锥约束变分不等式问题的数值方法的研究
|
摘要
变分不等式问题在运筹学、计算机科学、系统科学、工程技术、交通、经济与管理等许多方面有广泛应用。在二十世纪最后20年里,它受到许多学者的特别关注。另外,锥约束优化,尤其半定规划和二阶锥规划也是目前最优化领域的研究热点之一,而锥约束变分不等式的研究还很初步.本论文主要研究了锥约束变分不等式问题的数值方法的收敛性,包括求解二阶锥约束变分不等式的半光滑Newton方法与光滑函数方法,以及Hilbert空间中的变分不等式的迫近点算法的收敛性.
本论文所阐述的主要研究结果可概括如下:
1.第二章主要研究二阶锥约束变分不等式的半光滑Newton方法,运用Fischer-Burmeister函数将二阶锥约束变分不等式的Karush-Kuhn-Tucker条件转化为非光滑方程组问题Φ_(FB)=0,并给出了映射Φ_(FB)的Clark广义微分(?)Φ_(FB)的表达式。在一定条件下证明了该广义微分的非奇异性。提出采用Armijo线搜索的求解该非光滑方程组的修正牛顿法,证明了该算法的全局收敛性和局部超线性收敛性。最后给出数值例子验证了算法的有效性。
2.第三章主要用光滑化方法求解二阶锥约束变分不等式。运用光滑化的Fischer-Burmeister函数将二阶锥约束变分不等式的Karush-Kuhn-Tucker条件转化为光滑方程组问题E=0.在一定条件下,证明了映射E的Jacobin矩阵JE的非奇异性。运用
光滑的牛顿算法求解该光滑方程组问题,证明了算法的全局收敛性。最后给出数值例子验证了算法的有效性。
3.第四章主要研究了基于预解算子理论的一般变分不等式的求解方法。我们引入了Hilbert空间中一类新的单调算子,即M-单调算子,建立了一般变分不等式和一个不动点问题的等价性。为了求解该不动点问题,本文提出了一个迫近点算法。在一定条件下证明了该迫近点算法的全局收敛性。而后,将上述理论应用到半定矩阵空间中的变分不等式的求解。为了保证算法的可行性,还另外给出了求解不动点问题的沂似解方法。
It is well known that variational inequality problems have many important applications in operation research,computer science,system science,engineering technology,transportation, economics and management ect.In the last 20 years of the twentieth century,great attentions have been paid by many scholars in this direction.Meanwhile,conic optimization,especially semidefinite programming and second order cone programming,is an active field in optimization community.However,the study of variational inequality problems with cone constraints is not enough.Based on this observation,this dissertation focuses on the study of convergence analysis of numerical methods for variational inequality problems with cone constraints,including semismooth Newton methods and smoothing function methods for second-order cone constrained variational inequality problems and proximal point methods for variational inequality problems in Hilbert space.The main results of this dissertation can be summarized as follows:
1.In the second chapter,the Karush-Kuhn-Tucker system of a second order cone constrained variational inequality problem is transformed into a semismooth system of equations with the help of Fischer-Burmeister operators over second order cones.The Clarke generalized differential of the semismooth mapping is presented.A modified Newton method with Armijo line search is proved to have global convergence with local superlinear rate of convergence under certain assumptions on the variational inequality problem.An illustrative example is given to show how the globally convergent method works.
2.The third chapter mainly deals with the smooth method for second-order cone constrained variational inequality problems.The Karush-Kuhn-Tucker system of a second order cone constrained variational inequality problem is transformed into a smooth system of equations E=0 with the help of smoothing Fischer-Burmeister operators over second order cones.Under some conditions,we prove the Jacobin JE of E is nonsingular.A global convergent smooth Newton method is given for solving the smooth system of equations.
3.Chapter 4 discuss the method based on the resolvent operator for general variational inequality. A new monotonicity,M-monotonicity,is introduced.With the help of resolvent operator, an equivalence between the variational inequality VI(C,F+G) and the fixed point problem of a nonexpansive mapping is established.A proximal point algorithm is constructed to solve the fixed point problem,which is proved to have a global convergence under the condition that F in theⅥproblem is strongly monotone and Lipschitz continuous. Furthermore,a convergent path Newton method,which is based on the assumption that the projection mapping∏_C(·) is semismooth,is given for calculatingε-solutions to the sequence of fixed point problems,enabling the proximal point algorithm implementable.
