# How to Formulate a Constraint Which Ensures All Variables Have the Same Sign

I'm trying to include a constraint in my problem (to be solved by any convex optimization solver). Let {a,b,c,d ...} be a finite set of continuous variables. How to formulate a constraint which ensure all of these variables, in the same time interval, take either "positive or zero" or "negative or zero" values.

sincerely,

Mohan

• A simple $\mathrm(sign)$ function would be suffice right? Sep 21, 2018 at 6:12
• Thanks for the reply Maxtron but that doesn't answer my question. Here's what I was looking for orinanobworld.blogspot.com/2018/09/… Sep 24, 2018 at 5:02
• The way it is done in the blog you posted means the problem becomes Non Convex. So I'm not sure it would work for you.
– Royi
Jan 24, 2019 at 13:13
• @MohanLal, Could you please mark my answer or write what is missing?
– Royi
Apr 16 at 14:35

The solution from the blog you linked goes as following (Coordinating Variable Signs by Paul Rubin, Web Archive):

Someone asked me today (or yesterday, depending on whose time zone you go by) how to force a group of variables in an optimization model to take the same sign (all nonpositive or all nonnegative). Assuming that all the variables are bounded, you just need one new binary variable and a few constraints.

Assume that the variables in question are $${x}_{1}, {x}_{2}, \ldots, {x}_{n}$$ and that they are all bounded, say $$\forall \; 1 \leq i \leq n, \; {l}_{i} \leq {x}_{i} \leq {u}_{i}$$. If we are going to allow variables to be either positive or negative, then clearly we need $${l}_{i} < 0 < {u}_{i}$$. We introduce a new binary variable $$y \in \left\{ 0, 1 \right\}$$ and for each $$i$$ the constraints become:

$$\forall \; 1 \leq i \leq n, \; {l}_{i} \left( 1 - y \right) \leq {x}_{i} \leq {u}_{i} y$$

If $$y = 0$$ every original variable must be between its lower bound and 0, nonpositive. If $$y = 1$$ every original variable must be between 0 and its upper bound, nonnegative.

Note that trying to enforce strictly positive or strictly negative rather than nonnegative or nonpositive is problematic, since optimization models abhor strict inequalities. The only work around I know is to change "strictly positive" to "greater than or equal to $$> \epsilon$$" for some strictly positive $$\epsilon$$, which creates holes in the domains of the variables (making values between 0 and $$\epsilon$$ infeasible).

The problem with above approach is that it will make a Convex Optimization Problem into Non Convex Optimization Problem because of the multiplication between variables.

For many solvers I guess it will make no difference. Yet if you use Convex Optimization solver or even Disciplined Convex Programming (DCP) based solver (Like CVX) it won't work.

What I suggest doing, in this case, is take advantage that you have only 2 cases. One where all variables are Non Negative and the other when all of them are Non Positive.

Then all you need is to solve the problem with 2 different sets of constraints (Forming 2 different problems). When the constraints are:

• $$\forall \; 1 \leq i \leq n, \; {l}_{i} \leq {x}_{i} \leq 0$$
• $$\forall \; 1 \leq i \leq n, \; 0 \leq {x}_{i} \leq {u}_{i}$$

Then chose the solution with the lowest objective value.

The main advantage is while the solution using Binary Variable (Which probably will do the same) require special MILP / MIQP solver the approach I suggested, for Linear Least Squares problem only require Non Negative Least Squares solver which is easier to find (Usually faster).

## Example with Code

Let's say we want to solve the following problem:

\begin{aligned} \arg \min_{x} \quad & \frac{1}{2} {\left\| A x - b \right\|}_{2}^{2} \\ \text{subject to} \quad & \operatorname{sign} \left( {x}_{i} \right) = \operatorname{sign} \left( {x}_{j} \right) && \forall i, j \in \mathcal{S} \end{aligned}

Where $$\mathcal{S} = \left\{ i \mid {x}_{i} \neq 0 \right\}$$, namely the set of all indices for which the element of $$x$$ isn't zero.
So basically we say all elements should have the same sign or be zero.

Let's say we have a building block for solving Non Negative Least Squares. For instance one could use MATLAB's lsqnonneg().
To solve the problem above one could solve it one time for $$A$$ and the other for $$- A$$. Then take the argument which minimize the objective (If we take the argument of $$- A$$ we need to negate it).

So the MATLAB function would look like:

function [ vX ] = SolveLsSameSign( mA, vB )

[vX1, resNorm1] = lsqnonneg(mA, vB);
[vX2, resNorm2] = lsqnonneg(-mA, vB);

if(resNorm1 < resNorm2)
vX = vX1;
else
vX = -vX2;
end

end


I created a simple script to compare results to CVX using the Mixed Integer Quadratic Programming (MIQP) which is given by:

cvx_solver('Mosek');
% cvx_solver('Gurobi');

cvx_begin()
% cvx_precision('best');
variable vX(numCols, 1);
variable varY(1) binary;
minimize( norm(mA * vX - vB, 2) );
subject to
vX >= (1 - varY) * lowerBound;
vX <= varY * upperBound;
cvx_end


Pay attention that it requires MOSEK or Gurobi to be installed in order to solve this MIQP problem.

The full MATLAB code is available on my StackExchange Signal Processing Q52099 GitHub Repository (Look at the SignalProcessing\Q52099 folder).

• Great idea! I think you should show a sample code. Apr 18, 2020 at 20:14
• @David, I added a simple example.
– Royi
Apr 19, 2020 at 10:02