# What does arg min max mean?

For example, as seen in this article on support vector machines there's an optimization step like

$$\arg\min_{\mathbf{w},b } \max_{\boldsymbol{\alpha}\geq 0 } f(\mathbf{w},b,\boldsymbol{\alpha})$$

I think this means, first find every vector $\boldsymbol{\alpha}$ that produces a maximum of f. Of that domain find $\mathbf{w}$ and b that produce a minimum of f. Is this correct?

• It explains in the next line of your link: "that is we look for a saddle point."
– A.Γ.
Oct 16 '15 at 5:32

The space between "arg" and "min" is confusing; it would better be written "argmin". What the operator argmin does, when applied to a function, is pick out the point in the function's domain at which the function takes its minimum value (assuming that the point is unique). In this case, $\arg\min_{\mathbf{w},b } \max_{\boldsymbol{\alpha}\geq 0 } f(\mathbf{w},b,\boldsymbol{\alpha})$ is that value of $(\mathbf{w},b)$ which minimizes $\max_{\boldsymbol{\alpha}\geq 0 } f(\mathbf{w},b,\boldsymbol{\alpha})$.
• And just to be clear, $max_{α≥0}f(w,b,α)$ is the maximum value the function achieves, subject to the constraint alpha greater than or equal to 0 right?
• @Ben: Yes, but one should be clear that it depends on the variable vector $(w,b)$: given any $(w,b),$ it equals $f(w,b,\alpha_0)$, for some (maximizing) $\alpha_0\geqslant0$; that is, $f(w,b,\alpha_0)\geqslant f(w,b,\alpha)$ for all $\alpha\geqslant0$. Generally, $\alpha_0$ will depend on $(w,b)$ too. Oct 16 '15 at 22:43
• @rpmcruz: Wikipedia uses the LaTex default notation, which is indeed common. In the article you refer to, the subscript is display-style, centred under the "arg min", which links the "arg" and the "min". The notational issue arises with text-style subscripts, and with the wrongly typeset form in the OP's question, where the subscript attaches to the "min" only. This can easily be misread, because an expression such as $\min_{x\in A}f(x)$ is already syntactically complete. Nov 25 '15 at 17:38