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| Abstract |
A mathematical programming problem is called recursive programming with monotonicity if both objective and constraint functions are recursive functions with monotonicity, i.e., either with nondecreasi...ngness or with nonincreasingness (defined in Section 2) [12]. Thus recursive programming includes dynamic programming. This paper, viewed as a sequel to the previous one [9], studies systematically inequalities from the viewpoint of recursive programming with monotonicity. The theory of recursive programming is based upon two principles: the Bellman's Principle of Optimality and the Second Principle of Optimality. It is well known that the former is valid for the case when objective function is a recursive function with nondecreasingness, i.e., for dynamic programming [2, 14, 15]. On the other hand, it is shown that the latter is valid for the case when objective function is a recursive function with nonincreasingness [13; 15, p. 44]. However, both recursive and dynamic programmings are same in their spirits. We obtain inequalities between two recursive functions with strict monotonicity by direct applications of recursive programming like as we have done ones between two recursive functions with strict increasingness by applica tions of dynamic programming [9]. In Section 2 we consider six pairs of main and inverse problems with $ N $-variable recursive functions with strict monotonicity and characterize them in terms of optimal-value function and optimal-point function. As corollaries inequalities between such recursive functions are established. Similar results for 2-variable problems not belonging to above $ N $-variable problems are obtained in Section 3. In Section 4 we give a recursive programming method to calculate a pair of optimal-value function and optimal-point function of the underlying problem. The method are based upon either the Bellman's Principle of Optimality or the Second Principle of Optimality. In Section 5 only a few theorems in Sections 2, 3 and 4 are proved in detail, since others can be proved in a similar manner. Finally, in the last section, we obtain some inequalities by immediate applications of the results in Sections 2, 3 and 4.show more
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