1732-3916 The p-Factor Method for Nonlinear Optimization Szczepanik Ewa author Prusińska Agnieszka author Tret’yakov Alexey author Uniwersytet w Siedlcach Russian Academy of Education, Russia Correspondence to: Ewa Szczepanik Correspondence to: Agnieszka Prusińska Correspondence to: Alexey Tret’yakov 20 12 2012 Volume 21 2012 141 157 Copyright © 2012 2012 <p>We present the main concept and results of the p-regularity theory (also known as p-factor analysis of nonlinear mappings) applied to nonlinear optimization problems. The approach is based on the construction of p-factor operator. The main result of this theory gives a detailed description of the structure of the zero set of irregular nonlinear mappings. Applications include a new numerical method for solving nonlinear optimization problems and p-order necessary and sufficient optimality conditions. We substantiate the rate of convergence of p-factor method.<br /> </p> 1 Brezhneva O.A., Tret’yakov A.A.; Optimality conditions for degenerate extremum problems with equality constraints, SIAM J. Contr. Optim. 12, 2003, pp. 729–745. 2 Belash K.N., Tret’yakov A.A.; Methods for solving degenerate problems, USSR Com- put. Math. and Math. Phys. 28, 1988, pp. 90–94. 3 Korneva I.T., Tret’yakov A.A.; Application of the factor-analysis to the calculus of variations, Proceedings of Simulation and Analysis in Problems of Decision-Making Theory, Computing Center of Russian Academy of Sciences, Moscow, 2002, pp. 144– 162 (in Russian). 4 Prusi´nska A., Tret’yakov A.A.; P-order Necessary and Sufficient Conditions for Optimality in Singular Calculus of Variations, Discussiones Mathematicae, Differential Inclusions, Control and Optimization 30, 2010, pp. 269–279. 5 Szczepanik E., Tret’yakov A.A.; Methods for irregular equality-constrained optimization problems, Nonlinear Analysis 69, 2008, pp. 4241–4251. 6 Tret’yakov A.A., Marsden J.E.; Factor analysis of nonlinear Mappings: p-regularity theory, Computations on Pure and Applied Analysis 2(4), 2003, pp. 425–445. 7 Tret’yakov A.A.; Necessary and sufficient conditions for optimality of p-th order, Comput. Math. and Math. Phys. 24, 1984, pp. 123–127. 8 Tret’yakov A.A.; The implicit function theorem in degenerate problems, Russ. Math. Surv. 42, 1987, pp. 179–180. 9 Polyak B.T.; Introduction to optimization, Nauka, Moscow 1983 (in Russian). 10 Brezhneva O.A., Tret’yakov A.A.; New methods for solving nonlinear problems, Computing Center of the Russian Academy of Sciences, Moscow 2000 (in Russian).