Directional derivative of marginal function in mathematical programming (Q1080367)

This paper considers the convex programming problem \[ (P)\quad \text{Max} f(x),\quad s.t.\quad g_ i(x)\leq b_ i,\quad i=1,...,m,\quad x\in R^ n, \] where f is concave and each \(g_ i\) is convex, and its dual problem \[ (D)\quad Min_{x}\{f(x)+\lambda^ T(b-g(x))\},\quad \lambda \in R^ m. \] It is well known that if there exists \(x^ 0\in R^ n\) such that \(g_ i(x^ 0)\leq b_ i\), \(i=1,...,m\) (Slater's condition) and \(p(b)=\text{Max}\{f(x)|\) \(g_ i(x)\leq b_ i\), \(i=1,...,m\}\) is finite, then (D) has an optimal solution as a Lagrange multiplier of (P). Let \(\Lambda^*\) be the set of all solutions to (D). It is also well known that if Slater's condition holds, then \(\Lambda^*=\partial (-p(b))\), where \(\partial\) means the subdifferential [see e.g. \textit{J. P. Aubin}, ''L'analyse non linéaire et ses motivations economiques'' (1984; Zbl 0551.90001)\(p. 60]\). Hence \[ (*)\quad p'(b,u)=\lim_{\theta \to o^+}\frac{p(b+\theta u)-p(b)}{\theta}=\inf [\lambda^ Tu| \lambda \in \Lambda^*]. \] The main result of this paper is (*), even with a supplementary condition.