线性规划转对偶问题
有限制:
\(\forall i, \sum_j A_{i,j} \times x_j \ge B_i\)
\(min \sum_j c_j \times x_j\)
对偶问题:
\(\forall j, \sum_i y_i \times A_{i,j} \le C_j\)
\(max \sum_i y_i \times B_i\)
简略证明:
∵\(B_i \le \sum_j A_{i,j} \times x_j\),∴ \(~\sum y_i B_i \le y_i \sum_j A_{i,j} \times x_j\)
∵\(\sum A_{i,j} y_i \le c_j\), ∴ \(~\sum A_{i,j} y_i x_j\le \sum c_j x_j\)
所以\(\sum y_i B_i \le y_i \sum_j A_{i,j} \times x_j\le \sum c_j x_j\)
所以\(\sum y_iB_i \le \sum c_j x_j\)
大概可以感受到,要使\(\sum c_j x_j\)最小,\(\sum_i y_i \times B_i\)就要最大。
转载注意标注出处:
转自Cold_Chair的博客+原博客地址