Obtain Hessian matrix from a sum of squares expression

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Yannis Stamatiou
Yannis Stamatiou 2023-5-10
评论： Yannis Stamatiou 2023-5-11
Hi,
I have been trying to obtatin the Hessian matrix of a complex quadratic expression of 2-d variables so as to write the quadratic optimization problam I have in matrix form. However, no matter how hard I tried, I could not find a way to accomplish this.
The expression looks like
0.0010309*x(11, 7)*x(13, 9) + 0.0016748*x(15, 11)*x(13, 9) + 0.0025*x(12, 7)*x(14, 9) + 0.00049916*x(16, 11)*x(14, 9) + 9.6648e-06*x(13, 7)*x(15, 9) + 0.0024*x(14, 7)*x(16, 9) + 3.5338e-07*x(3, 12)*x(1, 10) + 0.0019432*x(4, 12)*x(2, 10) + 0.0025*x(1, 8)*x(3, 10) + 0.0017032*x(5, 12)*x(3, 10) + 3.4478e-07*x(2, 8)*x(4, 10)
but with variables indexed up to 64 in each dimension. This is stored in the problem.Objective field of an minimization problem I set up. I tried the coeff function but it needed a list of variable, which are too many to list. Any help would be greatly appreciated.
Thanks,
Yannis

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采纳的回答

Matt J
Matt J 2023-5-10
编辑：Matt J 2023-5-10
Ran in:
x=optimvar('x',[64,64]);
Objective=0.0010309*x(11, 7)*x(13, 9) + 0.0016748*x(15, 11)*x(13, 9) + 0.0025*x(12, 7)*x(14, 9) + 0.00049916*x(16, 11)*x(14, 9) + 9.6648e-06*x(13, 7)*x(15, 9) + 0.0024*x(14, 7)*x(16, 9) + 3.5338e-07*x(3, 12)*x(1, 10) + 0.0019432*x(4, 12)*x(2, 10) + 0.0025*x(1, 8)*x(3, 10) + 0.0017032*x(5, 12)*x(3, 10) + 3.4478e-07*x(2, 8)*x(4, 10);
prob=optimproblem('Objective',Objective);
Hessian=prob2struct(prob).H
Hessian =
(395,525) 0.0021 (655,525) 0.0033 (396,526) 0.0050 (656,526) 0.0010 (397,527) 0.0000 (398,528) 0.0048 (707,577) 0.0000 (708,578) 0.0039 (449,579) 0.0050 (709,579) 0.0034 (450,580) 0.0000
  2 个评论
Yannis Stamatiou
Yannis Stamatiou 2023-5-10
Hi Matt,
this is very helpful, thanks! I will try it on the whole expression later and get back to this thread to tell the result.
With best wishes,
Yannis

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更多回答（1 个）

Torsten
Torsten 2023-5-10
编辑：Torsten 2023-5-10
Each optimizer expects the unknowns as a vector, not as a 2d-matrix.
So choose an arrangement of your unknowns in a vector of size (64^2,1). Then use MATLAB's "hessian" function to build the Hessian of your expression. Once you've created it with the symbolic toolbox, you can write it to file for use in a numerical computation. Since your expression seems to be quadratic, your Hessian will be constant, I guess.
https://de.mathworks.com/help/symbolic/sym.hessian.html
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Yannis Stamatiou
Yannis Stamatiou 2023-5-10
Hi Torsten,
thanks again! This is exactly my issue. I believe that using Matlab to solve quadratic optimization problems works best with the matrix representation of the objective function instead of the explicit algebraic form. The 64x64 objective function takes minutes to solve. For the 128x128 problem I simply stop the computation after several minutes have passed with no result. I will try the Hesisian then and get back to this thread to tell you the result.
Thanks!
Yannis
Yannis Stamatiou
Yannis Stamatiou 2023-5-11
I also thank you Torsten for this approach.
With best wishes,
Yannis

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