Tracking local optimality for cost parameterized optimization problems

In this paper, a procedure for computing local optimal solution curves of the cost parameterized optimization problem is presented. We recast the problem to a parameterized nonlinear equation derived from its Lagrange function and show that the point where the positive definiteness of the projected Hessian matrix vanishes must be a bifurcation point on the solution curve of the equation. Based on this formulation, the local optimal curves can be traced by the continuation method, coupled with the testing of singularity of the Jacobian matrix. Using the proposed procedure, we successfully compute the energy diagram of rotating Bose–Einstein condensates.