with one inequality being strict, which is equivalent to the fact that f 1 (y) ≤ f 1 (x) and f 2 (y) ≤ f 2 (x), with one inequality being strict. And vice versa. Hence x is not in the Pareto set of (g 1 • f 1 , g 2 • f 2 ) if and only if it is not in the Pareto set of (f 1 , f 2 ), which shows that both problems have the same Pareto set.

From now on (f 1 , f 2 ) denote a bi-objective convex-quadratic problem. More precisely, let x 1 , x 2 be two different vectors in R n , and α, β > 0. Let Q 1 and Q 2 (in R n 2 ) be two positive definite matrices and consider the bi-objective minimization problem (f 1 , f 2 ) defined for x ∈ R n as

f 1 (x) = 1 α (x -x 1 ) Q 1 (x -x 1 ) , f 2 (x) = 1 β (x -x 2 ) Q 2 (x -x 2 ) . (1)
We denote this general bi-objective convex-quadratic problem by P, and assume that the optimization goal is to find (an approximation of) the Pareto set of P.

Pareto set

We characterize in this section the Pareto set of P. We use the linear scalarization method to obtain the whole Pareto set. This is doable, whenever f 1 and f 2 are strict convex functions (see [START_REF] Jahn | Vector optimization: Theory, applications and extensions[END_REF]). Then the Pareto set of P is described by the solutions of

min x∈R n (1 -t) f 1 (x) + tf 2 (x) , for t ∈ [0, 1]
. We prove in the next proposition that the Pareto set of P is a continuous and differentiable parametric curve of R n whose extremes are x 1 and x 2 .

Proposition 1. The Pareto set of P is the image of the function ϕ defined as

ϕ : t ∈ [0, 1] → [(1 -t)Q 1 + tQ 2 ] -1 [(1 -t)Q 1 x 1 + tQ 2 x 2 ] . (2)
The function ϕ is differentiable and verifies for any t in [0, 1]

(1 -t)Q 1 (ϕ(t) -x 1 ) = tQ 2 (x 2 -ϕ(t)) , (3)
t [(1 -t)Q 1 + tQ 2 ] ϕ (t) = Q 1 (ϕ(t) -x 1 ) . (4)
Hence, the Pareto set is a continuous (differentiable) curve of R n whose extremes are x 1 = ϕ(0) and x 2 = ϕ(1).

Proof. For any s in [0, 1], define g

s def = (1 -s)f 1 + sf 2 .
We observe that g s , like f 1 and f 2 , is strictly convex, differentiable, and diverges to ∞ when x goes to ∞ (where x denotes the Euclidean norm). Then its critical point minimizes g s . Let us now compute the gradient of g s times αβ for x in R n :

αβ∇g s (x) = (1-s)αβ ∇f 1 (x)+sαβ ∇f 2 (x) = 2(1-s)β Q 1 (x-x 1 )+2sα Q 2 (x-x 2 ) Thus, αβ∇g s (x) = 2 [(1 -s)β Q 1 + sα Q 2 ] x -2(1 -s)β Q 1 x 1 -2sα Q 2 x 2 .
Then it follows that for any s in [0, 1], the point that minimizes g s (its critical point), denoted by xs verifies

(1 -s)β Q 1 + sα Q 2 (1 -s)β + sα xs = (1 -s)β Q 1 x 1 + sα Q 2 x 2 (1 -s)β + sα . Since [0, 1] s -→ sα (1 -s)β + sα ∈ [0, 1] is bijective (its derivative is s -→ αβ ((1 -s)β + sα)
2 ), then it is equivalent to parametrize the Pareto set with

t def = sα (1 -s)β + sα
. Hence, the Pareto set is fully described by (ϕ(t)) t∈[0,1] such that:

