Is it possible to express inequality constraints as equality constraints?

Question

For quadratic programming, the objective function is:

$$\text{min } \frac{1}{2}x^TQx + c^Tx$$

Subject to: $$Ax \leq b$$

These QP-solvers that can only handle inequality constraints are the most simplest QP-solvers. But adding an equality constraints together with the inequality constraints, can turn the QP-solver to a much more advanced QP-solver.

I got therefore a question. What if I could do like this:

Subject to: $$Ax \leq b$$ $$Gx \leq d$$ $$-Gx \leq -d$$

For example if $G = 5$ and $d = 7$. Assume that we say $x = 0.5$

5*0.5 <= 7.0 // Yes
-5*0.5 <= -7.0 // No

If I say $x = 1.4$

5*1.4 <= 7.0 // Yes
-5*1.4 <= -7.0 // Yes

If I say $x = 1.5$

5*1.5 <= 7.0 // No
-5*1.5 <= -7.0 // Yes

Question:

By using the same inequality constraints, but adding a negative sign on both sides, does this turn two inequality constraints into an equality constraint?

$$Gx = d$$

Or is it important to have some kind of a small number $\epsilon$ ?

$$Gx \leq d + \epsilon$$ $$-Gx \leq -d - \epsilon$$

Answer 1

Yes it works!

Q = [1.200 -5.1000; -5.1000 26.0000];
c = [-2; -6];

% Inequality constraints
A = [1 1; -1 2; 2 1];
b = [2; 2; 3];

% "Equality" constraints
Aeq = [3 6; 3 1];
beq = [5; 1];

% Internal QP-solver for GNU Octave
[x, ~, info] = qp([], Q, c, Aeq, beq, [], [], [], A, b)

% Internal QP-solver for GNU Octave
[x, ~, info] = qp([], Q, c, [], [], [], [], [], [A;Aeq;-Aeq], [b;beq;-beq])


Output is:

x =

   0.066667
   0.800000

info =

  scalar structure containing the fields:

    solveiter = 1
    info = 0

x =

   0.066667
   0.800000

info =

  scalar structure containing the fields:

    solveiter = 3
    info = 0

>>

The only difference is that the QP-solver that using equality constraints as it should be used, was faster than the "trick".