Rephrase convex constraint to satisfy DCP rules

Question

_{Crossposted on Stack Overflow}

I have a convex objective function and a convex constraint of the form $$\frac{y^2}{x} + z \leq 1$$ on the domain $x \geq 0$. However, I'm not sure how to encode this constraint in a way that CVXPY will accept it. It seems to be rejecting it because it can't convince itself that the constraint is convex.

Here is a simple example code:

import numpy as np
import cvxpy as cp
n = 5
x = cp.Variable(n)
xtarget = np.random.normal(0, 1, n)
A = np.random.normal(0, 1, [n, n])
B = np.random.normal(0, 1, [n, n])
y = A @ x
z = B @ x
obj = cp.sum_squares(xtarget - x)
constraints = [x>= 0, y ** 2 / x + z <= 1]
prob = cp.Problem(cp.Minimize(obj), constraints)
prob.solve()

This gives the error:

DCPError: Problem does not follow DCP rules. Specifically:
The following constraints are not DCP

I have tried to replace y ** 2 / x + z <= 1 with cp.multiply(y ** 2, cp.pos_inv(x)) + z <= 1 so that it know that $x$ variable is non-negative. It still gives me the DCP error.

Is there a way to reformulate the problem such that it satisfies the DCP rules?

Addendum

Thanks, Rodrigo de Azevedo, for suggesting positive semidefinite programming idea. Here is the code I ended with. I didn't know how to do this cleanly, so here is a hacky way of doing it.

import numpy as np
import cvxpy as cp
n = 5
x = cp.Variable(n, pos = True)
xtarget = np.random.normal(0, 1, n)
A = np.random.normal(0, 1, [n, n])
B = np.random.normal(0, 1, [n, n])
y = A @ x
z = B @ x
obj = cp.sum_squares(xtarget - x)
constraints = [x>= 0, y ** 2 <= 1, 1 - z >= 0]
Ms = [cp.Variable([2, 2], symmetric = True) for i in range(n)]
for i in range(n):
    constraints.append(Ms[i] >> 0)
    constraints.append(Ms[i][0,0] == x[i])
    constraints.append(Ms[i][0,1] == y[i])
    constraints.append(Ms[i][1,0] == y[i])
    constraints.append(Ms[i][1,1] == 1 - z[i])
prob = cp.Problem(cp.Minimize(obj), constraints)
prob.solve()

JEK, I don't understand your code. Why do you define y and z in terms of x? — Rodrigo de Azevedo, Mar 21 '23 at 13:12

Rodrigo de Azevedo · Accepted Answer · 2023-03-05T20:59:09.117

2

Using the Schur complement and Sylvester's criterion, the given inequalities (plus the inequality $z \leq 1$) can be encapsulated in the following linear matrix inequality (LMI)

$$ \begin{bmatrix} x & y \\ y & 1 - z \end{bmatrix} \succeq 0$$

Can you now translate this to CVXPY? You might want to use PSD or >>.

edited Mar 05 '23 at 20:59

answered Mar 05 '23 at 17:17

Rodrigo de Azevedo

23,223

1

Thank you! That’s a really neat solution! I will try it on the computer tonight and once it works, I will accept as answer. – JEK Mar 05 '23 at 17:30
@JEK Happy to help. If it works, please consider sharing the code – Rodrigo de Azevedo Mar 05 '23 at 17:31
Yup. I will try to write it up tonight when I have access to my computer. – JEK Mar 05 '23 at 18:01
1

The atom quad_over_lin in cvxpy does the job. Then you can use all SOCP solvers and are not limited to SDP-aware solvers only. And if you want to reformulate it, rather than use a ready-made function, then it will be cheaper to reformulate it directly as SOCP (although an SDP solver will likely presolve a 2x2 SDP to SOCP, but still). – Michal Adamaszek Mar 06 '23 at 07:12
@MichalAdamaszek That comment would make a good answer. – Rodrigo de Azevedo Mar 06 '23 at 08:20
@MichalAdamaszek You can now consider converting your comment into an answer – Rodrigo de Azevedo May 05 '23 at 19:58

Rephrase convex constraint to satisfy DCP rules

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