[(1 -t)Q 1 + tQ 2 ] ϕ(t) = (1 -t)Q 1 x 1 + tQ 2 x 2 , (5)
(

1 -t)Q 1 (ϕ(t) -x 1 ) = tQ 2 (x 2 -ϕ(t)) . (6)
The

function t → [(1 -t)Q 1 + tQ 2 ]
-1 is differentiable as inverse of a differentiable and invertible matrix function. Then ϕ is differentiable. We differentiate [START_REF] Grodzevich | Normalization and other topics in multi-objective optimization[END_REF] and multiply by t

to obtain t [(1 -t)Q 1 + tQ 2 ] ϕ (t) = tQ 2 x 2 -tQ 1 x 1 +tQ 1 ϕ(t)-tQ 2 ϕ(t). Injecting in (6) gives t [(1 -t)Q 1 + tQ 2 ] ϕ (t) = Q 1 (ϕ(t) -x 1 ) , for any t ∈ [0, 1].
We obtain as corollary that when f 1 and f 2 have proportional Hessian matrices, then the Pareto set is the line segment between the optima of the functions f 1 and f 2 .

Corollary 1. In the case where f 1 and f 2 have proportional Hessian matrices, the Pareto set of P is the line segment between x 1 and x 2 .

Proof. In that case, their exists a real γ such that

Q 1 α = γ Q 2 β .
Then, Proposition 1 implies that for any t ∈ [0, 1],

ϕ(t) = (1 -t)γ α β Q 2 + tQ 2 -1 (1 -t)γ α β Q 2 x 1 + tQ 2 x 2 = γα(1 -t)x 1 + tβx 2 (1 -t)αγ + tβ , which is [x 1 , x 2 ], since [0, 1] t -→ tβ (1-t)αγ+tβ ∈ [0, 1] is a bijection.
Using Lemma 1, we directly deduce the following corollary.

Corollary 2. If f 1 and f 2 have proportional Hessian matrices, g 1 : Im(f 1 ) -→ R, g 2 : Im(f 2 ) -→ R are two strictly increasing functions, then the Pareto set of the problem (g

1 • f 1 , g 2 • f 2 )
is the line segment between x 1 and x 2 .

As an example, the double-norm problem defined as:

(x → x -x 1 2 , x → x -x 2 2 ) can be seen as: (g •f 1 , g •f 2 ) where g(x) = √ x, f 1 (x) = x -x 1 2 2 and f 2 (x) = x -x 2 2 2 . Then (g • f 1 , g • f 2
) has the same Pareto set than the double-sphere problem (f 1 , f 2 ), which is the line segment between x 1 and x 2 . Therefore the Pareto front of the double-norm problem is described by (t

x 2 -x 1 2 , (1 -t) x 2 -x 1 2 ) t∈[0,1] .
Thereby, the front is described by the function u -→ x 2 -x 1 2 -u. We recover the well-known result that the double-norm problem has a linear front.

Corollary 2 allows also to recover the Pareto set description for the one-peak scenario in the Mixed-Peak Bi-Objective Problem (see [START_REF] Kerschke | Search dynamics on multimodal multi-objective problems[END_REF] and [START_REF] Kerschke | Towards analyzing multimodality of continuous multiobjective landscapes[END_REF]). 1 In general, the Pareto set of a bi-objective convex-quadratic problem is not necessarily a line segment. Consider for instance for n = 2 the case where x 1 = (0, 0) , x 2 = (1, 1) and where we generate two different matrices Q 1 and Q 2 by randomly rotating a diagonal matrix with eigenvalues 1 and 10. Two resulting Pareto fronts associated to different random rotations are depicted in Figure 1.

For n = 10, we also define P 10 setting x 1 = (0, . . . , 0) , x 2 = (1, . . . , 1) and Q 1 and Q 2 as diagonal matrices such that for i = 1, . . . , 10

Q 1 (i, i) = 100 i-1 9 , and Q 2 (i, i) = 10 i-1 9 . (7)
The different coordinates of the Pareto set given in (3) are depicted in Figure 1.

