improving the speed of parallel optimization

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sensation 2018-6-4

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评论： sensation 2018-6-28

Hi, I am trying to optimize in parallel but the speed is increased just slightly by using parfor. Do you have any further recommendations? Thanks!

    parfor i = 1:M
    options = optimset('MaxFunEvals',Inf,'MaxIter',10,...
    'Algorithm','interior-point','Display','iter'); 
    startTime = tic; 
    [x(:,i),fval(:,i)] = fmincon(@(x)revenue(price(1:N,1),ro,g,eff,x,N,k1,init_s(i),inFlow(:,i),alpha_par(i),b_par(i)),x0(1:2*N,i),A,b(:,i),Aeq,beq(:,i),LB(:,i),UB(:,i),[],options);      
    time_fmincon_parallel = toc(startTime);
    fprintf('Parallel FMINCON optimization takes %g seconds.\n',time_fmincon_parallel);
    end

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Walter Roberson 2018-6-4

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Instead of running the fmincon calls in parallel, try running them in a loop, but using the option UseParallel to allow parallel estimation of the gradient.

Remember, it is common for Parallel processing to be slower than serial, depending on the amount of data to be transferred compared to the amount of work to be done per iteration, and taking into account that non-parallel workers can use the built-in parallelization of some operations on large "enough" matrices by calling into LaPACK / MKL.

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Walter Roberson 2018-6-6

Generalities:

if you can rewrite the problem as a linear or quadratic problem, you probably should do that and use an appropriate solver, as there are analytic methods to deal with those
nonlinear constraints are expensive; avoid them if you can. Fortunately for you, you do not have any
linear equality constraints are slower than linear inequalities
When you have a linear equality, examine your code. You might be able to reduce by one or more variables, rewriting the remaining variable internally in the objective function as (constant minus linear sum of the remaining variables). Reducing the number of variables also makes it faster to estimate gradient and hessian as the cost of estimating hessians grows with the square of the number of variables
Sometimes there are regions where certain variables have no effect on the outcome. It can sometimes make sense to break up the problem into sub-problems with different objective functions that deal only with the regions where the variables are meaningful
never load() inside your objective function; load first and pass the data in as part of the parameters to the objective function
Give some thought to vectorizing your objective function. Your objective function will be executed many many times, so you want it to be efficient.
if you have an efficient function for calculating gradients, then provide it in the options instead of having it estimate the gradients and hessian. But do keep in mind that having an analytic function to calculate the gradients or hessian does not always mean that you should calculate it analytically: getting all of the details right for exact solutions can be computationally expensive. The gradient and hessian estimation that is done is sort of like doing a truncated taylor expansion: because of the truncation it could end being much faster, but on the other hand the effective truncation can lead to some pretty misleading results. If your objective function is somewhat smooth at a macro level then gradient estimation can be enough, but if it has some pretty steep areas then you might need to provide a more analytic estimator.

Walter Roberson 2018-6-6

We would need to know depth() to be able to answer.

If you have the symbolic toolbox, then what can help is to create a symbolic vector the same of appropriate length and pass it into your objective function, hopefully getting back a symbolic expression for your objective function. Sometimes the result is just ugly messy, but other times the interaction with the inputs turns out to be fairly clean and you can use diff() and gradient() to come up with reasonable functions and then matlabFunction() to create appropriate function handles.

One trick with matlabFunction is to consider using the 'File' output option, which by default enables the optimization option. The symbolic engine then spends a bunch of time doing common sub-expression optimization to come out with a more efficient series of statements to do the calculation. But this process can take a lot of time; if the expression is longer and more complicated then it can take more than an hour, so it is only worth doing if the expression is not complicated or if the objective function will be executed a lot of times.

sensation 2018-6-21

Any idea Walter? Thanks!

Walter Roberson 2018-6-21

If I recall correctly, with N even close to that large, asking matlabFunction to optimize the code takes far far too long, so I do not think you are going to be able to take advantage of that.

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Matt J 2018-6-21

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编辑：Matt J 2018-6-21

在 MATLAB Online 中打开

This is a more optimal implementation of storage(),

    function S=storage(init_s,inFlow,x,N)
         D=inFlow-totalflow(x,N);
         D(1) = D(1) + ( init_s(1) + D(1) );  
         S=cumsum(D);
   end

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Matt J 2018-6-26

编辑：Matt J 2018-6-26

I have implemented those suggestions and its a bit faster

How fast is it now? It should have been a lot faster than what you were doing.

Do you know maybe how I can run it with quadprog instead of fmincon?

The problem doesn't look quadratic, except maybe when b_par=1.

sensation 2018-6-28

With N being 4000 time steps, the optimization is quite long. It is a bit faster 1.2 times.

I was thinking to run fmincon with supplied hessian and gradient maybe. I run it in parallel with parfor but still not efficient.

If you have any idea to make it feasible, would appreciate it a lot Matt. Thanks!

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