Suppose we have a typical minimization problem as follows:
Min $\sum\limits_{i=1}^{n}{\left( {{a}_{i}}{{x}_{i}}-{{b}_{i}}x_{i}^{2} \right)}$
$0\leq {x}_{i}\leq {c}_{i} \,\,\forall i$
$\sum\limits_{i=1}^{n}{{{x}_{i}}}=D$
here (${a}_{i}$, ${b}_{i}$, ${c}_{i}$) $\forall i=1,...,n$ and $D$ are positive parameters. ${x}_{i}$s are decision variables.
As seen, the only expression that makes the problem non-convex is the quadratic term in the objective function. In order to exploit the efficiency of convex optimization problems (having global solution), is there any way to convexify the objective function? For simplicity, suppose that ${a}_{i}$s have meaningful distances from ${b}_{i}$s, e.g. $\,\,\,\,{a}_{i} > 100{b}_{i}$.
Our expression it's $$\sum_{i=1}^na_ix_i-\sum_{i=2}^nb_ix_i^2-b_1\left(D-\sum_{i=2}^nx_i\right)^2,$$ concave of $x_i$ for any $i$ besides $1$, which says that the minimum occurs for $x_i\in\{0,c_i\}$ for $1<i\leq n$.