On a representation of convex vector functions and the maximal cyclic monotonicity of their subdifferential (Q1306562)

The author presents a~convex (with respect to a~cone) vector function \(f\) defined on a~nonempty relatively open set \(D\subset R^n\) with values in \(R^m\) in the following form \[ f(x)=f(x_0)+\sup \Biggl\{ \sum_{i=0}^{k-1}\langle A_i, x_{i+1}-x_i\rangle + \langle A_k,x-x_k\rangle: (x_i,A_i)\in \text{ graph }\partial f,\;i=1,\ldots ,k ,\;k\geq 1 \Biggr\} \] for arbitrary \(x\in D\) and \((x_0,A_0)\in \text{ graph }\partial f\). He has also proved the maximal cyclic monotonicity of the subdifferential of such a~function.