# Remove fixed variable from quadratic program

I have a convex quadratic optimization problem with $$n+1$$ variables $$x_i$$

$$\text{minimize}\,f(x)=x^Tc+\frac{1}{2}x^TQx$$ $$s.t.$$ $$Ax=a$$ $$Bx\leq b$$

with exactly two equality constraints $$x_1=-1$$ $$\sum_{i=2}^{n+1}x_i=1$$ and inequality constraints all being of the form $$l_i\leq x_i\leq u_i$$ for potentially all $$i\neq 1$$. I.e. we have that $$x_1$$ is a fixed variable which does not occur in any other constraint.

From computation it looks like the problem can be reduced by completely removing $$x_1$$ and all corresponding elements in $$c$$ and $$Q$$, namely $$c_1$$, $$Q_{1\cdot}$$ and $$Q_{\cdot 1}$$.

My question is: Is it indeed correct that the above problem is equivalent to a problem with everything related to $$x_1$$ being removed? Either a quick explanation or a link to some literature would be really helpful.

• If you were solving an ordinary system of equations and at some point deduced that $x_1=-1$ then you would substitute this value for $x_1$ in all other places. This situation is no different. – Michal Adamaszek Sep 12 at 12:48
• Note that $x_{1}$ appears in cross terms with $x_{2}$, $x_{3}$, etc. You can't simply drop those terms since these do depend on the remaining variables. – Brian Borchers Sep 12 at 13:47