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

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: \[ \inf_{x\in X}\{g(x)-h(x)\}=\inf_{x^*\in X^*}\{h^*(x^*)- g^*(x^*)\}. \]


90C30 Nonlinear programming
49N15 Duality theory (optimization)
Full Text: DOI