A general formula on the conjugate of the difference of functions. (English) Zbl 0608.90087

Hiriart-Urruty, J.-B.

Can. Math. Bull. 29, 482-485 (1986).

Given an arbitrary function \(g: X\to (-\infty,+\infty]\) and a lower semicontinuous convex function \(h: X\to (-\infty,+\infty]\), we give the general expression of the conjugate \((g-h)^*\) of g-h in terms of \(g^*\) and \(h^*\). As a consequence, we get Toland’s duality theorem: \[ \inf_{x\in X}\{g(x)-h(x)\}=\inf_{x^*\in X^*}\{h^*(x^*)- g^*(x^*)\}. \]

MSC:

90C30	Nonlinear programming
49N15	Duality theory (optimization)

Keywords:

difference of functions; lower semicontinuous convex function; conjugate; duality theorem