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Let $F \colon \mathbb{R}^{n} \times \mathbb{R}^{m} \to \mathbb{R}$ with partial Lipschitz continuous gradient, that is:

  • for any fixed $y \in \mathbb{R}^{m}$, $\nabla_{x} F \left( x , y \right)$ is Lipschitz continuous with Lipschitz constant $L_{1} \left( y \right)$,
  • for any fixed $x \in \mathbb{R}^{n}$, $\nabla_{y} F \left( x , y \right)$ is Lipschitz continuous with Lipschitz constant $L_{2} \left( x \right)$.

To my impression this type of function has the form $$ F \left( x , y \right) = \left\lVert x + \alpha y \right\rVert ^{2} + \beta \left\langle x , y \right\rangle + f_{1} \left( x \right) + f_{2} \left( y \right) $$ for some real number $\alpha , \beta$ and $f_{1} , f_{2}$ are functions whose gradient are Lipschitz continuous. And thus the partial gradient is a separable function. Is this correct? Otherwise can anyone give an example where the partial gradient is not separable?

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What about $F:\mathbb{R}\times\mathbb{R}\to\mathbb{R}:(x,y)\mapsto\sin(x+y)$? It satisfies the hypothesis, but it isn't of the form you suggest.

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