 # Algorithm

An iterative method that generates a sequence of the form $LaTeX: x^{k+1} = S_k (A_k (x^k))$, where $LaTeX: x^0$ is given (called the initial point). $LaTeX: A_k$ is an algorithm map that yields a set of policies, given the current point, $LaTeX: x^k$, and $LaTeX: S_k$ is a selection function (in case $LaTeX: A_k$ has more than one policy). Note that the algorithm map and selection function can depend on the iteration number $LaTeX: k$. A concrete example is of the form $LaTeX: x^{k+1} = x^k + s_k d^k$, where $LaTeX: s_k$ is a scalar parameter, called the tep size and $LaTeX: d^k$ is a vector, called the direction of change. The step size may require a selection from a line search if there are many optimal solutions; one is to select the least value (perhaps above some threshold). In practice, algorithms have a variety of tolerances to control its computer representation, progression and termination.
A stochastic algorithm is one whose algorithm map is a random variable. Unlike the meaning in computer science, most people in mathematical programming mean an algorithm whereby an iterate is selected by some random mechanism. One example is simulated annealing. When $LaTeX: x^k$ is a random variable, there is a meaning to convergence that differs from ordinary (deterministic) sequences.