Orateur
Description
We present a new algorithm for minmax optimization problems of the form
$\min_x \max_y \; \phi(x,y) - h(y)$, where $\phi$ has block-wise Lipschitz-continuous gradient,
semi-convex in $x$, concave in $y$ and either $\phi$ or $-h$ is strongly concave in $y$.
Such problems arise in image inverse problems where $\phi$ encodes regularity prior on the image to restore $x$.
State-of-the-art optimization methods for such problems rely on alternate explicit gradient descent-ascent
steps on the coupling term $\phi$. On the other hand, Plug and Play (PnP) approaches replace the explicit
gradient step on $\phi$ by a neural network that is at the same time an image denoiser, and a proximal operator
of a semi-convex potential. This proximal step does not fit into the current optimization framework for which
convergence is known to hold.
In this context, we here propose a new min-max optimization scheme with proximal steps on $x$, thus allowing
to provide convergence guarantees for some PnP applications. We derive sufficient conditions on the convergence
of the algorithm and showcase its interest for PnP image restoration.