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Suppose we have the following continuous optimization problem:

$$ \underset{x}{\mathrm{minimize}}f\left(x\right) $$ subject to $$ \exists X:\nabla X=Jac\left(X\right)=x $$ where $f$ is a function $f:M_{v}\left(_{n}D_{v}\right)\rightarrow M_{v}\left(_{n}D_{v}\right)$, where $M_{v}\left(_{n}D_{v}\right)$ is essentially a $v$-dimensional matrix space over polynomial functions $\mathbb{R}^{v}\rightarrow\mathbb{R}$.

With care and attention to detail, the matrices can be treated as vectors for certain purposes.

In other words, function $f$ is minimized subject to the constraint that $x$ is a conservative field.

Another way to write the constraint is by using the symmetry of second derivatives property, which is $$ D_{i}f_{jk}\left(x\right)=D_{k}f_{ji}\left(x\right) $$

How can this constraint be efficiently implemented in practice, for example in case of simple gradient descent?

It is assumed that the dimension $n$ is relatively high, such as $n=16$ and possibly higher, so that applying all the numerous constraints in brute-force fashion is undesirable.

There is an equivalent way to formulate this problem with objective function $\tilde{f}\left(X\right)$, implementation of which poses the same practical efficiency difficulties as the original problem.

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  • $\begingroup$ Wait, don't you mean $x$ is a function $x:\mathbb R^n\to\mathbb R^n$ and $f$ is a functional that maps $x$ to $\mathbb R$? Otherwise I can't make sense of your question. Anyway, why not just take the scalar potential $X$ as the optimization variable and minimize $f(\nabla X)$? $\endgroup$ – Rahul Nov 2 '14 at 17:13
  • $\begingroup$ @Rahul Okay, sorry, I oversimplified the problem statement, but now I corrected it. I have considered what you are suggesting, but in practice it is quite hard to implement in a computer program; it is essentially an equivalent problem with similar difficulties. I would be happy to hear a suggestion on a "trick" to efficiently solve the equivalent problem you mentioned as well. $\endgroup$ – Jake Nov 2 '14 at 17:35
  • $\begingroup$ Do you mean $\nabla X =f$ ? $\endgroup$ – guest Nov 2 '14 at 17:49
  • $\begingroup$ @guest: No, this is not the case. $f$ acts on $x$, where $x$ is a gradient/Jacobian of $X$. $\endgroup$ – Jake Nov 2 '14 at 17:50
  • $\begingroup$ So $X$ is a scalar field, $x$ is its gradient. How is then $x$ an element of a "matrix space"? $\endgroup$ – guest Nov 2 '14 at 17:52

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