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amount approximation assume beginning of period Bellman calculate calculus of variations Chapter component compute concave function consider constraints convex combination convex function convex set decision processes decision variables deﬁne deﬁnition depends derive determine diﬂicult discounted dynamic programming entries evaluate example expected extreme points ﬁnal ﬁnd ﬁnding ﬁnite ﬁrst ﬁve ﬁxed following problem formulation functional equations Hence income induction integer inventory investment Lagrange multipliers machine Markov process mathematical maximize maximum value method minimize minimum notation objective function obtain Operating cost optimal policy optimal return functions optimal solution optimal value optimization problem original problem possible principle of optimality probability proﬁt random variable recurrence relations recursion Replacement cost retum satisﬁes Section sequence simpliﬁes speciﬁed stage stochastic Substituting suﬂicient Suppose tabulation total return transformation transition variation vector wish to solve yields