zbMATH — the first resource for mathematics

Primal-dual nonlinear rescaling method for convex optimization. (English) Zbl 1129.90339
Summary: We consider a general primal-dual nonlinear rescaling (PDNR) method for convex optimization with inequality constraints. We prove the global convergence of the PDNR method and estimate the error bounds for the primal and dual sequences. In particular, we prove that, under the standard second-order optimality conditions, the error bounds for the primal and dual sequences converge to zero with linear rate. Moreover, for any given ratio \(0<\gamma<1\), there is a fixed scaling parameter \(k_\gamma>0\) such that each PDNR step shrinks the primal-dual error bound by at least a factor \(0<\gamma<1\), for any \(k\geq k_\gamma\). The PDNR solver was tested on a variety of NLP problems including the constrained optimization problems (COPS) set. The results obtained show that the PDNR solver is numerically stable and produces results with high accuracy. Moreover, for most of the problems solved, the number of Newton steps is practically independent of the problem size.

90C25 Convex programming
90C46 Optimality conditions and duality in mathematical programming
90C51 Interior-point methods
Full Text: DOI
[1] Wright, S., Primal-Dual Interior Points Methods, SIAM, Philadelphia, Pennsylvania 1997. · Zbl 0863.65031
[2] Megiddo, N., Pathways to the Optimal Set in Linear Programming, Interior Point and Related Methods, Edited by N. Megiddo, Springer Verlag, New York, NY, pp. 131–158, 1989.
[3] Mehrotra, S., On Implementation of a Primal-Dual Interior-Point Method, SIAM Journal on Optimization, Vol. 2, pp. 575–601, 1992. · Zbl 0773.90047 · doi:10.1137/0802028
[4] Potra, F., and Wright, S., Interior-Point Methods, Technical Report, Department of Mathematics and Statistics, University of Maryland Baltimore County, 1999. · Zbl 0967.65078
[5] Lustig, I., Marsten, R., and Shanno, D., Computational Experience with a Globally Convergent Primal-Dual Predictor-Corrector Algorithm for Linear Programming, Mathematical Programming, Vol. 66, pp. 23–135, 1994. · Zbl 0811.90068 · doi:10.1007/BF01581140
[6] Vanderbei, R., Symmetric Quasidefinite Matrices, SIAM Journal on Optimization, Vol. 5, pp. 100–113, 1995. · Zbl 0822.65017 · doi:10.1137/0805005
[7] Zhang, Y., Solving Large-Scale Linear Programs by Interior-Point Methods under MATLAB Environment, Department of Computational and Applied Mathematics, Rice University, 1996.
[8] Zhang, Y., Taptia, R., and Dennis, J., On the Superlinear and Quadratic Convergence of Primal-Dual Interior-Point Linear Programming Algorithms, SIAM Journal on Optimization, Vol. 2, pp. 304–324, 1992. · Zbl 0763.90066 · doi:10.1137/0802015
[9] Forsgren, A., and Gill, P., Primal-Dual Interior Methods for Nonconvex Nonlinear Programming, SIAM Journal on Optimization, Vol. 8, pp. 1132–1152, 1998. · Zbl 0915.90236 · doi:10.1137/S1052623496305560
[10] Gay, D., Overton, M., and Wright, M., A Primal-Dual Interior Method for Nonconvex Nonlinear Programming, Technical Report 97–4–08, Computing Science Research Center, Bell Laboratories, Murray Hill, New Jersey, 1997. · Zbl 0908.90236
[11] Nocedal, J., and Wright, S., Numerical Optimization, Springer, New York, NY, 1999. · Zbl 0930.65067
[12] Shanno, D.F., and Vanderbei, R.J., An Interior-Point Algorithm for Nonconvex Nonlinear Programming, Computational Optimization and Applications, Vol. 13, pp. 231–252, 1999. · Zbl 1040.90564 · doi:10.1023/A:1008677427361
[13] Polyak, R., Modified Barrier Functions, Mathematical Programming, Vol. 54, 177–222, 1992. · Zbl 0756.90085 · doi:10.1007/BF01586050
[14] Polyak, R., and Teboulle, M., Nonlinear Rescaling and Proximal-Like Methods in Convex Optimization, Mathematical Programming, Vol. 76 pp. 265–284, 1997. · Zbl 0882.90106
[15] Polyak, R., Griva, I., and Sobieski, J., The Newton Log-Sigmoid Method in Constrained Optimization, A Collection of Technical Papers, 7th AIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, Vol. 3, pp. 2193–2201, 1998.
