# Tag Info

3

From the looks of it (simple feasible set, convex objective, gradient available), Frank-Wolfe indeed makes a lot of sense here. I can point out that there exist many variants of the algorithm, and that several are implemented in this recent Julia package. This can give you a quick way to experiments various algorithmic configurations. If you do decide to ...

3

Using Calculus of Variations as an inspiration we have the lagrangian $$\mathcal{L}(f,\lambda) = \int_0^{x_{max}}\left(-\ln\left(\alpha(x)+f(x)\right)+\lambda\left(f(x)-\frac{1}{x_{max}}\right)\right)dx\ \ \ \text{s. t.}\ \ \ f(x)\ge 0$$ The variation regarding $f$ gives  -\frac{1}{\alpha(x)+f(x)}+\lambda = 0\Rightarrow f(x) = \frac{1}{\lambda}-\alpha(x)...

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The KKT conditions are necessary conditions for an optimum to your problem, so if you can find all feasible points satisfying them, the one with the best objective function will be your optimum. There is no need to consider the corners of the feasible region explicitly. If any of them is optimal, it will also be a KKT point.

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I hesitate to suggest it, since gradient methods tend to be faster than non-gradient methods, but the Nelder-Mead "simplex" algorithm is easy to code and might be worth a try.

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When you have only box constraints I don't think Frank Wofle is very efficient. Frank Wolfe can handle more complex constraints. You should try a quasi newton algorithm like l-bfgs-b or a truncated newton conjugate gradient algorithm with projection.

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I can't tell if your path is right (because it depends on that you want to do), but it is sensible. Your problem structure is quite similar to one occurring in the field of optimal control. I'm not aware of any closed form solution of your approach, so you will have to pick a numerical approach to get started. As you are probably aware, you can't express ...

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The property that some infinitesimal change in one of the constraints impacts the objective is called "a constraint being active" or "a constraint being in conflict with the objective". "if we increase $b$ arbitrarily small, we can also bound the change in the optimal objective value with an arbitrarily small change." This ...

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As far as i know you will have to resort to a bi-level optimzation problem: $\min_{A,B} x$ subject to ($\max_x$ subject to $x\leq A$ $x \leq B$) solve a series of non linear problems where you approximate the $\max$ term by something non-linear (a soft max) and let that converge against $\max$ be fine with an $\frac{1}{m}$ error and turn it into an Mixed ...

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