  In this paper, we propose two simple yet efficient computational algorithms
to obtain approximate optimal designs for multi-dimensional linear regression
on a large variety of design spaces. We focus on the two commonly used optimal
criteria, $D$- and $A$-optimal criteria. For $D$-optimality, we provide an
alternative proof for the monotonic convergence for $D$-optimal criterion and
propose an efficient computational algorithm to obtain the approximate
$D$-optimal design. We further show that the proposed algorithm converges to
the $D$-optimal design, and then prove that the approximate $D$-optimal design
converges to the continuous $D$-optimal design under certain conditions. For
$A$-optimality, we provide an efficient algorithm to obtain approximate
$A$-optimal design and conjecture the monotonicity of the proposed algorithm.
Numerical comparisons suggest that the proposed algorithms perform well and
they are comparable or superior to some existing algorithms.
