How to minimize a function with 112 variables?

I have a non-negative function(in the attachment) named `aim` with 112 variables to minimize, but even

     fminsearch('aim',random('Normal',0,3,1,112),optimset('MaxFunEvals',100^141))

output that

     Exiting: Maximum number of iterations has been exceeded
           - increase MaxIter option.
           Current function value: 1.977209

As I know, the minimum of `aim` should be 0. But I just do not know what to do then.

Then I increase the MaxIter option, and it is still running...

     fminsearch('aim',random('Normal',0,3,1,112),optimset('MaxFunEvals',10^9,'MaxIter',10^9))

Answer 1

[John D'Errico]

NEVER use a tool like fminsearch to minimize a function with more than about 10 variables. Really, about 6 unknowns is a good practical limit for that tool. 112 variables is just folly, as you have found.

Beyond that, you need to use a toolbox function like fminunc or fmincon from the optimization toolbox. If you don't have that toolbox and you do optimization with any frequency, BUY IT.

You can also use tools like simulated annealing, particle swarms, genetic algorithms, etc. These stochastic optimizers have some good properties for nasty problems with multiple local solutions or with problems like internal singularities.

Typically I'd suggest that for smooth functions, you use a tool like fminunc/fmincon. It will be efficient and will converge to an accurate solution more rapidly than will the stochastic tools I suggested. For problems with a problem like those I mentioned above, go with a stochastic tool.

You can find toolboxes for all of these things from the MathWorks, and at least a few of them on the file exchange. For example, there are many particle swarm toolboxes to be found on the FEX.

[Jhon]

Thanks! I'll try it。

[Jhon]

Each `fminunc` or `fmincon` may take more than half a minute, the whole program may run 10000 years

[John D'Errico]

So you have many millions of problems to solve, each of which has 112 variables, if each fminunc solution would run for on the order of 30 seconds?

Then guess how many millions of years do you think it would take fminsearch to converge for that many problems?

Just because your computer is fast does not make it a super computer. Solve smaller problems, or find some time on a seriously massive computer. Just wanting to solve a massive problem is not sufficient.

And a good idea is to choose your starting value more intelligently. Random starts will simply not suffice for a problem like this.

Answer 2

[Walter Roberson]

For something like that you would usually use a technique such as simulated annealing, and you would not expect to definitely find the global minimum.

[Jhon]

Could you kindly teach me how? Many thanks!