### November 2012

While this blog is intended to be totally job-related, this post will be unusual in that it will be a little bit personal. And indeed, this post is about the mix of work and non-work life.

A little more than half a year ago I started my life as a part-time mathematician. To be more precise, I am in Elternzeit, which is the German framework for parental leave. According to German law, parents have several rights related to their work when they get kids. One of them is to reduce their weekly hours of work when they have small children and this is what I did: For one year I reduced my weekly hours of work to 50%. The reason for the reduction is that my wife has a very busy year of work (for those who know: she is in her “Referendariat” to become a teacher – but I am not going to rant about the German Referendariat for teachers here…). There is also Elterngeld is Germany: That is, that you get some fraction of your salary if you stay at home to be with your kids, but this only applies in the first 14 months but my kids are older and hence, I just get about half the salary during the Elternzeit. But one of the first things I learned was that this is not totally true. In fact, my “parental-leave”-salary is higher than half of my usual salary. This has two reasons: First, there is “income-tax-progression” here, that is, for more salary, you pay a higher tax rate. Second, some parts of my salary are bonuses which are not halved.

Besides the financial issues there are other things about parental leave as a mathematician at a university which I find interesting and worth blogging about. I divided these issues into two parts: work-related and family related.

1. Work life

In this year of parental leave I am in my office for about four hours a day. I go there, when the kids are in school and at day-care and I leave early enough to prepare lunch for our schoolgirl. Four hours a day is not much and indeed, I get much less things done, than I used to do. But that was totally expected. To counteract a bit, I do a little less reviewing papers. Moreover, my teaching duties are halved. And also, blogging is a little bit reduced. But anyway, I do not make it to get all the things done, that I would like to. However, this is not much different from how it used to be. And with even less time, I get better in prioritizing.

But more important than the mere reduction of time for work is the fact that I am not able to be in the office in the afternoon, and I am not able to make appointments or to attend meetings in the afternoon. This does influence my daily work to some extend. While all my colleagues know that I am on part-time (and, as far as I experience, have no objections to that), they usually do not have it in mind when they make dates. If I am lucky, dates are found with doodle, and then it’s easy for me, but I have to remind people that I do not have time in the afternoon frequently. Moreover, some regular meetings are scheduled for the afternoon by default and I just miss them. Finally, there are exceptional dates (e.g. the ones related with job interviews and negotiations on which I have written here and here). In these cases, I also mentioned my limited possibilities for meetings and the severe restrictions. For the job interviews I even asked for a video-interview, but that did not work out. On one occasion, they tried to schedule my job interview for the weekend (which would be easier to arrange for me) but some members of the committee objected due to family reasons – which is totally OK in my opinion. For the negotiations, I tried to have them during school holidays but this did not work out either. Hence, we had to arrange additional child care (our children have four grand parents and always some of them could step in – what I didn’t try was, to get reimbursed for the train tickets for the grandmas; I think this would be reasonable but would cause serious irritation in the administration).

Another severe change in my work-life is that I can not attend conferences and meetings with the only exception when they are during school holidays. Here we arranged holidays of the kids at their grandparents and this way I could attend some nice meetings this year.

2. Family life

In my parental leave my wife and I changed roles. It is not that I did not care about family life, household and such when I was working full-time but still, I was working full-time and my wife was working half-time or less. Since she has to work a LOT, I am in charge of everything here at home now. But, as before, it is not that she is not doing anything in that respect, but the important thing is that I am in charge. And that makes more of a difference than I anticipated. The pure amount of work doing shopping, cooking, cleaning and so on is OK because I have the time to do so (although – it is work nonetheless). But to be in charge is a bit more. Especially, if there is something to be done, I can not say “Let’s wait for one day, maybe it will be done by the time…”. If I will not do it, nobody will (again: Not totally true, but this is the way of thinking which I employ). Moreover, there are some things which come with being “house husband” which did not come to my mind when I planned my part time year: organizing children’s birthday parties, arranging new afternoon activities and weekend activities or trips, taking care of seasonal decoration, thinking of Christmas presents early enough or arranging routine medical check-ups on time are just a few of them. All this is also not hard, but harder than I thought, when you are in charge.

In total, all this role change and house-keeping is a very refreshing experience. Being a house husband is not something be scared of. It is rewarding, it’s mostly fun, it’s a lot of work and it is quite some responsibility.

