Currently LikelihoodProfiler relies on NLopt.
Internally LikelihoodProfiler utilizes :LN_AUGLAG algorithm from NLopt to construct an augmented objective function. Then the augmented objective function with no constraints is passed to an optimization algorithm, which is defined by the keyword argument local_alg (see LikelihoodProfiler.get_interval). Default is :LN_NELDERMEAD, which is the most reliable for the current problem among the derivative-free algorithms. The following gradien based algorithms have also shown good relsults on test problems (see /test/test_grad_algs.jl): :LD_MMA, :LD_SLSQP, :LD_CCSAQ.