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Strategic Optimization

How to use the Nelder Mead - Simplex Method to solve many nonlinear problems


 Function 'fminsearch' in Matlab is used for strategic optimization or utilized to solve, almost magically, a number of nonlinear problems.

fminsearch is the Nelder-Mead implementation of the simplex method, which is utilized for the minimization of functions.

If you can formulate your problem in such a way that its minimum value represents the solution to your problem, 

then it's very likely that you can solve it by utilizing the fminsearch built-in function, in Matlab. You can solve nonlinear systems, curve fitting and optimization design, among others.


There are three steps that you have to include or elaborate to solve these types of problems:

  1. You have to be able to run your parameterized function. This can be a mathematical function or maybe you are running an external simulator that's delivering a response that you need to improve.
  1. Then, you have to prepare an objective function (OF). This OF must be able to run the function, deliver new parameters each time and somehow evaluate or measure the results. Maybe you are trying to match two responses (a goal and the one that you're working with) or maybe you need a maximum error in a particular range. You could be working with a minimax algorithm, for example. I mean, you could be minimizing the maximum error in a particular region of interest.
  1. Now, you can use function fminsearch (the Nelder-Mead implementation of the simplex algorithm) to iteratively run your OF and automatically change the parameters in order to deliver the minimum value possible.

It's important to mention that this strategic optimization method delivers local minima, not global ones. This means that the algorithm can give you a solution, but not necessarily the best one.

The article named Circuit Simulator shows how you can run an electronic simulator -it's Winspace- from Matlab. You can change parameters in the circuit and read the different responses. This means that your function to be optimized can be delivered from an external source, and you can use Matlab to optimize it, as long as you can interact with the external source.

The article named Mathematical Optimization shows how the fminsearch function finds different minimum values, depending on your starting point. You can get many different solutions and you must be aware of this behavior. There's another useful function named fminbnd (using the Golden Section search algorithm) that can be used for optimizations, but it only accepts one variable.

I wrote an article to solve Nonlinear Systems, where I show you how to implement the three steps mentioned above: express your function(s), plan an OF (considered the goal) and use fminsearch to iteratively modify the parameters to get the minimum value of your OF. You must verify always how good your response is, since you know now that you're working with local minima, which are not always the best responses.

Another article is about Curve Fitting. You have some empirical data and want to find an analytical expression for those numbers. You also implement the three steps above and find the correct expression, when you know in advance the type of function that you want to fit.

Those are some of the applications of this type of strategic optimization that you may use for many purposes...


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