Further Optimization

Recently I've tried to make up my mind about how to maximize benefits of control in indeterministic network. I mean how to find a best solution for equation that is not mathematical. Let's say that some problems have strict mathematical structure. Those problems can be optimized mathematically (and all those problem are hopefully solved nowadays) . Thanks God most of surrounding us world is far more complex than mathematical. 

Can We model Transport Network mathematically?

Urban Passenger Transport Network is hardly modeled mathematically and stochastic attempts to model our travel behavior simply fail. And no, we do not travel to fulfill Nash's eulibrium, or Wardrop principle ( http://en.wikipedia.org/wiki/Wardrop_equilibrium, http://en.wikipedia.org/wiki/Traffic_assignment#Equilibrium_assignment). Let's say that each of us is separate agent in network, with his individual behavior, intelligence, and reacts differently to changing conditions. And when we realize that transport assignment is result of hundreds of users, and each of them is individual agent, human being with intelligence, than for me it's ridiculous to approximate this process with equilibrium assignment, or shortest path search. It's far more complex and we will not be able to define it with simple model. 

If it's impossible to model transport urban network with simple mathematical model, than let's introduced slightly more complicated mathematical model. In fact that's solution, but for those who like "parametrization". In practice it most likely looks like this: 

it's nothing more than making simple model more parametrized. I'd assume that we are unable to handle more complex models, especially series of those models with constrains, and we do not see dependencies between equations and parameters.

If not what shell we do?

 Reasonable solution is to benefit from Mr Nash's and others for their contribution in understanding transport networks. But we shall:
1. understand their models fully 
2. use them in their essence
3. do not parametrize them excessively

those models are fantastic in their simplicity. When we analyze whole city as a set of rational agents willing only to fulfill global equilibrium so that: "I cannot change my behaviour because when I change it I'll worsen our global optimum" (System Optimum Equilibrium) . It's elegant, simple, rational, and gives us precious information. But this model will never perfectly fit to reality it's just a simplification. Great for macro scale analysis, so what if we want something more detailed (ie for real time controlling)?

 Non-analytical AI methods  

 Let's say that Genethic Algorythm is great to find an optimum without knowing the structure of problem. Only things that we need to know is Input and Output. We need to have evaluation model testing whot values of do population. Let's say we have model that represents reality and we seek for the perfect solution:
 1.We generate population to be tested and 
2. choose best ones to survive and 
3. survivors give birth to new generation (being genetical combination of survivors) 
4. we test new generation of population and so on... until converged.

"Great for lazy cowboys who don't like gradients jacobians" critics would say and I agree. I'm not very familiar with this dark area of very long series of strange sings on blackboard. But I'd say more: it's far more than maths - it's genetic. 

Let's say neural network  is perfect for finding the hidden structure. Neural network is generated as an empty entity with learning potential. Se send the network to school. In learning process we show multiple examples with solutions (we give inputs, and outputs - numerical). We teach the network until it can solve any example. Then we graduate it and it's postgradual is solving the problems but not on scholat examples. 

See how it works:

Let's generate database of DowJones indexes fro past 500 days, and database of any economical factors from past 500 days. 

We show to our pupil (neural network) inputs, and tell him to predict DowJones for the certain day. If neural network succeeds for 500 past days we can say teaching process has completed - we can say that with prob -> 1 it will predict DowJones for tommorow. 

Mathematical framework for this algorithm is not that complicated it's just guessing paramters and their weights in complex network  (http://en.wikipedia.org/wiki/Neural_network)


For me this is the future of Transport Modeling and Optimization, soon I'll try to show why I think so. 

Many things can be done in this area.