Convexity of the Pareto front

Corollary 1 proves that in the case where we have proportional Hessian matrices in problem P, the Pareto set is a line segment. Then it is reasonable to expect a simple analytic expression for the corresponding Pareto front. In what follows, we will express the Pareto front of a bi-objective problem as a one-dimensional 1 In that scenario, we set f1(x) = (x -c) Σ (x -c), f2(x) = (x -c ) Σ (x -c ) (f1 and f2 are seen as squares of the Mahalanobis distance to the optima, with respect to the Hessian matrices), g1(u) = 1 -h1

1 + √ u r 1 , g2(u) = 1 - h2 1 + √ u r 2 .
function u ∈ R → g(u). Formally, if t ∈ R → ϕ(t) ∈ R n is a parametrization of the Pareto set, then the function g satisfies f 2 (ϕ(t)) = g(f 1 (ϕ(t)). It is well- known that when (f 1 , f 2 ) is the double-sphere, that is f 1 (x) = 1 n n i=1 x 2 i and f 2 (x) = 1 n n i=1 (x i -1) 2 , then the Pareto front expression is given by g(u) = (1 - √ u) 2 [4].
In the next proposition, we show that this expression of the Pareto front holds (up to a normalization) for all bi-objective convex-quadratic problems, provided the Hessians of f 1 and f 2 are proportional.

Proposition 2. When we have proportional Hessian matrices in the problem P, the Pareto front is described by the following continuous and convex function:

u ∈ [0, κ α ] → κ β 1 - u κ α 2
, where

       κ α = (x 2 -x 1 ) Q 1 (x 2 -x 1 ) α κ β = (x 2 -x 1 ) Q 2 (x 2 -x 1 ) β (8)
Proof. Denote

u def = f 1 •ϕ and v def = f 2 •ϕ, where ϕ : [0, 1] t -→ (1-t)x 1 +tx 2 ∈ [x 1 , x 2 ] is the line segment between x 1 and x 2 . For any t ∈ [0, 1], u(t) = f 1 (ϕ(t)) = 1 α (x 2 -x 1 ) Q 1 (x 2 -x 1 ) t 2 , v(t) = f 2 (ϕ(t)) = 1 β (x 2 -x 1 ) Q 2 (x 2 -x 1 ) (1 -t) 2 .
It follows that for any t ∈ [0, 1]:

v(t) = (x 2 -x 1 ) Q 2 (x 2 -x 1 ) β 1 - αu(t) (x 2 -x 1 ) Q 1 (x 2 -x 1 ) 2 .
From Proposition 2, we deduce that if we set κ α = κ β = 1, then the Pareto front will be independent from the Hessian matrix and will be described by the front of the double-sphere problem:

u → (1 - √ u) 2 .
We investigate now the general case where the Hessians of the functions f 1 and f 2 are not necessarily proportional. Yet, before digging into the general convex-quadratic problems, we show a result on the shape of the Pareto front of a larger class of bi-objective problems.

Theorem 1. Let f 1 : R n -→ R and f 2 : R n -→ R be strict convex differentiable functions such that the problem (f 1 , f 2 ) has, as Pareto set, the image of a differentiable function ϕ :

[0, 1] -→ R n . Assume that: (i) f 1 • ϕ is strictly monotone, (ii) lim t→0 (f 1 • ϕ) (t) t = 0 and (iii) lim t→1 (f 2 • ϕ) (t) 1 -t = 0.
Then, the Pareto front is a convex curve, with vertical tangent at t = 0 and horizontal tangent at t = 1.

Proof. Denote by

u def = f 1 • ϕ and v def = f 2 • ϕ.
Then the Pareto front is described by the parametric equation (u(t), v(t)) , for t ∈ [0, 1]. We will show that u vu v > 0 which implies the convexity of the curve.

By linear scalarization (see [START_REF] Jahn | Vector optimization: Theory, applications and extensions[END_REF], or weighted sum method in [START_REF] Grodzevich | Normalization and other topics in multi-objective optimization[END_REF]), as in the proof of Proposition 1, we have (1 -t)∇f 1 (ϕ(t)) + t∇f 2 (ϕ(t)) = 0. If we take the scalar product of the former equation with ϕ (t), we obtain that

(1 -t) ∇f 1 (ϕ(t)) , ϕ (t) + t ∇f 2 (ϕ(t)) , ϕ (t) = 0. (9)
Moreover, for any differentiable function f with suitable domains,

(f • ϕ) (t) = d (f • ϕ) t (1) = df ϕ(t) (dϕ t (1)) = ∇f (ϕ(t)) , ϕ (t) . (10)
Inserting this in [START_REF] Kerschke | Search dynamics on multimodal multi-objective problems[END_REF] shows (1