本论文所阐述的主要研究结果可概括如下:
1.第二章主要研究二阶锥约束变分不等式的半光滑Newton方法,运用Fischer-Burmeister函数将二阶锥约束变分不等式的Karush-Kuhn-Tucker条件转化为非光滑方程组问题Φ_(FB)=0,并给出了映射Φ_(FB)的Clark广义微分(?)Φ_(FB)的表达式。在一定条件下证明了该广义微分的非奇异性。提出采用Armijo线搜索的求解该非光滑方程组的修正牛顿法,证明了该算法的全局收敛性和局部超线性收敛性。最后给出数值例子验证了算法的有效性。
2.第三章主要用光滑化方法求解二阶锥约束变分不等式。运用光滑化的Fischer-Burmeister函数将二阶锥约束变分不等式的Karush-Kuhn-Tucker条件转化为光滑方程组问题E=0.在一定条件下,证明了映射E的Jacobin矩阵JE的非奇异性。运用
光滑的牛顿算法求解该光滑方程组问题,证明了算法的全局收敛性。最后给出数值例子验证了算法的有效性。
3.第四章主要研究了基于预解算子理论的一般变分不等式的求解方法。我们引入了Hilbert空间中一类新的单调算子,即M-单调算子,建立了一般变分不等式和一个不动点问题的等价性。为了求解该不动点问题,本文提出了一个迫近点算法。在一定条件下证明了该迫近点算法的全局收敛性。而后,将上述理论应用到半定矩阵空间中的变分不等式的求解。为了保证算法的可行性,还另外给出了求解不动点问题的沂似解方法。
It is well known that variational inequality problems have many important applications in operation research,computer science,system science,engineering technology,transportation, economics and management ect.In the last 20 years of the twentieth century,great attentions have been paid by many scholars in this direction.Meanwhile,conic optimization,especially semidefinite programming and second order cone programming,is an active field in optimization community.However,the study of variational inequality problems with cone constraints is not enough.Based on this observation,this dissertation focuses on the study of convergence analysis of numerical methods for variational inequality problems with cone constraints,including semismooth Newton methods and smoothing function methods for second-order cone constrained variational inequality problems and proximal point methods for variational inequality problems in Hilbert space.The main results of this dissertation can be summarized as follows:
1.In the second chapter,the Karush-Kuhn-Tucker system of a second order cone constrained variational inequality problem is transformed into a semismooth system of equations with the help of Fischer-Burmeister operators over second order cones.The Clarke generalized differential of the semismooth mapping is presented.A modified Newton method with Armijo line search is proved to have global convergence with local superlinear rate of convergence under certain assumptions on the variational inequality problem.An illustrative example is given to show how the globally convergent method works.
2.The third chapter mainly deals with the smooth method for second-order cone constrained variational inequality problems.The Karush-Kuhn-Tucker system of a second order cone constrained variational inequality problem is transformed into a smooth system of equations E=0 with the help of smoothing Fischer-Burmeister operators over second order cones.Under some conditions,we prove the Jacobin JE of E is nonsingular.A global convergent smooth Newton method is given for solving the smooth system of equations.
3.Chapter 4 discuss the method based on the resolvent operator for general variational inequality. A new monotonicity,M-monotonicity,is introduced.With the help of resolvent operator, an equivalence between the variational inequality VI(C,F+G) and the fixed point problem of a nonexpansive mapping is established.A proximal point algorithm is constructed to solve the fixed point problem,which is proved to have a global convergence under the condition that F in theⅥproblem is strongly monotone and Lipschitz continuous. Furthermore,a convergent path Newton method,which is based on the assumption that the projection mapping∏_C(·) is semismooth,is given for calculatingε-solutions to the sequence of fixed point problems,enabling the proximal point algorithm implementable.
引文
[1]Signorini A.Sopra alcune questioni di elastostatica.Atti.Soc..ltal.per il progesso delle Sciercze,1933.
[2]Fichera G.Problemi elastostatici con vincoli unilaterali:il problema di Signorini ambigue condizione al contorno.Attem.Acad.Naz.Lincei.Mem.C1.Sci.Nat.Sez.la,1963/64,7(8):91-140.
[3]Stampacchia G.Forms bilinearires coercitives sur les ensembles convexs.C.R.Acad.Sci.Paris,1964,258:4413-4416.
[4]Lion J L,Stampacchia G.Variational inequalities.Comm.Pure Appl.Math.,1967,20:493-519.
