A converse duality theorem in multiple objective programming (Q1101016)

The author considers the multiobjective problem: \[ (P)\text{ Minimize \(f(x)\) such that }g(x)\leq 0 \] where \(f: R^ n\to R^ k\) and \(g: R^ n\to R^ m.\) The author has associated a Wolfe type multiobjective (that is the new point) dual problem \[ (D)\text{Maximize \(f(x)+y^ Tg(x)e\) such that }\nabla \lambda \quad Tf(x)+\nabla y^ Tg(x)=0, \] \(y\geq 0\), \(\lambda\in \Lambda^+\), \(e=(1,1,...,1)\in R^ k\) and \(\Lambda^+=\{\lambda \in R^ k| \lambda >0\), \(\lambda^ Te=1\}\). In a previous paper [J. Aust. Math. Soc., Ser. A 43, 21-34 (1987; Zbl 0616.90077)] the author has shown that, under some constraint qualifications, to a properly efficient \(x^*\) corresponds a pair \((y,\lambda)\) such that \((x^*,y,\lambda)\) is properly efficient for (D), with, moreover, the equality of the objective functions. In this paper the author shows that, conversely, with the positivity of the Hessian matrix, if \((x^*,y^*,\lambda^*)\) is a properly efficient solution of (D) then \(x^*\) is a properly efficient solution for (P).