Now we're going to understand the specific behavior in distributed power control,

from two angles now. And in that sense we're using DPC as a way

also to introduce us to the terminology, and the basic notions in two powerful

mathematical modeling languages. One is, the language for, constraint

decision making called optimization theory.

And the other, is the language for strategic thinking by intelligent agent

called game theory. Now lets start with optimization theory.

This is a word that we use in our daily language.

Alright. We try to optimize something.

We try to optimize our time, our, holiday schedule, our work schedule.

And if you think about it. In that optimization, you have some degree

of freedom, which we will mathematically call optimization variables.

You have some objectives you wanna maximize or minimize.

For example, maximize your employability. Okay.

Your happiness. Minimize the cost it takes to finish some

task. And then you have some constraints.

Without constraints the problem will be too good to be true.

This could be constraints, on the time you have on the money you have on energy you

have. For example when you look at your schedule

this weekend. You may say, well I can pick my variable

to be spend the time taking a course, like this one.

Or spend the time to watch a movie. An objective function may be, well, make

me as happy as possible. And the constraint, say, is you only have

24 hours in each day. Now we will see a mathematically precise

unambiguous language called optimization theory.

And in each optimization problem, there are four main data fields.

One is, of course, the objective. Now, in the case of transmit power control

in cellular networks, the objective is to minimize the sum of the individual

transmit powers. So we will write, minimize the sum of the

PIs over Is. So, this is a power minimal configuration,

subject to the constraints that you have to achieve, the target SIRs for all the

users. The SIRs for each user indexed by i must

be no smaller than the target gamma i, and you need this to be hold, to be held not

just for single i but for all i. And then you need to vary the degree of

freedom, in this case obviously it is the set of transmit powers.

And everything else are constants including the channel condition GIJs.

The noises ni for each receiver and the target SIRs gamma.

Towards end of this lecture we also see what will happen if these target SIR

values become variables as well. But for now they are held constant.

Now once you have an objective function, a set of constraints.

And you know what is variable and what is constant, then you have an optimization

problem. Pictorially, what we are looking at in

this case of transmitter power control is visualizable in the following cartoon.

Now suppose we have just two users, so can draw in the 2-dimensional plane in a piece

of paper or slide. And the first user SIR is on the X axis,

the second on the Y axis. And we have a region, okay?

So shaded region here, that denotes a set of feasible SIR values.

Now ideally you want SIR for both users to be high.

But as you know, because of interference, that is not possible.

So let's just look at those SIR1, SIR2 values that are possible.

And we call these the feasible SIR values. So any point inside this region, okay, is

a feasible point. But there are also inferior points because

I could find a way to increase SIR for both users.

Okay. So any points in this region will be

strictly better than this point. At the same time, every point outside the

boundary is infeasible. What about the points exactly on the

boundaries? We call those points Pareto optimal.

Now clearly there are infinite number of these points on the boundary and they are

all Pareto optimal. In some sense, they are not directly

comparable. Points inside are feasible, but inferior.

Points outside are infeasible, so don't even worry about those.

Our job is to find a point that's at least on the pareto optimal boundary.

That means you cannot increase one dimension without hurting the other

dimension. Those points formulate from the boundary

on this region. Now which point to pick then.

That would depend on the objective function.

You have to impose some other objective function in order to pick exactly the

point that you like most. Now later we will see, this theme of trade

off. Okay, tradeoff in this case between two

users, SIR. And in general, the tradeoff between any

competing users in a social, economic, or technological networks.

And we will always want to operate on the parietal optimal boundary.

At least pick a point that is parietal optimal.

You cannot make one user happier without making, another user, less happy.