Saturday, February 10, 2018

Singular system of nonlinear equations

For a project I am faced with solving large, sparse systems of nonlinear equations\[F(x)=0\] In some cases I get a a singular system: the Jacobian at the feasible point is not invertable. These models are also solved using a simple iterative scheme, which tolerates such a condition. How would standard solvers handle this?

I generated a small test problem as follows:

where \(F(x)=0\) has no issues, but \(G(y)=0\) is singular. I simply used:\[\begin{align}&y_1+y_2=1\\&y_1+y_2=1\end{align}\] Let's see what feedback we get:

Solver Modeltype Model Status Feedback/Notes
Conopt CNS Locally Infeasible
** Error in Square System: Pivot too small.

**** ERRORS/WARNINGS IN EQUATION g(k1)
     1 error(s): Pivot too small.

**** ERRORS/WARNINGS IN VARIABLE y(k1)
     1 error(s): Pivot too small.
Ipopt CNS Solved Feasible solution
Knitro CNS Solved Feasible solution
Miles MCP Intermediate Infeasible
Failure to converge
Minos CNS Unbounded
EXIT - The problem is unbounded (or badly scaled).
Minos CNS+basis Solved Feasible solution
Path CNS Solved Feasible solution
Path MCP Optimal Feasible solution
Snopt CNS Solved Feasible solution

Some solvers will report a feasible solution (without a further hint). Conopt does not, but gives a good indication where things go wrong. We are not given the set of dependent equations, but Conopt at least points us clearly to one culprit.

Not related to my problem, but another question is: what if singular but no solution exists? E.g. use: \[\begin{align}&y_1+y_2=1\\&y_1+y_2=2\end{align}\] Conopt gives the same results as above. The other solvers will typically report "infeasible". In most cases a solution corresponding to a phase I "min sum of infeasibilities" objective is provided where \(x\) is feasible and \(y\) is infeasible. Some solvers just give back a solution with many infeasibilities.