In my personal studies/exploration I have come across the following problem: $$\inf_{a \geq 0} \sup_{\substack{0 \leq z \leq 1\\t > 0}} z\sqrt{b(a^2 + s^2)} - \dfrac{at}{2} - \dfrac{az^2}{2}\left(\dfrac{1}{a\sqrt{\delta} + t}\right) + \dfrac{ty\sqrt{\delta}}{2\left(a\sqrt{\delta} + t\right)}$$ I can solve this problem numerical using any of a few numerical optimization algorithms (grid search, golden-section search have attempted an analytic solution, but have made no headway whatsoever. Despite that, I feel an analytic solution is better (and cheaper) for my purposes, but I have made no headway whatsoever towards one. I would appreciate any nudges towards the right approach. Thank you!
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Start by computing an analytic expression for $f(a,t) = \sup_{0 \le z \le 1} \cdots$ where the $\cdots$ is replaced with the inner expression from your problem. You should be able to compute this analytically: it is the maximum of a quadratic function (quadratic in $z$; all other variables are considered constant), so you can solve for $z$ (using the quadratic formula and a case analysis) and then plug that value of $z$ into the expression. You will find that optimal $z$ is either a constant times $(a \sqrt{\sigma}+t)$ or a constant, depending on a case analysis involving the values of $a$, $t$, and other constants.
Next, compute an analytic expression for $g(a) = \sup_{t > 0} f(a,t)$. This too is doable, as $f(a,t)$ is a quadratic function of $t$ (when we consider $a$ as a constant). So you should be able to solve for $t$ (using the quadratic formula), plug in, do a case analysis, and find an analytic expression for $g(a)$.
Now check whether you can compute an analytic expression for $\inf_{a \ge 0} g(a)$. That will depend on the form of $g(a)$. I haven't tried to work through the details to see what form $g(a)$ has; I will leave that to you.
D.W.
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