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.