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I'm solving a set of similar optimization problems(quadratic programming). We have $n$ variables $x_i$s. And $n$ positive constnats $a_i$ and another three positve constants $c$, $k$ and $X$. The objective function I'm trying to maximize is like:

$$ \sum_{i} a_i x_i(c-kx_i) $$

And there are $2n$ constraint $$ \sum_i x_i \leq X $$ and $$ x_i \geq 0 $$

I already have a program to solve these problems and I'm looking for some systematical ways to approach this problem theoretically. More specifically, I'm also looking for things like how $x_i$ affects objective function value when $x_i$ is close to $\beta$ or $\frac{\beta}{2}$. I tried to analyze this by looking at some extreme cases but I didn't get much result. I don't have any background in convex analysis and I think there may be some theorems or concepts can help me analyze this problem more systematically.

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    Sensitivity analysis is typically your friend for these problems, but the number of questions you can answer is typically limited, and restricted to the neighborhood of a solution. – 2017-01-11

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Note that $a x (c - x)$ is a concave function of $x$, so your objective is a concave function of $x_1,\ldots, x_n$. This is good if you're trying to maximize the objective: it means, among other things, that any local maximum is a global maximum.

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    Thank you! I just add that I am trying to maximizing the objective function. I should mention it! – 2017-01-11