- Convergence to the optimal value for barrier methods combined with Hessian Riemannian gradient flows and generalized proximal algorithms Felipe Alvarez(falvarezdim.uchile.cl) Julio Lopez(jclopezdim.uchile.cl) Abstract: 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 $g_j(x)\le0$, then endowing $\{x\in\R^n\mid Ax=b, x\in C\}$ with the Riemannian structure induced by the Hessian of an essentially smooth convex function $h$ such that $C=\hbox{int}(\dom h)$, and finally considering the flow generated by the Riemannian penalty gradient vector field. Under minimal hypotheses, we investigate the well-posedness of the resulting ODE and we prove that the value of the objective function along the trajectories, which are strictly feasible, converges to the optimal value. Moreover, the value convergence is extended to the sequences generated by an implicit discretization scheme which corresponds to the coupling of an inexact generalized proximal point method with parametric barrier schemes. Specializations and simple illustrations of the general results are given for the positive orthant, the unitary simplex and the second-order cone. Keywords: Hessian Riemannian gradient,proximal point algorithm, Riemannian metric Category 1: Convex and Nonsmooth Optimization (Convex Optimization ) Citation: CMM-B-09/09-230 Download: [PDF]Entry Submitted: 01/06/2010Entry Accepted: 01/06/2010Entry Last Modified: 01/06/2010Modify/Update this entry Visitors Authors More about us Links Subscribe, Unsubscribe Digest Archive Search, Browse the Repository Submit Update Policies Coordinator's Board Classification Scheme Credits Give us feedback Optimization Journals, Sites, Societies Optimization Online is supported by the Mathematical Programming Society.