### Mixed Algorithm

The mixed algorithm is the recursive algorithm, with it's recursive call replaced by a call to the iterative algorithm. It has an expected runtime of O(d^{2}n) + (d^{2}log n)O(d)^{d/2 + O(1)} + O(d^{4}n^{1/2}log n), for more info on this see run time analysis. For more info on the two algorithms used in the mixed algorithm see:

The recursive algorithm and The iterative algorithm
function x_{m}^{*}(S : set_of_halfspaces)
V^{*} = {} // the empty set
C_{d} = 9d^{2}
if n <= C_{d} then return SimplexMethod(S)
repeat
choose R subset of S \ V^{*} at random, |R| = dn^{1/2}
x^{*} = **x**_{i}^{*}(R union V^{*})
V = {H exits in S | x^{*} violates halfspace H}
if |V| <= 2n^{1/2} then V^{*} = V^{*} union V
until V = {}
return x^{*}

function x_{i}^{*}(S : set_of_halfspaces)
for H exits in S do w_{H} = 1 od;
C_{d} = 9d^{2}
if n <= C_{d} then return SimplexMethod(S)
repeat
choose R subset of S at random, |R| = C_{d}
x^{*} = SimplexMethod(R)
V = {H exits in S | x^{*} violates halfspace H}
if w(V) <= 2w(S)/(9d - 1)
then for H in V do w_{H} = 2w_{H} od
until V = {}
return x^{*}