I have a quadratic binary optimization problem of the form

\begin{align} &\max x^TQx \cr &\text{subject to }x\in\mathcal{X}\subseteq\{0,1\}^n, \end{align}

where $\mathcal{X}$ is the feasible area defined by linear constraints and binary requirement on the variables. I can assume that all entries in the matrix $Q$ are non-negative. My question is as follows: is it possible to write the function $x^TQx$ as a "combination" of two non-negative linear functions in the following way:

$$ x^TQx = aF(x)^2 + bG(x)^2+cF(x)G(x) + dF(x) + eG(x) \tag{1} $$

where $F(x)=\sum_{i=1}^nf_ix_i + f_0$ and $G(x)=\sum_{i=1}^ng_ix_i + g_0$ are linear funtions, and $a,b,c,d,e\geq 0$ are real/rational constants. Or stated in another way; can I find values for $f_i,g_i,\ i=0,\dots,n$ and $a,b,c,d,e$ such that (1) is true?

Thanks in advance

I have a quadratic binary optimization problem of the form

\begin{align} &\max x^TQx \cr &\text{subject to }x\in\mathcal{X}\subseteq\{0,1\}^n, \end{align}

where $\mathcal{X}$ is the feasible area defined by linear constraints and binary requirement on the variables. I can assume that all entries in the matrix $Q$ are non-negative. My question is as follows: is it possible to write the function $x^TQx$ as a "combination" of two non-negative linear functions in the following way:

$$ x^TQx = aF(x)^2 + bG(x)^2+cF(x)G(x) + dF(x) + eG(x) \tag{1} $$

where $F(x)=\sum_{i=1}^nf_ix_i + f_0$ and $G(x)=\sum_{i=1}^ng_ix_i + g_0$ are linear funtions, and $a,b,c,d,e\geq 0$ are real/rational constants. Or stated in another way; can I find values for $f_i,g_i,\ i=0,\dots,n$ and $a,b,c,d,e$ such that (1) is true?

I have a quadratic binary optimization problem of the form $\max x^TQx$ subject to $x\in\mathcal{X}\subseteq\{0,1\}^n$, where $\mathcal{X}$ is the feasible area defined by linear constraints and binary requirement on the variables. I can assume that all entries in the matrix $Q$ are non-negative. My question is as follows: is it possible to write the function $x^TQx$ as a "combination" of two non-negative linear functions in the following way:

$x^TQx = aF(x)^2 + bG(x)^2+cF(x)G(x) + dF(x) + eG(x)\ \ \ \ \ \ \ \ \ \ \ \ (1)$

where $F(x)=\sum_{i=1}^nf_ix_i + f_0$ and $G(x)=\sum_{i=1}^ng_ix_i + g_0$ are linear funtions, and $a,b,c,d,e\geq 0$ are real/rational constants. Or stated in another way; can I find values for $f_i,g_i,\ i=0,\dots,n$ and $a,b,c,d,e$ such that (1) is true?

Thanks in advance

I have a quadratic binary optimization problem of the form

\begin{align} &\max x^TQx \cr &\text{subject to }x\in\mathcal{X}\subseteq\{0,1\}^n, \end{align}

where $\mathcal{X}$ is the feasible area defined by linear constraints and binary requirement on the variables. I can assume that all entries in the matrix $Q$ are non-negative. My question is as follows: is it possible to write the function $x^TQx$ as a "combination" of two non-negative linear functions in the following way:

$$ x^TQx = aF(x)^2 + bG(x)^2+cF(x)G(x) + dF(x) + eG(x) \tag{1} $$

where $F(x)=\sum_{i=1}^nf_ix_i + f_0$ and $G(x)=\sum_{i=1}^ng_ix_i + g_0$ are linear funtions, and $a,b,c,d,e\geq 0$ are real/rational constants. Or stated in another way; can I find values for $f_i,g_i,\ i=0,\dots,n$ and $a,b,c,d,e$ such that (1) is true?

Thanks in advance