As a final remark, I think, that this year of my parental leave does also mean something for our children. Seeing that dad is at home and being the one which is always there, and that both parents can do this job and that, the other way round, mom is at work most of the day, comes home late and needs rest and such, is an experience for them which shows them that there are different ways to organize a family. It probably also influences them in the way they see gender roles.

In conclusion: If you think about staying at home for kids and the laws of your country give you the opportunity to do so, then I can recommend it. Even being fully in charge and with a full-time working wife – probably even more so.

There are several answers to the following question:

1. What is a convex set?

For a convex set you probably know these definitions:

Definition 1 A subset ${C}$ of a real vector space ${V}$ is convex if for any ${x,y\in C}$ and ${\lambda\in[0,1]}$ it holds that ${\lambda x + (1-\lambda)y\in C}$.

In other words: If two points lie in the set, then every convex combination also lies in the set.

While this is a “definition from the inside”, convex sets can also be characterized “from the outside”. We add closedness as an assumption and get:

Definition 2 A closed subset ${C}$ of a real locally convex topological vector space ${V}$ is convex if it is the intersection of closed halfspaces (i.e. sets of the form ${\{x\in V\ :\ \langle a,x\rangle\geq c\}}$ for some ${a}$ in the dual space ${V^*}$ and ${c\in {\mathbb R}}$).

Moreover, we could define convex sets via convex functions:

Definition 3 A set ${C\subset V}$ is convex if there is a convex function ${f:V\rightarrow {\mathbb R}}$ such that ${C = \{x\ :\ f(x)\leq 0\}}$.

Of course, this only makes sense once we have defined convex functions. Hence, we could also ask the question:

2. What is a convex function?

We can define a convex function by means of convex sets as follows:

Definition 4 A function ${f:V\rightarrow{\mathbb R}}$ from a real vector space into the real numbers is convex, if its epigraph ${\text{epi} f = \{(x,\mu)\ :\ f(x)\leq \mu\}\subset V\times {\mathbb R}}$ is convex (as a subset of the vector space ${V\times{\mathbb R}}$).

The epigraph consists of the points ${(x,\mu)}$ which lie above the graph of the function and carries the same information as the function.

(Let me note that one can replace the real numbers here and in the following with the extended real numbers ${\bar {\mathbb R} = {\mathbb R}\cup\{-\infty,\infty\}}$ if one uses the right extension of the arithmetic and the obvious ordering, but we do not consider this in this post.)

Because epigraphs are not arbitrary convex sets but have a special form (if a point ${(x,\mu)}$ is in an epigraph, then every ${(x,\lambda)}$ with ${\lambda\geq \mu}$ is also in the epigraph), and because the underlying vector space ${V\times {\mathbb R}}$ comes with an order in the second component, some of the definitions for convex sets from above have a specialized form:

From the definition “convex combinations stay in the set” we get:

Definition 5 A function ${f:V\rightarrow {\mathbb R}}$ is convex, if for all ${x,y\in V}$ and ${\lambda\in [0,1]}$ it holds that

$\displaystyle f(\lambda x + (1-\lambda)y) \leq \lambda f(x) + (1-\lambda)f(y).$

In other words: The secant of any two points on the graph lies above the graph.

From the definition by “intersection of half spaces” we get another definition. Since we added closedness in this case we add this assumption also here. However, closedness of the epigraph is equivalent to lower-semicontinuity (lsc) of the function and since lsc functions are very convenient we use this notion:

Definition 6 A function ${f:V\rightarrow{\mathbb R}}$ is convex and lsc if it is the pointwise supremum of affine functions, i.e., for some set ${S}$ of tuples ${(a,c)\in V^*\times {\mathbb R}}$ it holds that

$\displaystyle f(x) = \sup_{(a,c) \in S} \langle a,x\rangle + c.$

A special consequence if this definition is that tangents to the graph of a convex function lie below the function. Another important consequence of this fact is that the local behavior of the function, i.e. its tangent plane at some point, carries some information about the global behavior. Especially, the property that the function lies above its tangent planes allows one to conclude that local minima of the function are also global. Probably the last properties are the ones which give convex functions a distinguished role, especially in the field of optimization.

