cplexquadratic-programmingconvex

CPLEX says a sum of squares is not convex

I want to solve a maximisation problem with objective

f(x, y, z) = 0.5 * (x^2 + y^2 + z^2)

with x, y, z in [0, 1].

However when feeding this problem to CPLEX with the code below, it prompts an error saying that my objective is non-convex.

Also I should say that I checked the .lp file to be sure that I have correctly stated the problem.

import cplex


p = cplex.Cplex()
p.objective.set_sense(p.objective.sense.maximize)

p.variables.add(ub=[1, 1, 1],
                names=["x", "y", "z"])


qmat = [[[0], [1.]],
        [[1], [1.]],
        [[2], [1.]]]
p.objective.set_quadratic(qmat)

p.write("qpex.lp")

p.solve()

I don't understand why since my function f is definitely convex (the Hessian is the identity matrix). What am I doing wrong?

Solution

  • Notice that when you specify the objective 

    p.objective.set_sense(p.objective.sense.maximize)

    You have specified it to be a maximization problem.

    To make it a convex optimization problem, you should be solving a minimization problem instead.

    Minimizing a convex function is a convex optimization problem, maximizing a convex function in general is not a convex optimization problem.