-t) (f 1 • ϕ) (t) + t (f 2 • ϕ) (t) = 0
, which is the same as:

(1 -t)u (t) + tv (t) = 0, for any t ∈ [0, 1]. [START_REF] Nocedal | Numerical optimization[END_REF] Since lim [START_REF] Nocedal | Numerical optimization[END_REF] implies that:

t→0 (f 1 • ϕ) (t) t exists,
v (t) = 1 - 1 t u (t), for any t ∈ [0, 1]. ( 12
)
By deriving [START_REF] Powell | The NEWUOA software for unconstrained optimization without derivatives[END_REF] and multiplying by u (t) in a suitable way, we obtain

u (t)v (t) = 1 t 2 u (t) 2 + 1 - 1 t u (t)u (t), for any t ∈ [0, 1]. ( 13
)
Using ( 12) in [START_REF] Tušar | COCO: the bi-objective black box optimization benchmarking (bbob-biobj) test suite[END_REF] gives

u (t)v (t) = 1 t 2 u (t) 2 + v (t)u (t)
. Thanks to the assertions on f 1 •ϕ, we have that u (t)v (t)-u (t)v (t) > 1 t 2 u (t) 2 > 0, for any t ∈ [0, 1]. Thus, the Pareto front is a convex curve. Evaluating (11) at t = 0 and at t = 1 implies that u (0) = 0, v (1) = 0. And if we divide (11) by t (resp. 1 -t) and take the limit to 0 (resp. 1), it follows that v (0) = 0 (resp. u (1) = 0). Thereby we also obtain the derivative assumptions on the extremal points.

Remark 1. Note that the above result about the tangents in the extremal points have additional consequences: according to [START_REF] Auger | Theory of the hypervolume indicator: optimal µ-distributions and the choice of the reference point[END_REF], the assumptions of Theorem 1 imply that the extremal points are never included in any optimal µ-distributions of the Hypervolume indicator.

We now deduce the convexity of the Pareto front for convex-quadratic biobjective problems and characterize the derivatives at the extremes of the front.

Corollary 3. For the problem P, the Pareto front is a convex curve, with vertical tangent at (0, f 2 (x 1 )) and horizontal tangent at (f 1 (x 2 ), 0).

Proof. We will show that f 1 •ϕ verifies the assumptions of Theorem 1. From [START_REF] Kerschke | Towards analyzing multimodality of continuous multiobjective landscapes[END_REF] we know that

(f 1 • ϕ) (t) = ∇f 1 (ϕ(t)) , ϕ (t) . (14)
In addition, ∇f 1 (ϕ(t)) = 2 α Q 1 (ϕ(t) -x 1 ) and Eq. ( 4) Proposition

1 gives t [(1 -t)Q 1 + tQ 2 ] ϕ (t) = Q 1 (ϕ(t) -x 1 ) . Multiplying (14) by t ∈ [0, 1] shows t (f 1 • ϕ) (t) = 2 α [(1 -t)Q 1 + tQ 2 ] -1 Q 1 (ϕ(t) -x 1 ) , Q 1 (ϕ(t) -x 1 ) .(15) Since [(1 -t)Q 1 + tQ 2 ] -1 is a positive definite matrix, then t (f 1 • ϕ) (t) ≥ 0. Let us prove that ϕ(t) = x 1 , for t ∈ (0, 1]. By contradiction, assume that there exists t ∈ (0, 1] such that ϕ(t) = x 1 . Then Equation (3) in Proposition 1 shows that: tQ 2 (x 2 -ϕ(t)) = (1-t)Q 1 (ϕ(t) -x 1 ) = 0, which implies that x 2 = ϕ(t) = x 1 : that is impossible since x 1 = x 2 .
Hence, by reductio ad absurdum, ϕ(t) = x 1 , for t ∈ (0, 1]. From (15), it follows that

(f 1 • ϕ) (t) > 0, for any t ∈ (0, 1]. ( 16
)
If we use again the relation from Proposition 1, we obtain lim

t→0 Q 1 (ϕ(t) -x 1 ) t = Q 2 (x 2 -ϕ(0)) = Q 2 (x 2 -x 1 )
. Injecting this result in (15), it follows that:

lim t→0 (f 1 • ϕ) (t) t = 2 α Q -1 1 Q 2 (x 2 -x 1 ) , Q 2 (x 2 -x 1 ) > 0, since (Q -1 1 is a positive definite matrix) (17)
In the same way as above, we obtain that

lim t→1 (f 2 • ϕ) (t) 1 -t = - 2 β Q -1 2 Q 1 (x 1 -x 2 ) , Q 1 (x 1 -x 2 ) < 0 . (18)
Equations ( 16), (17), and (18) allow us to apply Theorem 1.

