On the class of \(\mathbb{H}\)-convex functions. (English. Russian original) Zbl 0822.26009

Russ. Acad. Sci., Dokl., Math. 48, No. 1, 95-97 (1994); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 331, No. 4, 391-392 (1993).
The authors investigate the class \(A\) consisting of all functions \(f: \mathbb{R}^ n_ +\to \mathbb{R}\) that satisfy the following conditions:
(i) \(f(x)\geq f(y)\) whenever \(x, y\in \mathbb{R}^ n_ +\) and \(x- y\in \mathbb{R}^ n_ +\);
(ii) for each \(x\in \mathbb{R}^ n_ +\), the function \(\lambda\in \mathbb{R}_ +\mapsto f(\lambda x)\in \mathbb{R}\) is convex.
They associate with \(\mathbb{R}^ n_ +\) a set \(\mathbb{H}\) of real-valued functions and claim that a function \(f: \mathbb{R}^ n_ +\to \mathbb{R}\) is \(\mathbb{H}\circ \mathbb{I}\)-convex (in the sense introduced by S. S. Kutateladze and the second author [Minkowski duality and its applications (Russian), Izdat. “Nauka”, Sibirsk. Otdel., Novosibirsk (1976; M.R. 58 30048)]) if and only if \(f\in A\). An application of this result yields a minimax theorem for a modified Lagrange function assigned to the following optimization problem: minimize \(F(z)\) subject to \(z\in V\) and \(g(z)\in \mathbb{R}^ n_ +\), where \(V\) is a subset of \(\mathbb{R}^ m_ +\).


26B25 Convexity of real functions of several variables, generalizations
90C30 Nonlinear programming