I've been working with the following optimization problem:

$$ \max \int \left(\frac{1}{2}\left\| x \right\|^2 + \tilde{u}(x)\right) \,ds(x) + \int \left(\frac{1}{2}\left\| y \right\|^2 + \tilde{v}(y) \right) \,dt(y) $$ s.t. $\forall x,y,\quad \tilde{u}(x)+\tilde{v}(y) \leq -\langle x,y\rangle.$

Currently, I'm trying to understand stability/sensitivity of optimal solutions to small changes in $s$ and $t$.

I haven't found anything similar in the literature that covers this, so I'm wondering where I can begin. If you know a good reference, feel free to send that my direction.


  • 1
    $\begingroup$ Several stupid questions: 1) are $x,y\in \mathbb R^n$? 2) are $s,t$ given positive measures and $\tilde u,\tilde v$ unknown functions? 3) How exactly is the smallness of the change in measure is measured? $\endgroup$ – fedja Jun 10 at 19:41
  • $\begingroup$ Yep! $x,y\in \mathbb{R}^n$ and $s,t$ are given positive measures with $\tilde{u},\tilde{v}$ unknown. $\endgroup$ – Glassjawed Jun 10 at 19:44
  • $\begingroup$ The norms in the objective could be omitted. Also: For sure you aware of the fact that this is exactly the dual of the optimal transport problem with quadratic cost, right? You could check out stability results for linear programming... $\endgroup$ – Dirk Jun 11 at 12:41
  • $\begingroup$ Yes yes I know. For some reason my original post was edited (it had an extra term in there). $\endgroup$ – Glassjawed Jun 12 at 5:20

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