  In this paper we present a new steepest-descent type algorithm for convex
optimization problems. Our algorithm pieces the unknown into sub-blocs of
unknowns and considers a partial optimization over each sub-bloc. In quadratic
optimization, our method involves Newton technique to compute the step-lengths
for the sub-blocs resulting descent directions. Our optimization method is
fully parallel and easily implementable, we first presents it in a general
linear algebra setting, then we highlight its applicability to a parabolic
optimal control problem, where we consider the blocs of unknowns with respect
to the time dependency of the control variable. The parallel tasks, in the last
problem, turn "on" the control during a specific time-window and turn it "off"
elsewhere. We show that our algorithm significantly improves the computational
time compared with recognized methods. Convergence analysis of the new optimal
control algorithm is provided for an arbitrary choice of partition. Numerical
experiments are presented to illustrate the efficiency and the rapid
convergence of the method.
