非线性规划的QP-free方法

Abstract

Abstract:

In this paper, A QP-free feasible method is proposed to obtain the local convergence under some weaker conditions for the minimization of a smooth function subject to smooth inequalities. Based on the solutions of linear systems of equation reformulation of the KKT optimality conditions, this method uses the 3-1 NCP function^[1]. The method is iterative, which means each iteration can be viewed as a perturbation of a Newton or Quasi Newton on both the primal and dual variables for the solution of the equalities in the KKT first order conditions of optimality, and the feasibility of all iterations is ensured in this method. In particular, this method is implementable and
globally convergent without assuming the strict complementarity condition, the isolation of the accumulation point and the linear
independence of the gradients of active constrained functions. The method has also superlinear convergence rate under some mild
conditions which are the same as those in[2]. Some preliminary numerical results indicate that this new QP-free feasible method is quite promising.

Key words: Filter, QP-free method, Constrained function, Convergence, NCP function

CLC Number:

90C30

JIANG Ai-Ping. QP-free Method for Nonlinear Programming Problems[J].Acta mathematica scientia,Series A, 2011, 31(1): 103-116.

Trendmd

References

[1] Han J Y, Xiu N H, Qi H D. Nonlinear Omplementary Theory and Algorithm. Shanghai: Shanghai Science and Technology Press, 2006

[2] Qi H, Qi L. A new QP-free, globally convergent, locally superlinearly convergent algorithm for inequality constrained optimization. SIAM J Optim, 2000, 11: 113--132

[3] Ferris M C, Pang J S. Engineering and economic applications of complementarity problems. SIAM Review, 1997, 39: 669--713

[4] Fukushima M. Merit Functions for Variational and Complementarity Problems//Di Pillo G, Giannessi F, eds. Nonlinear Optimization and Applications. New York: Plenum Press, 1996: 155--170

[5] Harker P T, Pang J S. Finite-dimensional variational and nonlinear complementarity problems: a survey of theory, algorithm and applications. Mathematical Programming, 1990, 48: 161--220

[6] Pang J S. Complementarity Problems//Horst R, Pardalos P, eds. Handbook of Global Optimization. Boston: Kluwer Academic Publishers, 1995: 271--338

[7] Facchinei F, Kanzow C. A nonsmooth inexact Newton method for the solution of large nonlinear complementarity problems. Mathematics Programming, 1997, 76: 493--512

[8] Herskovits J N. A two-stage feasible direction algorithm for nonlinear constrained optimization. Math Programming, 1986, 36: 19--38

[9] Herskovits J N. Feasible direction interior-point technique for nonlinear optimization. J Optim Theory Appl, 1998, 99: 121--146

[10] Panier E R, Tits A L, Herskovits J N. A QP-free, globally, locally superlinear convergent method for the inequality constrained optimization problems. SIAM J Control Optim, 1988, 36: 788--811

[11] Sun D. A regularization Newton method for solving nonlinear complementarity problems. Applied Mathematics and Optimization, 1999, 40: 315--339

[12] Urban T, Tits A L, Lawrence C T. A Primal-dual Interior-point Method for Nonvex Optimization with Multiple Logarithmic Barrier
Parameters and with Strong Convergence Properties. Maryland: Department of Electrical Engineering and Institute for Systems Research,
University of Maryland, College Park, 1998

[13] Panier E R, Tits A L. A superlinear convergent feasible method for the solution of inequality constrained optimization problems. SIAM
J Control Optim, 1987, 25: 934--950

[14] Clarke F H. Optimization and Nonsmooth Analysis. Philadelphia: SIAM, 1990

[15] Chen X, Qi L, Yang Y F. Lagrangian globalization methods for nonlinear complementarity problem. Optim Theory Appl, 2002, 112: 77--95

[16] Pu D, Zhang J. An inexact generalized Newton method for second order C-differentiable optimization. J of Computational and Applied Mathematics, 1998, 93: 107--122

[17] Qi L. Convergence analysis of some method for solving nonsmooth equations. Marhematics of Operations Research, 1993, 18: 227--243

