Minimization of a function with unknown gradient but known sparsity pattern of its hessian

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Jan Valdman
Jan Valdman 2019-10-29
回答： Jan Valdman 2019-10-30
Dear colleagues,
is there a fmincon option to minimize a function without the knowledge of its gradient but providing a sparsity pattern of a Hessian?
My function comes from a FEM formulation of an energy in nonlinear mechanics of solids and it is too difficult to differentiate analytically.
However the sparsity pattern of the hessian is easily available though a FEM connectivity of variables.
Is there a way to exploit it efficiently? If I run with 'Algorithm','quasi-newton', it seems not to accept 'HessPattern' option. An alternative would be to obtain an appriximative gradient (can you suggest one?) and use 'Algorithm','trust-region' insteady. Does anyone have experience with it?
Best wishes,
Jan

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Alan Weiss
Alan Weiss 2019-10-29
Sorry, I am afraid that the available options don't work efficiently for your case. The HessPattern option is available only for the 'trust-region-reflective' algorithm, but for that algorithm you need to supply a derivative.
I am not sure what to suggest that you probably have not yet tried. For the default 'interior-point' algorithm you can try using the HessianApproximation option set to 'lbfgs' or {'lbfgs',Positive Integer}, but that does not directly use the sparsity pattern that you know. Or, and this seems crazy, you could code a finite difference gradient in your objective funtion, bypassing MATLAB's internal one, and then you could use the 'trust-region-reflective' algorithm with the HessPattern option. I am not sure that the 'trust-region-reflective' algorithm would satisfy you anyway, as it accepts only bound constraints or only linear equality constraints.
Sorry.
Alan Weiss
MATLAB mathematical toolbox documentation
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Catalytic
Catalytic 2019-10-29
编辑：Catalytic 2019-10-29
One possibility might be to use a 1-iteration call to fmincon itself to return the gradient. This uses Matlab's finite differencer and so might be faster than 3rd party implementations,
function [f,numerical_grad]=myObjective(x)
f=...
if nargout>1
options=optimoptions('fmincon','MaxIter',1,'SpecifyObjectiveGradient',false);
[~,~,~,~,~,numerical_grad] = fmincon(@myObjective,x,[],[],[],[],...
[],[],[],options);
end
end
Jan Valdman
Jan Valdman 2019-10-29
Thank you, can you add please add your numerical gradient to the code attached below and beat 'quasi-newton' times? Alan's limited BFGS also works fine, but it is not that fast. Maybe one might even use the sparsity pattern to generate a gradient faster, since the functional is a sum of values (in this example and in my computations as well).

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Jan Valdman
Jan Valdman 2019-10-30
Dear colleague,
based on our discussions from yesterday, I implemented a finite difference scheme for a gradient approximation exploiting a particular sum form of the function f. It enables me to compute an approximate gradient very quicky and then I can feed fminunc with it in both variants 'quasi-newton' and 'trust-region-reflective'. Please fiind resulting two files attached.
For n=1000, 'quasi-newton' is still faster.
For n=20000, 'trust-region-reflective' beats 'quasi-newton'.
For n=40000, 'trust-region-reflective' converges and 'quasi-newton' fails due to memory issues.
So,an efficient evaluation of an approximative gradient might be a way to my speedup in FEM formulations. I will look further into it. Thank you.

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