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We study an optimization problem with the feasible set being a real algebraicvariety $X$ and whose parametric objective function $f_u$ is gradient-solvablewith respect to the parametric data $u$. This class of problems includesEuclidean distance optimization as well as maximum likelihood optimization. Forthese particular optimization problems, a prominent role is played by the EDand ML correspondence, respectively. To our generalized optimization problem weattach an optimization correspondence and show that it is equidimensional. Thisleads to the notion of algebraic degree of optimization on $X$. We apply theseresults to $p$-norm optimization, and define the $p$-norm distance degree of$X$, which coincides with the ED degree of $X$ for $p=2$. Finally, we derive aformula for the $p$-norm distance degree of $X$ as a weighted sum of the polarclasses of $X$ under suitable transversality conditions.

Algebraic degree of optimization over a variety with an application to $p$-norm distance degree