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

### MSC:

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