Given $x_1, x_2, \ldots, x_n$, (each $x_i\in\mathbb{R}^m$), we have $f(x_1, x_2, \ldots, x_n)$.
We also know that $f(x_1, x_2, \ldots, x_n)$ is convex with respect to each $x_i$ when $x_1,\ldots,x_{i-1},x_{i+1},\ldots,x_n$ are fixed.
I want to solve the problem that finds $x_i$'s minimizing $f(x_1,x_2,\ldots,x_n)$.
Hence, I am finding some materials related to this problem. However, it is too hard to find because I do not know how to call this function.
Is there any specific term that describes this function $f$ ?
- $f$ has multiple variables, each of them is vector.
- $f$ is differentiable with respect to each vector variable.
- $f$ is convex with respect to each vector variable, i.e., for $y$ and $z\in \mathbb{R}^m$, $$f(x_1,\ldots,x_{i-1},\theta y+(1-\theta)z,x_{i+1},\ldots,x_n) \le \theta f(x_1,\ldots,x_{i-1}, y,x_{i+1},\ldots,x_n) + (1-\theta)f(x_1,\ldots,x_{i-1}, z,x_{i+1},\ldots,x_n)$$ with any $\theta\in[0,1]$.