Can I use QR decomposition as a last step in a Constrained Orthonormal Matrix Optimization problem?

Question

I have the optimization problem: minimise $f(V)$, where $V$ is $N\times N$, subject to

1. $V$ is orthonormal
2. All entries of the first column of $V$ are $1/\sqrt{N}$
3. $V \cdot D \cdot V^T \cdot \mathbf{1}_N = \mathbf{0}_N$ (I do not optimize over $D$)

Because of (1), I resorted to the Manopt package in Matlab where I optimize over a Stiefel Manifold to obtain an Orthonormal Matrix.

But due to (2) and (3), I resorted to the Augmented Lagrangian Method to solve the problem, where I modified my original optimization problem to include (2) and (3).

However, too many iterations are required to nicely obey (2) and (3).

So I am doing some intuitive tricks (with no theoretical justification) after I obtain $V$ via optimizing over the Stiefel manifold:

• Step 1: Solve Augmented Lagrangian Method.
• Step 2: Obtain a $V$.
• Step 3. Replace the first column of $V$ with all $1/\sqrt{N}$ to obtain a modified V
• Step 4. Perform QR decomposition over this modified $V$, to obtain another Orthonormal $V$.

In steps 3 and 4, I have doubts whether these are the right steps to obey my constraints. The surprising thing is that they seem to work, but seems unethical to me, because:

1. In step 3, I am replacing the first column of the $V$ with all $1/\sqrt{N}$, now at this point in time my $V$ may not be orthonormal anymore!
2. Then I do a QR decomposition over this modified $V$ which may not be orthonormal. So this QR decomposition doesn't seem to make sense.

Please let me know whether my trick is right or wrong. If correct, please kindly try to give a proof or theoretical justification. If wrong, then please tell me it's wrong. Also in case of any better alternative, please also tell me that.