Constraint on the sum of the absolute values in QP?

[Adrian Sarno]

I am coding qp() solver to maximizes a quadratic objective.

The x vector is constrained between -1 and +1 by the inequality constraint.

Is it possible to constrain the sum of absolute values to be equal to 1?

like in:        sum_i( abs(x_i)) == 1

thanks in advance

[Martin]

No, it's a non-convex constraint, but the constraint sum_i(abs(x_i)) <= 1 is convex. 

[Adrian Sarno]

thanks for your response, inequality would work fine. I would need a bit of help to code it.

I can't figure out how to code the abs() inside the sum()

The closest I get is to code an inequality constraint on the linear sum_i(x_i) <= 1 

but not sum_i(abs(x_i)) <= 1

  # Create inequality constraint for sum_i(x_i) <= 1

    G = opt.matrix(1.0, (1, n))  #   row vector for sum_i(), with dot product.

    h = opt.matrix(1.0, (n ,1))   # column vector of constraints 

[Martin]

You can express the constraint in terms of the following linear inequality constraints:

and

[Adrian Sarno]

I understand the logic, but I can't find a place in the cvxopt function for quadratic programming to include the t variables.

http://abel.ee.ucla.edu/cvxopt/userguide/coneprog.html#quadratic-programming

How can you add those variables to the model?

[Martin]

You can define a new variable so that

[-I -I ][x] <= [0]
[ I -I ][t]    [0]
[ 0 e’ ]       [1]

corresponds to the inequality in the QP and . In other words, you let the vector in the QP correspond to in your problem formulation.

[Adrian Sarno]

that worked!

thanks a lot, I learned what I needed to know.

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