  We introduce a new geometric approach that constructs a transition kernel of
Markov chain. Our method always minimizes the average rejection rate and even
reduce it to zero in many relevant cases, which cannot be achieved by
conventional methods, such as the Metropolis-Hastings algorithm or the heat
bath algorithm (Gibbs sampler). Moreover, the geometric approach makes it
possible to find not only a reversible but also an irreversible solution of
rejection-free transition probabilities. This is the first versatile method
that can construct an irreversible transition kernel in general cases. We
demonstrate that the autocorrelation time (asymptotic variance) of the Potts
model becomes more than 6 times as short as that by the conventional
Metropolis-Hastings algorithm. Our algorithms are applicable to almost all
kinds of Markov chain Monte Carlo methods and will improve the efficiency.