We illustrate the previous corollary by taking three random instances of our general problem P, with the scalings always chosen as α = β = max (f 1 (x 2 ), f 2 (x 1 )). The Pareto fronts are presented in Figure 2. We observe that the Pareto fronts are convex and their derivatives are infinite on the left and zero on the right.

New Classes of bi-objective test functions

Bi-objective problems using convex-quadratic functions have been used to test MO algorithms (see for example [START_REF] Igel | Covariance matrix adaptation for multi-objective optimization[END_REF]). Problems where both Hessian matrices have the same eigenvalues have been used in particular. Yet, test problems considered so far do not explore full possibilities of properties that can be tested. We therefore extend the test problems from the literature to be able to capture more properties. To do so we present seven classes of bi-objective convex-quadratic problems where the eigenspectra of both Hessian matrices are equal. A natural extension of these classes is to use in each objective different eigenspectra, ∆, which leads in general to a nonlinear Pareto set.

The proposed construction parametrizes, apart from search space translations, all bi-objective convex-quadratic functions with identical Hessian eigenspectrum in seven classes with increasing difficulty. The particular focus is on problems with a linear Pareto set in five of the seven classes. Some classes represent essentially different problems, hence we do not expect uniform performance over all problems within each class. Independently of the given construction, invariance to search space rotation can be tested by applying an orthogonal transformation to the input argument.

We start from a diagonal matrix ∆ with positive entries that define a separable convex-quadratic function f (x) = 1 α x ∆x. For instance, ∆ can be equal to the identity and we recover the sphere function. If ∆(1, 1) = 1, ∆(n, n) = 10 8 and ∆(i, i) = 10 4 , we recover the separable cig-tab function and if ∆(i, i) = 10 6 i-1 n-1 , we recover the separable ellipsoid function.

In the sequel, O and O 2 denote orthogonal matrices. O 1 is either a permutation matrix, or an orthogonal matrix, depending on the context. The classes of problems proposed are summarized in Table 1 and Table 2.

The Sep problem classes We define the Sep-k class by considering two separable functions and place the optimum of f 1 in 0 and of f 2 in the k th unit vector:

f sep-k 1,∆ (x) = 1 α (x -x 1 ) ∆ (x -x 1 ) and f sep-k 2,∆ (x) = 1 β (x -x 2 ) ∆ (x -x 2 )
, where x 1 = (0, . . . , 0) and x 2 = (0, . . . , 0, √ n, 0, . . . , 0) where √ n is at coordinate k. According to Corollary 1, the Pareto set of this class of problems is the line segment between the optima of the single-objective problems. These problems allow to test the performance on separable problems with a Pareto set aligned with the coordinate axis and check the sensibility with respect to different axes (by varying k).

For the Sep-O class, we only change the location of the optimum of the second objective by taking x 2 = O(1, . . . , 1) . If O has elements ∈ {-1, 0, 1}, the Pareto set is not anymore aligned with the coordinate system, but the objectives f 1 and f 2 themselves remain separable. Comparing with class Sep-k, we can test whether having the Pareto set not aligned with the coordinate axis has an influence on the performance of the algorithm.

For the Sep-Two-O class, we define

f sep-Two-O 1,∆ (x) = 1 α (x -x 1 ) ∆ (x -x 1 ) and f sep-Two-O 2,∆ (x) = 1 β (x -x 2 ) O 1 ∆O 1 (x -x 2 )
where O 1 is a permutation matrix, x 1 = (0, . . . , 0) and x 2 = O(1, . . . , 1) . The matrix O 1 ∆O 1 is also diagonal, and thereby each function is separable. Yet the Pareto set is generally not a line segment anymore since we have different Hessian matrices. We can test here the difficulty of having a nonlinear Pareto set on separable functions.

