Is there an example of a multivalued maximal monotone operator that is not the convex subdifferential of a proper convex lower semicontinuous? Besides, among these type of operators, are there any physically important? (describing any non-smooth dynamics of the real world). Thank you!
1 Answer
My prime example of such an operator comes from saddle point problems of the form $$ \min_x\max_y F(x) + \langle Kx,y\rangle - G(y) $$ with $F,G$ being two proper, convex, lower-semicontinuous functions defined on Hilbert spaces $X$ and $Y$, respectively, and $K:X\to Y$ linear and bounded. The Fenchel-Rockafellar optimality system is $$ 0\in \begin{pmatrix} \partial F(x) & -K^*y\\ Kx & \partial G(y)\end{pmatrix}. $$ The operator on the right hand side is indeed monotone as the sum of a subdifferential and a skew-symmetric one and it's also maximally so, since he both are and the second one is defined everywhere. It is not a subgradient of any function on $X\times Y$ since it's not symmetric.
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$\begingroup$ Thank you very much. In which applications, there is a need to treat this type of saddle point problems? $\endgroup$ Feb 23, 2023 at 10:35
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$\begingroup$ Tons of! If you evaluate the inner max you get $\min_x F(x) + G^*(Kx)$ (with the convex conjugate $G^*$ of $G). These problem are ubiquitous. For an old account with examples from the natural science consider Ekeland/Temam "Convex Analysis and Variational Probelms". For a more recent one with application mosty from mathematical imaging check Chambolle/Pock "An introduction to continuous optimization for imaging". $\endgroup$– DirkFeb 23, 2023 at 14:28