I'm interested in understanding whether there is a nice recursive-type relationship between solutions of linear optimizations problems of lower and higher dimensions.
E.g., Fix $a,b,c,d,e,f\in \mathbb{R}$. Suppose I have the solution $(\hat{x}, \hat{y})$ to the 2D linear optimization problem of maximizing $ax+cy$ such that $bx+dy\ge 0, x,y\in [0,1]$. How does this solution relate to the 3D solution $(x^*, y^*, z^*)$ of maximizing $ax+cy+ez$ such that $bx+dy+fz\ge 0, x,y,z\in [0,1]$ ?
Given the 2D solution, is there an optimizing method (or closed form solution) that allows the 3D solution to be attained?
Any reference suggestions or links would be greatly appreciated!