非线性规划在证明不等式中的应用(英文)

Application of Nonlinear Programming to the Proof of a Inequality

我们在讨论完全t部图的色等价问题时 ,需要确定变量 αt =-lTQB 在约束 ( β+b) TB( β+b) <α2 t 下的变化范围 ,其中αt ∈R ,l ,b ,β∈Rt-1 ,Q ,B∈R(t-1 )×(t-1 ) (αt,l,b ,Q ,B)的定义见正文中的定理 ) .我们利用非线性规划的方法 ,证明了如下不等式 :ct-dt-1 at<αt <ct+dt-1 at,其中dt-1 =2 (t - 1) /t,ct = t- 1i=1(ni-nt) /t .

When discussing the chromatic equivalence problem of a complete \$t\$-partite graph \$K(n\-1,n\-2,\:,n\-t)\$ we need to determine the range of the variable \$α\-t=-l\+TQβ\$ of which \$β\$ subjects to the constraint \$(β+b)\+TB(β+b)<a\+2\-t\$,where \$a\-t∈R,l,b,β∈R\+\{t-1\} \$and\$ Q,B∈R\+\{(t-1)×(t-1)\}\$(the definitions of \$a\-t,l,b,Q\$ and B see the below Theorem). By the method of nonlinear programming, we prove that \$c\-t-d\-\{t-1\}a\-t<α\-t<c\-t+d\-\{t-1\}a\-t,\$ where \$c\-t=(1/t)(t-1j=1(n\-j-n\-t))\$ and \$d\-\{t-1\}=(2(t-1)/t)\+\{1/2\}\$.