Duality in nonlinear programs using augmented Lagrangian functions (Q1090612)

The problem to determine \(V_ p=\min \{f(x)\); \(x\in S\subset R^ n\), \(g_ i(x)=0\), \(i=1,...,m\}\) is considered, by means of the generalized Lagrangian \(L(x,y,r)=f(x)+\phi (g(x),y,r)\), \((y,r)\in R^ m\). Conditions on the augmented multiplier function \(\phi\) are derived, which imply the duality relation \(V_ p=\sup_{y,r} \inf_{x\in S} L(x,y,r)\). These conditions are verified for some particular functions \(\phi\).