  New upper bounds on the smallest size t_{2}(2,q) of a complete arc in the
projective plane PG(2,q) are obtained for 853<= q<= 2879 and q=3511,4096,
4523,5003,5347,5641,5843,6011. For q<= 2377 and q=2401,2417,2437, the relation
t_{2}(2,q)<4.5\sqrt{q} holds. The bounds are obtained by finding of new small
complete arcs with the help of computer search using randomized greedy
algorithms. Also new sizes of complete arcs are presented.