[16] Polyak, R., Log-Sigmoid Multipliers Method in Constrained Optimization, Annals of Operations Research, Vol. 101, pp. 427–460, 2001. · Zbl 0996.90088 · doi:10.1023/A:1010938423538
[17] Polyak, R., Nonlinear Rescaling vs. Smoothing Technique in Convex Optimization, Mathematical Programming, Vol. 92A, pp. 197–235, 2002. · Zbl 1022.90014 · doi:10.1007/s101070100293
[18] Bondarenko, A.S., Bortz, D.M., and More, J.J, COPS: Large-Scale Nonlinearly Constrained Optimization Problems, Technical Report ANL/MCS-TM-237, Mathematics and Computer Science Division, Argonne National Laboratory, 1999.
[19] Auslender, A., Cominetti, R., and Haddou, M., Asymptotic Analysis for Penalty and Barrier Methods in Convex Inequalities and Linear Complementary Problems, Mathematical Programming, Vol. 71, pp. 51–69, 1995.
[20] Chen, C., and Mangasarian, O., Smoothing Methods for Convex Inequalities and Linear Complementary Problems, Mathematical Programming, Vol. 71, pp. 51–69, 1995. · Zbl 0855.90124 · doi:10.1007/BF01592244
[21] Fiacco, A., and McCormick, G., Nonlinear Programming: Sequential Unconstrained Minimization Techniques, SIAM Classics in Applied Mathematics, SIAM, Philadelphia, Pennsylvania, 1990. · Zbl 0713.90043
[22] Kort, B. W., and Bertsekas, D. P., Multiplier Methods for Convex Programming, Proceedings of IEEE Conference on Decision and Control, San Diego, California, pp. 428–432, 1973.
[23] Ben-tal, A., Yuzefovich, I., and Zibulevski, M., Penalty/Barrier Multiplier Method for Minimax and Constrained Smooth Convex Problems, Research Report 9/92, Optimization Laboratory, Faculty of Industrial Engineering and Management, Technion, Israel, 1992.
[24] Breitfelt, M., and Shanno, D., Experience with Modified Log-Barrier Method for Nonlinear Programming, Annals of Operations Research, Vol. 62, 439–464, 1996. · Zbl 0848.90108 · doi:10.1007/BF02206826
[25] Ng, E., and Peyton, B. W., Block Sparse Cholesky Algorithms on Advanced Uniprocessor Computers, SIAM Journal on Scientific Computing, Vol. 14, pp. 1034–1056, 1993. · Zbl 0785.65015 · doi:10.1137/0914063
[26] Polyak, R., and Griva, I., Nonlinear Rescaling in Constrained Optimization: Primal-Dual approach, Technical Report SEOR–02–05, Fairfax, Virginia, 2002; see http://mason.gmu.edu/!rpolyak/papers.html. · Zbl 1129.90339
[27] Smale, S., Newton’s Method Estimates from Data at One Point, The Merging of Disciplines in Pure, Applied, and Computational Mathematics, Edited by R.E. Ewing, Springer, New York, NY, pp. 185–196, 1986.
[28] Tichonov, A., and Arseniev, V., Methods de Resolution de Problemes Mal Posés, MIR, Moscow, Russia, 1974.
[29] Debreu, G., Definite and Semifinite Quadratic Forms, Econometrica, Vol. 20, pp. 295–300, 1952. · Zbl 0046.24401 · doi:10.2307/1907852
[30] Kantorovich, L.V., Functional Analysis and Applied Mathematics, Uspekhi Mathematicheskikh Nauk, Vol. 3, pp. 89–185, 1948. · Zbl 0034.21203
[31] Nesterov, Yu., and Nemirovsky, A., Self-Concordant Functions and Polynomial-Time Methods in Convex Programming, CEMI Academy of Sciences, Moscow, Russia, 1989.
[32] Polyak, B. T., Introduction to Opitmization, Software, New York, NY, 1987.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.