## PURPOSE

Test AD for a real optimization problem on a product manifold (struct)

## DESCRIPTION

` Test AD for a real optimization problem on a product manifold (struct)`

## CROSS-REFERENCE INFORMATION

This function calls:
• manoptAD Preprocess automatic differentiation for a manopt problem structure
• spherefactory Returns a manifold struct to optimize over unit-norm vectors or matrices.
• trustregions Riemannian trust-regions solver for optimization on manifolds.
• checkgradient Checks the consistency of the cost function and the gradient.
• checkhessian Checks the consistency of the cost function and the Hessian.
• productmanifold Returns a structure describing a product manifold M = M1 x M2 x ... x Mn.
This function is called by:

## SOURCE CODE

```0001 function realtest_AD1()
0002 % Test AD for a real optimization problem on a product manifold (struct)
0003
0004     % Verify that Manopt was indeed added to the Matlab path.
0005     if isempty(which('spherefactory'))
0006         error(['You should first add Manopt to the Matlab path.\n' ...
0008     end
0009
0010     % Verify that the deep learning tool box was installed
0011     assert(exist('dlarray', 'file') == 2, ['Deep learning tool box is '...
0012     'needed for automatic differentiation.\n Please install the'...
0014     ' R2021b if possible.'])
0015
0016     % Generate the problem data.
0017     n = 100;
0018     A = randn(n);
0019     A = .5*(A+A');
0020
0021     % Create the product manifold
0022     S = spherefactory(n);
0023     manifold.x = S;
0024     manifold.y = S;
0025     problem.M = productmanifold(manifold);
0026
0027     % Define the problem cost function
0028     problem.cost  = @(X) -X.x'*(A*X.y);
0029
0030     % Define the gradient and the hessian via automatic differentiation
0032
0033     % Numerically check gradient and Hessian consistency.
0034     figure;
0036     figure;
0037     checkhessian(problem);
0038
0039     % Solve.
0040     [x, xcost, info] = trustregions(problem);          %#ok<ASGLU>
0041
0042     % Test
0043     ground_truth = svd(A);
0044     distance = abs(ground_truth(1) - (-problem.cost(x)));
0045     fprintf('The distance between the ground truth and the solution is %e \n',distance);
0046
0047
0048 end```

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