Some of the previous definitions allow for generalizations into several direction and the quest for the abstract core of the notion of convexity has lead to the field of abstract convexity.

3. Abstract convexity

When searching for abstractions of the notion of convexity one may get confused by the various different approaches. For example there are generalized notions of convexity, e.g. for function of spaces of matrices (e.g. rank-one convexity, quasi-convexity or polyconvexity) and there are also further different notions like pseudo-convexity, invexity or another form ofquasi-convexity. Here we do not have generalization in mind but abstraction. Although both things are somehow related, the way of thinking is a bit different. Our aim is not to find a useful notion which is more general than the notion of convexity but to find a formulation which contains the notion of convexity and abstracts away some ingredients which probably not carry the essence of the notion.

In the literature one also finds several approaches in this direction and I mention only my favorite one (and I restrict myself to abstractly convex functions and not write about abstractly convex sets).

To me, the most appealing notion of an abstract convex function is an abstraction of the definition as “pointwise supremum of affine functions”. Let’s look again at the definition:

A function ${f:V\rightarrow {\mathbb R}}$ is convex and lsc if there is a subset ${S}$ of ${V^*\times {\mathbb R}}$ such that

$\displaystyle f(x) = \sup_{(a,c)\in S} \langle a,x\rangle + c.$

We abstract away the vector space structure and hence, also the duality, but keep the real-valuedness (together with its order) and define:

Definition 7 Let ${X}$ be a set and let ${W}$ be a set of real valued function on ${X}$. Then a function ${f:X\rightarrow{\mathbb R}}$ is said to be ${W}$-convex if there is a subset ${S}$ of ${W\times {\mathbb R}}$ such that

$\displaystyle f(x) = \sup_{(w,c)\in S} w(x) + c.$

What we did in this definition was simply to replace continuous affine functions ${x\mapsto \langle a,x\rangle}$ on a vector space by an arbitrary collection of real valued functions ${x\mapsto w(x)}$ on a set. On sees immediately that for every function ${w\in W}$ and any real number ${c}$ the function ${w+c}$ is ${W}$ convex (similarly to the fact that every affine linear function is convex).

Another nice thing about this approach is, that it allows for some notion of duality/conjugation. For ${f:V\rightarrow{\mathbb R}}$ we define the ${W}$-conjugate by

$\displaystyle f^{W*}(w) = \sup_{w\in W} \Big(w(x)- f(x) \Big)$

and we can even formulate a biconjugate

$\displaystyle f^{W**}(x) = \sup_x \Big(w(x) - f^{W*}(w)\Big).$

We naturally have a Fenchel inequality

$\displaystyle w(x) \leq f(x) + f^{W*}(w)$

and we may even define subgradients as usual. Note that a conventional subgradient is an element of the dual space which defines a tangent plane at the point where the subgradient is taken, that is, ${a\in V^*}$ is a subgradient of ${f}$ at ${x}$, if for all ${y\in V}$ it holds that ${f(y) \geq f(x) + \langle a,y-x\rangle}$ or

$\displaystyle f(y) - \langle a,y\rangle\geq f(x) - \langle a,x\rangle.$

A ${W}$-subgradient is an element of ${W}$, namely we define: ${w\in\partial^{W}f(x)}$ if

$\displaystyle \text{for all}\ y:\quad f(y) -w(y) \geq f(x) - w(x).$

Then we also have a Fenchel equality:

$\displaystyle w\in\partial^W f(x)\iff w(x) = f(x) + f^{W*}(w).$

One may also take dualization as the starting point for an abstraction.

4. Abstract conjugation

We could formulate the ${W}$-conjugate as follows: For ${\Phi(w,x) = w(x)}$ we have

$\displaystyle f^{W*}(x) = \sup_w \Big(\Phi(w,x) - f(x)\Big).$

This opens the door to another abstraction: For some sets ${X,W}$ (without any additional structure) define a coupling function ${\Phi: X\times W \rightarrow {\mathbb R}}$ and define the ${\Phi}$-conjugate as

$\displaystyle f^{\Phi*}(w) = \sup_x \Big(\Phi(w,x) - f(x)\Big)$

and the ${\Phi}$-biconjugate as

$\displaystyle f^{\Phi**}(x) = \sup_x \Big(\Phi(w,x) - f^{W*}(w)\Big)$