6

It is possible to apply the notions of stationary points and the second derivative of a function to functionals. For $|\varepsilon|\ll1$ and and a differentiable function $h$, we can write, using Taylor series,$$F(x,y+\varepsilon h,y'+\varepsilon h')=F(x,y,y')+\varepsilon\mathcal{I}(\Delta[y,h])+\frac{\varepsilon^2}2\mathcal I(\Delta_2[y,h])+\mathcal O(\...


6

Coming from the world of optimal control, I tend to view the calculus of variations from a Pontryagin point of view. The conditions stated by Pontryagin are necessary, and sufficient under certain hypotheses (mainly related to the convexity of the function $F$). I used to refer to this article during my PhD: O. L. Mangasarian, Sufficient Conditions for the ...


5

You can try adding a constraint forcing one of the affected variable to be nonzero. If the model becomes infeasible, you can try to find the conflicting constraints. If the model stays feasible, this means that your objective function represents other priorities than you expected.


3

I only skimmed your model so others may be better able to point to the error directly, but here are some reasons this may occur: Constraints: as you mention, perhaps they're set so that it's not possible to apply the treatment. E.g, is the budget accidentally set too tightly so it can't afford it? (There are several sums in your budget_rule - I'd double ...


2

Note that $V(O)$ is simply of the form $\sf Q_1/L_1$ where $\sf Q_1$ is a quadratic and $\sf L_1$ is a linear function of $p$. This can be written as ${\sf{L_2}}+c/\sf{L_1}$ where $\sf L_2$ is also linear in $p$ and $c$ is a constant. Letting ${\sf L_1}:=mp+n$, the second derivative becomes $$V''(O)=c[(mp+n)^{-1}]''=-cm[(mp+n)^{-2}]'=\frac{2cm^2}{(mp+n)^3}$$ ...


1

I suggest to take a look at "Foundations of Optimization" written by Osman Guler and edited by Springer in 2010. The 3rd chapter is wholly dedicated to Variational Principles and in 4.5 "Optimization on Convex Sets" is stated that: One of the most important and basic proprieties of convex functions is the fact that any local minimizer on ...


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