What is the min-max argument in mathematics?

Question

In the proof of a theorem the author says that he would prove a special case using the min-max argument. After reading the proof I could not infer what the min-max argument actually does. Could someone throw some light on what it could actually mean. I am attaching a screen shot of the proof

Answer 1

The minimax argument is the following. For any $μ\ge 0$

$$E\left[{M^*_t-μ\langle M\rangle_L^{r/2}}\right]+μE[\langle M\rangle]_L^{r/2}\le\inf_{μ>0}{\left\{E\left[\sup_{t\ge0}{(M^*_t-μ\langle M\rangle_L^{r/2})}\right]+μE[\langle M\rangle]_L^{r/2}\right\}}$$

The expression on the LHS is bounded by a min-max (inf-sup) expression on the RHS. Specifically, the proof above consists of the following steps:

1. Obvious equation: $E[M_t^*]=E\left[{M^*_t-μ\langle M\rangle_L^{r/2}}\right]+μE[\langle M\rangle]_L^{r/2}$ for any $μ\ge 0$.
2. Taking the $\sup$ over $t$ on the RHS turns the $=$ to $\le$ for the LHS $$E[M_t^*]\le E\left[\sup_{t\ge0}{\left(M^*_t-μ\langle M\rangle_L^{r/2}\right)}\right]+μE[\langle M\rangle]_L^{r/2}$$
3. Some manipulation on $M_t^*$ (where $B^*_t$ is introduced) and equation $(13.5)$ is reached. This is equivalent to the equation in Step 2. here.
4. Some more manipulation on the RHS (scaling property etc.). Steps 3. and 4. are irrelevant to the min-max argument.
5. The min-max argument: Since $(13.5)$ (modulo the manipulation) holds for any $μ\ge 0$ then is holds also for the $\inf_{μ>0}$.

Step 5. leads to the penultimate equation which can be written as (if we ignore the manipulation and in order to highlight the inf-sup expression on the RHS): \begin{align}E[M_L^*]&\le\inf_{μ>0}{\left\{E\left[\sup_{t\ge0}{(B^*_t-μt^{r/2})}\right]+μE[\langle M\rangle]_L^{r/2}\right\}}\\[0.2cm]\iff E\left[{M^*_t-μ\langle M\rangle_L^{r/2}}\right]+μE[\langle M\rangle]_L^{r/2}&\le\inf_{μ>0}{\left\{E\left[\sup_{t\ge0}{(M^*_t-μ\langle M\rangle_L^{r/2})}\right]+μE[\langle M\rangle]_L^{r/2}\right\}}\end{align}