[18] Schittkowski K. More Test Examples for Nonlinear Programming Codes. New York: Springer-Verlag, 1988

QP-free Method for Nonlinear Programming Problems

PDF (PC)

Abstract

Cite this article

share this article

References

Related Articles 15

Metrics

Comments

Recommended 10

[1]	Zhu Zhifeng, Zhang Shaoyi. Study on the Convergence of Nonhomogeneous Markov Chains with Probability Distance [J]. Acta mathematica scientia,Series A, 2018, 38(5): 963-969.
[2]	Wang Xiaoyan, Ding Yiming. A Filtering Algorithm for Removing Cauchy Noise in Images [J]. Acta mathematica scientia,Series A, 2018, 38(4): 823-832.
[3]	Yang Xuan, Ruan Xiaoe, Wang Peng. Convergence Analysis of Iterative Learning Control for Linear Continuous-Time Switched Systems with Arbitrary Time-Driven Switching Rules [J]. Acta mathematica scientia,Series A, 2018, 38(3): 599-612.
[4]	Chen Hongbin, Gan Siqing, Xu Da, Peng Yulong. A Formally Second-Order BDF Compact ADI Difference Scheme for the Two-Dimensional Fractional Evolution Equation [J]. Acta mathematica scientia,Series A, 2017, 37(5): 976-992.
[5]	Fu Zongkui, Wu Qunying. A General Result of Precise Asymptotics in Complete Moment Convergence for ρ-mixing Sequence [J]. Acta mathematica scientia,Series A, 2017, 37(3): 544-552.
[6]	Wang Xuewu. Convergence and Stability of Ishikawa Type Iterative Sequences for the φ-Hemi Pseudocontractive Operator Pair [J]. Acta mathematica scientia,Series A, 2017, 37(2): 248-256.
[7]	Qiu Dehua, Duan Zhenghua. Complete Moment Convergence for Weighted Sums of ρ-Mixing Random Variable Sequences [J]. Acta mathematica scientia,Series A, 2017, 37(2): 265-277.
[8]	Cai Gang, Yekini Shehu. Viscosity Iterative Algorithm for Variational Inequality Problems and Fixed Point Problems of Strict Pseudo-Contractions in q-Uniformly Smooth and Uniformly Convex Banach Spaces [J]. Acta mathematica scientia,Series A, 2016, 36(4): 623-638.
[9]	Zhang Houchao, Shi Dongyang. Superconvergence Analysis of a Lower Order Mixed Finite Element Method for Nonlinear Fourth-Order Hyperbolic Equation [J]. Acta mathematica scientia,Series A, 2016, 36(4): 656-671.
[10]	Li Wei. Complete Convergence for Weighted Sums Under END Setup [J]. Acta mathematica scientia,Series A, 2016, 36(3): 448-455.
[11]	Yao Qingliu. Existence and Successively Iterative Method of Singular Solution to a Nonlinear Fractional Differential Equation [J]. Acta mathematica scientia,Series A, 2016, 36(2): 287-296.
[12]	Wang Zhi, Jing Huiting, Yan Litan. Approximation of Multidimensional Parameter Fractional Brownian Sheet in Skorokhod Space [J]. Acta mathematica scientia,Series A, 2016, 36(1): 168-175.
[13]	Liu Yuanxing, Zhou Haiyun, Wang Peiyuan, Zhou Yu. Three Kinds of Iterative Algorithms for Solving the Constrained Split Common Fixed Point Problem in Hilbert Spaces [J]. Acta mathematica scientia,Series A, 2015, 35(6): 1115-1126.
[14]	Yang Liqin, Deng Fangwen. Random Iteration of Holomorphic Self-Maps on Non-Smooth Bounded Convex Domain [J]. Acta mathematica scientia,Series A, 2015, 35(6): 1127-1135.
[15]	Li Yingyi, Zhang Haibin, Gao Huan. A Modified Proximal Gradient Method and Its Convergence Rate [J]. Acta mathematica scientia,Series A, 2015, 35(6): 1136-1145.