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This document available on link https://cran.r-project.org/web/packages/Rcplex/Rcplex.pdf for RCplex might be helpful for you. Page 7 & 8 has an example of QCP.

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Yes. Your feasible region is a convex (bounded) polytope, which means that every point in the feasible region can be written as a convex combination of extreme points. Let's say that your function is $f()$, the maximum occurs at $\bar{x}\in [0,1]^n$, and $\bar{x} = \sum_{i=1}^k \alpha_i x^{(i)}$, where the $x^{(i)}\in [0,1]^n$ are extreme points of the ...

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The problem is infeasible: $(c_1,c_2) \Rightarrow x_1=1$ $(c_3,c_4) \Rightarrow x_2+x_3=2$ $(c_1,c_2,c_7) \Rightarrow x_4 + x_5 \le 2$ $(c_5,c_6) \Rightarrow -(x_2+x_3)+0.4(x_4+x_5) \ge 2 \text{ and so } (c_3,c_4,c_5,c_6)\Rightarrow x_4 + x_5 \ge 10$ The last two lines are not consistent. Note that we also obtain a conflict with $(c_8,c_9)$: \$(c_8,c_9) \...

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