Convergence to the optimal value for barrier methods combined with Hessian Riemannian gradient flows and generalized proximal algorithms

We consider the problem $\min_{x\in\R^n}\{f(x)\mid Ax=b, \ x\in\overline{C},\ g_j(x)\le0,\ j=1,\ldots,s\}$, where $b\in\R^m$, $A\in\R^{m\times n}$ is a full rank matrix, $\overline{C}$ is the closure of a nonempty, open and convex subset $C$ of $\R^n$, and $g_j(\cdot)$, $ j=1,\ldots,s$, are nonlinear convex functions. Our strategy consists firstly in to introduce a barrier-type penalty for the constraints $g_j(x)\le0$, … Read more