[5]Duvaut G,Lions J L.Inequalities in mechanics and physics.Berlin:Speinger-verlag,1976.
[6]Chipot M.Variational inequalities and flow in porous media.New York:Springer-Verlag,1984.
[7]Friedman A.Variational principle and flee-boundary problems.New York:John Wiley,1982.
[8]Glowinski R.Numercial methods for nonlinear variational Problems.New York:Springer-Verlag,1984.
[9]Glowinski R,Lions J L,Tremolieres R.Numerical analysis of variational inequalities.Amsterdam:North-Holland,1981.
[10]Han W,Reddy B D.Plasticity:Mathemaical theory and numercial analysis.New York:Springer-Verlag,1999.
[11]Han W,Sofonea M.Quasistatic contact problems in viscoelasticity and viscoplasticit.American Mathmatical Society and International Press,2002.
[12]Kinderlehrer D,Stampacchia G.An introduction to variational inequalities and their applications.New York:Academic Press,1980.
[13]Panagiotopoulos P D.Inequalities problems in mechanics and applications.Boston:Birkhauser,1985.
[14]Rodrigues J F.Obstacle problems in mathematical physics.Amsterdam:North-Holand,1987.
[15]Huang N J,Deng C X.Auxiliary principle and iterative algorithms for generalized set-valued strongly nonlinear mixed variational-like inequalities.J.Math.Anal.Appl.,2001,256:345-359.
[16]Ding X P.Preditor-corrector iterative algorithms for solving generalized mixed quasi-variational-like inclusion.Journal of Computational and Applied Mathematics,2005,182(1):1-12.
[17]Verma R U.On generalized variational inequalities involving relaxed Lipschitz and relaxed monotone operators.J.Math.Anal.Appl.,1997,213:387-392.
[18]Liu Z,Ume J S,Kang S M.General stongly nonlinear quasi-variational inequalities with relaxed Lipschitz and relaxed monotone mappings.J.Optim.Theory Appl.,2002,114:639-656.
[19]张石生.变分不等式和相补问题理论与应用.上海:上海科学技术文献出版社,1991.
[20] Isac G, Bulavaski V, Klashnikov V. Exceptional families, topological degree and complementarity problems. Journal of Global Optimization, 1997,10(2): 207-225.
[21] Zhao Y B, Han J Y, Qi H D. Exceptional families and existence theorems for variational inequality problems. Journal of Optimization Theory and Applications, 1999, 101(2): 475-495.
[22] Zhou S Z. Bai M R. A new exceptional family of elements for a variational inequality problem on Hilbert space. Appl. Math. Lett., 2004, 17(4):423-428.
[23] Cottle R W, Pang J, Stone R. The linear complementarity problems. Boston: Academic Press, 1992.
[24] Cottle R W. Nonlinear programs with positively bounded Jacobians. Journal of the Society for Industrial and Applied Mathematics, 1966, 14: 147-158.
[25] Habetler G J, Kostreva M M. On a direct algorithm for nonlinear complemen- tarity problems. SIAM Journal on Control and Optimization, 1978, 16: 504-511.
[26] Todd M J. Computation of fixed points and applications. Vol. 124 of Lecture Notes in Economics and Mathematical Systems, Springer Berlag, Heidelberg, 1976.
[27] Kojima M, Mizuno S, Noma T. A new continuation method for complementarity problems with uniform P-functions. Mathematical Programming, 1989,43: 107-113.
[28] Chen B, Harker P T. A non-interior continuation method for linear complementarity problems. SIAM Journal on Matrix Analysis, 1993, 14: 1168-1190.
[29] Kojima M, Shindo S, Hara S, Interior-point methods for the monotone semideffinite linear complementarity problem in symmetric matrices. Research Report on Information Sciences B-282, Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, Tokyo, Japan, 1994.
[30] Wang T, Monteiro R D C, Pang J S. An interior point potential reduction method for constrained equations. Manuscript, Department of Mathematical Sciences, The Johns Hopkins University, Baltimore, Maryland, 1994.
[31] Wright S J. An infeasible-interior-point algorithm for linear complementarity problems. Mathematical Programming, 1994, 67: 29-51.
[32] Lustig I J, Marsten R E, Shanno D F. Interior point methods for linear programming: computational state of the art. ORSA Journal of Computing, 1994, 6(1): 1-38.
