On the Implementation of a Primal-Dual Interior Point Method

Abstract

This paper gives an approach to implementing a second-order primal-dual interior point method. It uses a Taylor polynomial of second order to approximate a primal-dual trajectory. The computations for the second derivative are combined with the computations for the centering direction. Computations in this approach do not require that primal and dual solutions be feasible. Expressions are given to compute all the higher-order derivatives of the trajectory of interest. The implementation ensures that a suitable potential function is reduced by a constant amount at each iteration.There are several salient features of this approach. An adaptive heuristic for estimating the centering parameter is given. The approach used to compute the step length is also adaptive. A new practical approach to compute the starting point is given. This approach treats primal and dual problems symmetrically.Computational results on a subset of problems available from netlib are given. On mutually tested problems the results show that the proposed method requires approximately 40 percent fewer iterations than the implementation proposed in Lustig, Marsten, and Shanno Tech. Rep. TR J-89-11, Georgia Inst. of Technology, Atlanta, 1989]. It requires approximately 50 percent fewer iterations than the dual affine scaling method in Adler, Karmarkar, Resende, and Veiga [Math. Programming, 44 (1989), pp. 297-336], and 35 percent fewer iterations than the second-order dual affine scaling method in the same paper. The new approach for estimating the centering parameter and finding the step length and the starting point have contributed to the reduction in the number of iterations. However, the contribution due to the use of second derivative is most significant.On the tested problems, on the average the implementation shown was found to be approximately two times faster than OB1 (version 02/90) described in Lustig, Marsten, and Shanno and 2.5 times faster than MINOS 5.3 described in Murtagh and Saunders A comparison with the results reported in the literature (on mutually tested problems) shows that the method developed in this paper takes approximately 50 percent fewer iterations than the dual affine scaling method as implemented by Adler, Karmarkar, Resende, and Veiga 1 ], 40 percent fewer iterations than the primal-dual method implemented in Lustig, Marsten, and Shanno [18], and 55 percent fewer iterations than the logarithmic barrier function method implemented in Gill, Murray, and Saunders [8]. It requires 35 percent fewer iterations than the second-order dual affine scaling method implemented in Adler, Karmarkar, Resende, and Veiga 1 and 20 percent fewer iterations than the "optimal three-dimensional method" implemented by Domich, Boggs, Donaldson, and Witzgall [5].An efficient preliminary implementation of the proposed approach was developed. On average, it was found to be two times faster than OB1 (version 02/1990) [18]. On average, it was also found to be 2.5 times faster than MINOS 5.3.

Advances in the simulation of viscoplastic fluid flows using interior-point methods

“…As mentioned before, the choice of γ represents a trade-off between improving centering and reducing the complementarity gap. One major improvement in the efficiency of primal-dual IPM has been realized by Mehrotra [30] who proposed an adaptive choice of the centering parameter by using a predictor-corrector scheme. This scheme takes advantage of the fact that factorizing the KKT matrix A of equation (21) is the most demanding step for each iteration.…”

Application of Interior-Point Methods to Model Predictive Control

“…Feasibility can often be recovered by increasing the horizon length N, but when the initial state is not stabilizable, the feasible region will continue to be empty for all N. The existence of a feasible N can easily be checked by solving the following linear program: min u;x;r e T r; 15 where e is the vector whose entries are all 1 subject to the constraints A positive solution to the linear program indicates that a feasible solution does not exist and the horizon length N must be increased. If the feasibility c heck fails for some user supplied upper bound on the horizon length, then current state is not constrained stabilizable for the speci ed regulator.…”

Primal—dual methods for linear programming

“…The important point is to be able to fall back on a guaranteed method should the predictor-corrector direction fail to be a suitable descent direction for the merit function. A similar view was adopted by Mehrotra [Meh90].…”