We introduce a new barrier function to solve a class of
Semidefinite Optimization Problems (SOP) with bounded variables.
That class is motivated by some (SOP) as the minimization of the
sum of the first few eigenvalues of symmetric matrices and graph
partitioning problems. We study the primal-dual central path
defined by the new barrier and we show that this path is analytic,
bounded and that all cluster points are optimal solutions of the
primal-dual pair of problems. Then, using some ideas from
semi-analytic geometry we prove its full convergence. Finally, we
introduce a new proximal point algorithm for that class of
problems and prove its convergence.