The One and the One-O problem classes. We now consider non-separable problems with a line segment as Pareto set. We define

f one 1,∆ (x) = 1 α (x -x 1 ) O 1 ∆O 1 (x -x 1 ) and f one 2,∆ (x) = 1 α (x -x 2 ) O 1 ∆O 1 (x -x 2 )
, where O 1 is an orthogonal matrix, x 1 = (0, . . . , 0) and x 2 = (1, . . . , 1) . We replace x 2 by Ox 2 to obtain the One-O problems.

These two problem classes allow to test the performance on non-separable problems that have a line segment as Pareto set comparing in particular to class Sep-O. Up to a reformulation, the problems ELLI1 and CIGTAB1 from [START_REF] Igel | Covariance matrix adaptation for multi-objective optimization[END_REF] are from the One-O problem class. Generally, we do not expect different performance over all problems of the One vs the One-O class.

The Two and the Two-O problem classes. For these classes, we rotate each function independently; then the Pareto set is generally not a line segment anymore. We define f two 1,∆ (x) = 1 α (x -x 1 ) O 1 ∆O 1 (x -x 1 ) and f two 2,∆ (x) =

Summary

We have presented an analytic description of the Pareto set for quadratic biobjective problems. We have shown that the Pareto set is a line segment when both objectives have proportional Hessian matrices and deduced a complete description of the Pareto front in that case. We have also proven that some properties of the double-sphere are conserved in a wider framework that includes the general quadratic bi-objective problem: the Pareto front remains convex and its vertical and horizontal tangents remain at the extremal points of the front. Such assumptions on the derivatives imply that when looking at the optimal µ-distributions of the Hypervolume indicator, the extremal points are always excluded [START_REF] Auger | Theory of the hypervolume indicator: optimal µ-distributions and the choice of the reference point[END_REF]. We have also presented several classes of problems, where each one tests a specific capability of the multiobjective algorithm.

Fig. 1 :

 1 Fig. 1: Left: Two Pareto sets for n = 2 represented in R 2 with Q 1 and Q 2 randomly sampled and different. Right: Pareto set for n = 10 with matrices given in (7) represented as the function of the parameter t given in (3). The coordinates are ordered, the first one is on top and last one below.

Fig. 2 :

 2 Fig. 2: Left: Two Pareto fronts for n = 2 represented in R 2 with Q 1 , Q 2 randomly sampled and different. Right: Pareto front for n = 10 with matrices given in (7).

1 α (x -x 2 )

 12 O 2 ∆ O 2 (x -x 2 ), with O 1 orthogoanal, x 1 = (0, . . . , 0) and x 2 = (1, . . . , 1) . The corresponding O problems are obtained with Ox 2 replacing x 2 . All presented classes are subsets of the Two-O class. ELLI2 and CIGTAB2 from[START_REF] Igel | Covariance matrix adaptation for multi-objective optimization[END_REF] fall within the Two-O class. Compared to the respective One classes, we can test the impact of having a nonlinear Pareto set.

Table 1 :Table 2 :

 12 Unconstrained quadratic bi-objective test problems: ∆ is a positive diagonal matrix, O is an orthogonal matrix, O 1 is a permutation matrix. Unconstrained quadratic bi-objective test problems: ∆ is a positive diagonal matrix, O, O 1 and O 2 are three independent orthogonal matrices. . . . , 0) (0, . . . , 0) (0, . . . , 0) (0, . . . , 0) x2 (1, . . . , 1) O(1, . . . , 1) (1, . . . , 1) O(1, . . . , 1)Q1, Q2 O 1 ∆O1, O 1 ∆O1 O 1 ∆O1, O 1 ∆O1 O 1 ∆O1, O 2 ∆O2 O 1 ∆O1, O 2 ∆O2Level sets

Acknowledgments The Ph.D. of Cheikh Touré is funded by Inria and Storengy. We particularly thank F. Huguet and A. Lange from Storengy for their strong support, practical ideas and expertise.