Abstract

One of the main drawbacks associated with Interior Point Methods (IPMs) is the perceived lack of an efficient warmstarting scheme which would enable the use of information from a previous solution of a similar problem. Recently there has been renewed interest in the subject. A common problem with warmstarting for IPM is that an advanced starting point which is close to the boundary of the feasible region, as is typical, might lead to blocking of the search direction. Several techniques have been proposed to address this issue. Most of these aim to lead the iterate back into the interior of the feasible region—we classify them as either “modification steps” or “unblocking steps” depending on whether the modification is taking place before solving the modified problem to prevent future problems, or during the solution if and when problems become apparent. A new “unblocking” strategy is suggested which attempts to directly address the issue of blocking by performing sensitivity analysis on the Newton step with the aim of increasing the size of the step that can be taken. This analysis is used in a new technique to warmstart interior point methods: we identify components of the starting point that are responsible for blocking and aim to improve these by using our sensitivity analysis. The relative performance of a selection of different warmstarting techniques suggested in the literature and the new proposed unblocking by sensitivity analysis is evaluated on the warmstarting test set based on a selection of NETLIB problems proposed by [Benson and Shanno, Comput. Optim. Appl., 38 (2007), pp. 371–399]. Warmstarting techniques are also applied in the context of solving nonlinear programming problems as a sequence of quadratic programs solved by interior point methods. We also apply the warmstarting technique to the problem of finding the complete efficient frontier in portfolio management problems (a problem with 192 million variables—to our knowledge the largest problem to date solved by a warmstarted IPM). We find that the resulting best combined warmstarting strategy manages to save between 50 and 60% of interior point iterations, consistently outperforming similar approaches reported in current optimization literature.

MSC codes

  1. 90C51
  2. 90C20
  3. 65K05

Keywords

  1. interior-point methods
  2. warm-start
  3. quadratic programming

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References

1.
H. Y. Benson and D. F. Shanno, An exact primal-dual penalty method approach to warmstarting interior-point methods for linear programming, Comput. Optim. Appl., 38 (2007), pp. 371–399.
2.
H. Y. Benson and D. F. Shanno, Interior-point methods for nonconvex nonlinear programming: Regularization and warmstarts, Comput. Optim. Appl., 40 (2008), pp. 143–189.
3.
J. Fliege, An efficient interior-point method for convex multicriteria optimization problems, Math. Oper. Res., 31 (2006), pp. 825–845.
4.
J. Gondzio, HOPDM $($version $2.12)$—a fast LP solver based on a primal-dual interior point method, European J. Oper. Res., 85 (1995), pp. 221–225.
5.
J. Gondzio, Warm start of the primal-dual method applied in the cutting plane scheme, Math. Program., 83 (1998), pp. 125–143.
6.
J. Gondzio and A. Grothey, Reoptimization with the primal-dual interior point method, SIAM J. Optim., 13 (2003), pp. 842–864.
7.
J. Gondzio and A. Grothey, Parallel interior point solver for structured quadratic programs: Application to financial planning problems, Ann. Oper. Res., 152 (2007), pp. 319–339.
8.
J. Gondzio and J.-P. Vial, Warm start and $\varepsilon$-subgradients in cutting plane scheme for block-angular linear programs, Comput. Optim. Appl., 14 (1999), pp. 17–36.
9.
A. L. Hipolito, A weighted least squares study of robustness in interior point linear programming, Comput. Optim. Appl., 2 (1993), pp. 29–46.
10.
E. John and E. A. Yildirim, Implementation of warm-start strategies in interior-point methods for linear programming in fixed dimension, Comput. Optim. Appl., 41 (2008), pp. 151–183.
11.
I. J. Lustig, R. E. Marsten, and D. F. Shanno, Computational experience with a globally convergent primal-dual predictor-corrector algorithm for linear programming, Math. Program., 66 (1994), pp. 123–135.
12.
I. Maros and C. Mészáros, A repository of convex quadratic programming problems, Technical report DOC 97/6, Department of Computing, Imperial College, London, U.K., 1997.
13.
S. Mehrotra, On the implementation of a primal–dual interior point method, SIAM J. Optim., 2 (1992), pp. 575–601.
14.
M. Steinbach, Markowitz revisited: Mean-variance models in financial portfolio analysis, SIAM Rev., 43 (2001), pp. 31–85.
15.
E. A. Yildirim and S. J. Wright, Warm-start strategies in interior-point methods for linear programming, SIAM J. Optim., 12 (2002), pp. 782–810.

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Published In

cover image SIAM Journal on Optimization
SIAM Journal on Optimization
Pages: 1184 - 1210
ISSN (online): 1095-7189

History

Submitted: 19 December 2006
Accepted: 23 April 2008
Published online: 19 November 2008

MSC codes

  1. 90C51
  2. 90C20
  3. 65K05

Keywords

  1. interior-point methods
  2. warm-start
  3. quadratic programming

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