[Ipopt] Ipopt for Convex Problems

[Bailin Deng]

Hi all,

A few years back, on the mailing list there was a post  (http://list.coin-or.org/pipermail/ipopt/2009-November/001733.html) saying that Ipopt can find the global optimal solution for the following problem if f(x), d(x) are convex and c(x) is linear:

min  f(x)
s.t. c(x) = 0
      d(x) <= 0


Now if d(x) is not convex, but d(x) <=0 defines a convex feasible set, this is still a convex optimization problem. Can Ipopt still find the global minimum? One example is the following constraints:

x_1^2 + x_2^2 - x_3^2 <=0,     (1)
x_3 >=0.                                  (2)

This defines a quadratic cone, which is convex. But the function x_1^2 + x_2^2 - x_3^2 itself is not convex. 


Interestingly, we can also write these constraints as

\sqrt{ x_1^2 + x_2^2 } - x_3 <=0.

In this case, function \sqrt{ x_1^2 + x_2^2 } - x_3 is convex, but it is not differentiable at points where x_1 = x_2 = 0. So to use it in Ipopt, we would wnat to write the constraints in the form of Equations (1) and (2). But then it is unclear to me whether Ipopt will still find the global minimum. Thanks a lot for your help!


Best,
Bailin