An optimal adaptive algorithm for the approximation of concave functions

We consider the piecewise linear approximation of a concave function representing the value function of a parameterized convex program. Using dynamic programming, we derive a procedure for selecting the knots at which an oracle provides the function value and one supergradient. The procedure is adaptive in that the choice of a knot is dependent on the choice of the previous knots. It is also optimal in that the approximation error, in the integral sense, is minimized in the worst case.