An arc-search interior-point algorithm for nonlinear constrained optimization

This paper proposes a new arc-search interior-point algorithm for the nonlinear constrained optimization problem. The proposed algorithm uses the second-order derivatives to construct a search arc that approaches the optimizer. Because the arc stays in the interior set longer than any straight line, it is expected that the scheme will generate a better new iterate than a line search method. The computation of the second-order derivatives requires to solve the second linear system of equations, but the coefficient matrix of the second linear system of equations is the same as the first linear system of equations. Therefore, the matrix decomposition obtained while solving the first linear system of equations can be reused. In addition, most elements of the right-hand side vector of the second linear system of equations are already computed when the coefficient matrix is assembled. Therefore, the computation cost for solving the second linear system of equations is insignificant and the benefit of having a better search scheme is well justified. The convergence of the proposed algorithm is established. Some preliminary test results are reported to demonstrate the merit of the proposed algorithm.