## 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 $$\mathbb{R}^ m_ +$$.

### MSC:

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