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Let $\alpha_i$ and $\beta_i$ be given positive constants for $i\in\{1,2\}$. Also let $B$ be a given positive constant. I need to solve the following problem \begin{align} \max_{x_1,x_2}~&~\alpha_1x_1^2+\alpha_2x_2^2 \\ &s.t.~~\beta_1x_1+\beta_2x_2\leq B\\&~~~~~~~~~x_i\geq 0~,~\forall i\in\{1,2\}\end{align} Is there an analytic solution to this? If not, can we have a numerical method which will solve this.

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Yes. The maximum of a convex function on a convex set, if exists, is always attained at an extreme point of the feasible set. In your case, your feasible set is the triangle whose vertices (extreme points) are $A = (0, 0)$, $B = (\tfrac{B}{\beta_1}, 0)$, and $C = (0, \tfrac{B}{\beta_2})$. It is compact, so the maximum indeed exists.

The point $A$ clearly does not achieve the maximum objective value, and therefore either $B$ or $C$ must be a maximizer. You chose among the points by comparing the objective function value these two points.

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