# Parametric research for the global minimum in a family of polynomial multivariable functions on closed domains

Consider the family of functions:

$$V(\{x_j\},\{y_j\})=\sum_{j=1}^L\left[\frac{1}{2} x_j^2+ \frac{\beta^2}{2}y_j^2 + \alpha\beta\, x_jy_j \right]$$ Each member of the family is therefore specified by the specific value of $$L=2,3,\dots$$ which, in turn, determines the number of variables $$x_j$$'s and $$y_j$$'s, i.e. $$L+L=2L$$. $$\alpha$$ and $$\beta$$ are real parameters such that $$\alpha \in (-\infty,+\infty)$$ and $$\beta\in(0,1)$$.

Having fixed a certain $$L$$, variables $$x_j$$'s and $$y_j$$'s are subject to the following constraints: $$0\le x_j\le 1 \qquad \forall j$$ $$0\le y_j\le 1 \qquad \forall j$$ $$\sum_{j=1}^L x_j =1$$ $$\sum_{j=1}^L y_j =1$$

My goal is to find the global minimum of $$V$$ on the domain specified by the above constraints, as a function of parameters $$\alpha$$ and $$\beta$$. The tricky aspect of the problem is that, depending on parameters $$\alpha$$ and $$\beta$$, such minimum can fall on the domain boundaries.

I've solved the problem for $$L=2$$ and $$L=3$$ by fully exploring the domain of $$V$$ and, especially, its boundaries. Already with $$L=3$$, the search for the global minimum of $$V$$ is rather long and complicated because the relevant domain is 4D and the boundary is made up of 3D, 2D, 1D and 0D objects (they are obtained, respectively, fixing the value of 1,2,3 and 4 variables). The interesting aspect of this research is that one can eventually identify regions in the $$(\alpha,\beta)$$ plane such that, in each region, the global minimum of $$V$$ has a specific and characteristic functional dependence on $$\alpha$$ and $$\beta$$.

For $$L=2$$ the number of regions in the $$(\alpha,\beta)$$ is 3;

For $$L=3$$ the number of regions in the $$(\alpha,\beta)$$ is 7;

Increasing $$L$$, the dimension of the domain of $$V$$ increases and the complexity linked to the exhaustive exploration thereof explodes.

Question: is this a well-known problem in some branch of Mathematics? Is there a clever way to determine the number and the structure of the regions in the $$(\alpha,\beta)$$ plane for a generic $$L$$? I repeat -to be more clear- that in each such region, the global minumum of $$V$$ on its closed domain exhibits a specific dependence on parameters $$\alpha$$ and $$\beta$$.