Title: Lipschitz stability of  least-squares problems regularized by functions with $C^2$-cone reducible conjugates

Abstract: Stability analysis in optimization is the study of how the optimal solutions (or  the optimal value) of an optimization problem varies as the problem-defining data (parameters) changes. In this talk, I will present new results about  Lipschitz stability of solution mappings for least-squares regularized problems at which the Fenchel conjugates of convex regularizers are $\mathcal{C}^2$-cone reducible. The condition about $\mathcal{C}^2$-cone reducibility is not restrictive, as many important regularizers satisfy it such as the $\ell_1$ norm, the $\ell_1\ell_2$ norm, and the nuclear norm.  Our approach by using Robinson's strong regularity on the dual problem allows us to obtain new characterizations for Lipschitz stability that rely solely on first-order information, bypassing the need to explore the second-order curvature of the regularizer. We also show that these solution mappings are automatically Lipschitz continuous around the points in question whenever they are locally single-valued.