I have a relatively simple convex optimization problem that involves less than 100 variables but contains a terribly ill-conditioned matrix. I have tried CVX and CPLEX; even though both can typically solve the problem in about 1 second, both fail when the condition number of the matrix becomes very large. An arbitrary-precision solver would be able to solve this problem quickly and accurately. Does any such implementation exist?
Note: The conditioning of the problem has been considered in detail and is not part of this question. I'm just asking about software.