Working with square roots

One of the strengths in YALMIP is the built-in convexity analysis and modelling framework for conic optimization. The command sqrt is implemented in YALMIP as a convexity aware nonlinear operator. If you use the positive increasing concave operator sqrt in a way that satisfies the implemented rules for preserving convexity (maximize it, bound it from below,…), sqrt will be implemented using second-order cone constraints,

\[\sqrt{x} \geq t \Leftrightarrow x\geq t^2 \Leftrightarrow \left\| \begin{matrix}1-x\\2t\end{matrix}\right\|\leq 1+x\]

However, in many cases you may want to use a square-root in a nonconvex fashion, and solve the problem using a general-purpose nonlinear solver, such as FMINCON. In these situations, you do not want YALMIP to fail due to convexity propagation problems. In other cases, it might be the case that your model satisfies all convexity requirements, but you still want to solve the problem using FMINCON, and in those cases it would be silly to use a second-order cone model for a simple nonlinear function.

The remedy to these situations is to use the operator sqrtm instead. The sqrtm operator is implemented as an callback nonlinear operator, and will therefore not generate any second-order cone constraints. Furthermore, the sqrtm operator does not require convexity.

An alternative which often is even better is to square the constraint as this in many cases turns a general nonlinear non-smooth nonconvex constraint into a nonconvex quadratic constraint which there are more alternatives for if you are aiming for a globally optimal solution.

As an example, consider the following problem. This model will be converted to and solved as a second-order cone program, since sqrt is used in compliance with convexity rules.

x = sdpvar(3,1);
A = randn(3,3);A = A*A';

obj = x'*A*x - sqrt(sum(x));
F   = [1 <= x <= 2];

The following problem will however fail.

obj = x'*A*x + sqrt(sum(x));
F   = [1 <= x <= 2];

By using the sqrtm operator instead, a general nonlinear problem is set up, and a general nonlinear programming solver is called.

obj = x'*A*x + sqrtm(sum(x));
F   = [1 <= x <= 2];

Finally, we also note that square roots can be implemented using powers, such as x^0.5. This should be reserved for situations where you aim for a geometric programming approach.

Another related command is cpower which implements a conic model of powers,obviously reserved to convex cases where you intend to use a conic solver.