Let $a,b \in \mathbb{R}$ with $a<b$ and $f:\left(a,b\right)^n \to \mathbb{R}$ be a (possibly non-linear) function. I want to maximize $f$ on the region : $$S=\left\{\left(x_1,\cdots,x_n\right) \in \left(a,b\right)^n : x_1 \leq \cdots \leq x_n\right\}$$
i.e., I want to find :
$$\max_{\left(x_1,\cdots,x_n\right) \in S}{f\left(x_1,\cdots,x_n\right)}$$
Question : What are some general methods to work with such maximization problems?
What I know : In my multivariable calculus course, I have learnt about unconstrained maxima problems (with the second derivative tests, i.e. taking gradient of $f$ to be $0$ and showing that the hessian matrix at the critical point to be negative definite and so on). Also, I have learnt about constrained maxima problems with lagrange multiplier method.
I think that one needs some other methods to solve such irregular ($S$ looks pretty irregular to me!) optimization problem which I have not learnt yet. If anyone knows any method to tackle this and can share that, it would be great. Thank you.