正值幂函数乘积的极小和问题

Minimizing the Sum of the Product Problems of Positive Power Functions

研究下述非线性规划ｍｉｎｘ∈Ｘｓｊ＝１ｋｉ＝１ｆｐｉｊｉｊ（ｘ）这里ｆｉｊ：Ｘ→Ｒ＋，ｐｉｊ≥０，ｋｉ＝１ｐｉｊ＝１，ｉ＝１，２，…，ｋ，ｊ＝１，２，…，ｓ．Ｘ是Ｒｎ中非空紧集．借助加权平均值不等式将问题转化为含参数函数之和的极小化问题．证明了最优参数只需取一些特定的值．特别当ｆｉｊ是线性齐次函数，Ｘ为凸多面体时，其最优解必定可以在Ｘ的顶点达到．同时给出了可行点为最优解的Ｋｕｈｎ－Ｔｕｃｋｅｒ型最优性条件．

The non linear programming min x∈Xsj=1ki=1f p ij ij (x)is considered,where f ij :X→ R +,p ij ≥0,ki=1p ij =1,i=1,2,…,k,j=1,2,…,s.X is a nonempty compact set in R n.By means of the weighted average inequality,the problem can be converted into a minimization problem of the sum of functions containing parameters,and it is shown that the optimal parameters only take special values.Especially,when every f ij is linear and homogeneous and X is a polytope,there is the optimal solution to be the vertex of X. The Kuhn Tucker type optimal conditions are given for a feasible point to be an optimal solution.