There is huge branch of mathematical optimization theory, but it mostly considers the finding optimal parameter values for the predefined structures. There are variational calculus and optimal control that considers the optimal, time-dependent evolution of some parameters for the predefined sturctures.
My question is - is there branch of mathematics that tries to optimizes the sturctures or designs themselves. I.e. finding no only some parameters of some predefined structure, but to find the optimal structure itself? Where can I find such research? Any terms, keywords, any review articles or books can be helpful.
There can be lot of applications for such kind of optimization. Especially in robotics - try to find the optimal configuration of robot (number of joints, conditions on serial or parallel lines, and so on) and not only to find some parameters of already complete robot?
Another application is optimization of business processes - make processes optimal (number and kind of steps, optimize branching conditions) and not only optimize parameters of already existing processes.
Another application it optimization of biological systems. System is controlled by genetical DNS and system should be optimized for possibility to live healthy and long life. This can be the most important task for the personalized medicine (it is already known that biological life is strongly determined by which genes are activated or passivated, not only by the DNS structure itself, so there is necessary to update the system - activate or passivate some genes, e.g. environment can cause activization or cancer genes, medicine can suppress them in optimal way).
Another application is software design optimization - number of software layers in business application, the best modularity.
I guess that some important points can be made about such endeavour:
it is necesary to somehow encode the stucture or design for structural problem and to build target function utility=f(structure/design). Apparently such function can be greatly nonlinear, sometimes discrete and so on. There should be some kind of theory for such functions;
optimization on categories can be necessary to develop. E.g. one can consider each structure/design as object of category, the morphisms among objects can encode ordering of utility of such structures and different morphisms can encode structural similarity or possible evolution of structures. Then the optimization task could be stated as finding some special object in category. But I am not aware of any developments in category theory along these lines. I have asked in n-category cafe about Eulear-Lagrange equations for categories and there are some developments but those articles were quite obscure for me; One can esily imagene that sturctures can have different levels - parameter -> structure -> meta-structure -> meta-meta-structures. So the optimization should be made in different levels. Apparently - higher-categories can be answer to this hierarchy of abstraction.
graph theory can be of help - e.g. find optimal (in some sense) graph among others.
I guess that for some this kind of thinking seems to be too abstract, but business demand such approach here and now. I can be of great help to industries, medicine and intelligent systems.
Thanks for any hints in advance!