How can I use a lambda function in a constraint with CVXPY?

While CVXPY doesn't directly support lambda functions within constraints, you can achieve similar results by defining a function separately and using it in your constraints. CVXPY requires the constraint to be a valid DCP (Disciplined Convex Programming) expression, which lambda functions, in their raw form, often are not.

Here’s how you can effectively work around this limitation:

1. Define a Function: Instead of using a lambda function directly, define a regular Python function that encapsulates the logic you want to use in your constraint.

2. Use the Function in Your Constraint: Pass the variables you are using in the optimisation problem through this function, and add the constraint with the return of this function.

3. Example:

Let's say you want a constraint to enforce that x squared, plus y, should be less than 10. You can’t do cp.constraints = [lambda x, y: x2 + y <= 10], but here is an example that works.

import cvxpy as cp

def my_constraint_function(x, y):
  return x2 + y

# Define variables
x = cp.Variable()
y = cp.Variable()

# Define constraints using the function
constraints = [my_constraint_function(x, y) <= 10, x >= 0, y >= 0]

# Define the objective function (example: minimize x+y)
objective = cp.Minimize(x + y)

# Define and solve the problem
problem = cp.Problem(objective, constraints)
problem.solve()

print("Optimal value: ", problem.value)
print("x: ", x.value)
print("y: ", y.value)


In this example, my_constraint_function calculates x2 + y, and the CVXPY constraint is my_constraint_function(x, y) <= 10.

Important Considerations:

- DCP Compliance: Ensure that your function, when used in the constraint, forms a valid DCP expression. CVXPY won't be able to solve your problem if the constraints don’t adhere to the DCP rules. Most of the time, you will need convex functions.

- Alternative Approaches: If your constraints are more complex, consider if they could be defined using CVXPY's built-in functions or by a sequence of other operations. If your problem doesn't fit into a convex optimization frame, you would need to use alternative tools and methods.

By defining a standard Python function, you gain the flexibility you’d expect from a lambda while maintaining compatibility with CVXPY’s requirements. This approach allows you to express a wide range of constraints without compromising the optimization process.

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