[33] Cryer C W. The methods of Christopherson for solving free boundary problems for infinite journal bearings by means of finite differences. Mathematics of computation, 1971,25: 435-443.
[34] Mangasarian O L. Solution of symmetric linear complementarity problems by iterative methods. Journal of Optimization Theory and Applications, 1977,22: 465-485.
[35] Ahn B H. Solution of nonsymmetric linear complementarity problems by iterative methods. Journal of Optimization Theory and Applications, 1981, 33: 175-185.
[36] Mangasarian O L, De Leone R. Parallel successive overrelaxation methods for the symmetric linear complementaaity problems by iterative methods. Journal of Optimization Theory and Applications, 1987,54: 437-446.
[37] De Leone R, Mangasarian O L. Asynchronous parallel successive overrelaxation for symmetric linear complementarity problem. Mathematics Programming, 1988, 42: 347-361.
[38] Oleaty D, White R. Multisplittings of matrices and parallel solution of linear systems. SLAM Journal of Algebraic Discrete Methods, 1985, 6: 630-640.
[39] Frommer A, Mayer G. On the theory and practice of multisplitting methods in parallel computation. Computing, 1992,49: 63-74.
[40] Frommer A, Szyld D B. H-splittings and two-stage iterative methods. Numer. Math., 1992,63: 345-356.
[41] Frommer A, Schwandt H. A unified representation and theory of algebraic additive Schwarz and multi- splitting methods. SIAM J. Martrix Anal. Appl., 1997, 18: 893-912.
[42] Frommer A, Schwandt H, Szyld D B. Asynchronous weighted additive Schwarz methods. Elec. Traus. Numer. Anal., 1997, 5: 48-61.
[43] Frommer A, Szyld D B. Weighted max norms, splitting, and overlapping additive Schwarz iterations. Numer. Math., 1999, 83: 259-278.
[44] Benassi M, White R. Parallel numerical solution of variational inequalities. SLAM Journal of Numerical Mathematic and Analysis, 1994, 31: 813-830.
[45] Bai Z Z, Evans D J. Matrix multisplitting relaxation methods for linear complementarity problems. International Journal of Computer Mathematics, 1997, 63:309-326.
[46] Bai Z Z. On the convergence of the multisplitting methods for the linear complementarity problem. SIAM Journal of Matrix Analysis and Application, 1999, 21(1): 67-78.
[47] Bai Z Z, Evans D J. Matrix multisplitting methods with applications to linear complementarity problems: Parallel synchronous and chaotic methods. Calculatears Pasalleles, 2001, 13(1): 125-154.
[48] Bai Z Z, Evans D J. Matrix multisplitting methods with applications to linear complementaaity problems: Parallel asynchronous methods. International Journal of Computer Mathematics, 2002, 79(2): 205-232.
[49] Bai Z Z, Huang Y G. A class of asynchronous parallel multisplitting relaxation methods for the large sparse linear complementarity problems. Journal of Compu- tational Mathematics, 2003, 21(6), 773- 790.
[50] Mangasarian O L. Equivalence of the complementarity problem to a system of nonlinear equations. SIAM Journal of Applied Mathematics, 1976, 31: 89-92.
[51] Watson L T. Solving the nonlinear complementarity problem by a homotopy method. SIAM Journal on Control and Optimization, 1979,17: 36-46.
[52] Chen X, Qi L, and Sun D. Global and superlinear convergence of the smoothing Newton method and its application to general box constrained variational inequalities. Mathematics of Computation, 1998, 67(222): 519-540.
[53] Chen C. Smoothing methods in mathematical programming. PhD thesis, Computer Sciences Department, University of Wisconsin, Madison, Wisconsin, 1995.
[54] Kanzow C. Some equation-based methods for the nonlinear complementarity problem. Optimization Methods and Software, 1994, 3: 327-340.
[55] Pang J S. Newton's method for B-differentiable equations. Mathematics of Oper- ations Research, 1990, 15: 311-341.
[56] Pang J S. A B-differentiable equation based on globally and locally quadratically convergent algorithm for nonlinear programs, complementarity and variational in- equality problems. Mathematical Programming, 1991, 51: 101-132.
[57] Pang J S, Qi L. Nonsmooth equations: Motivation and algorithms. SIAM Journal on Optimization, 1993, 3: 443-465.
[58] Facchinei F, Soares J. Testing a new class of algorithms for nonlinear complemen tarity problems. Variational Inequalities and Network Equilibrium Problems, Plenum Press, New York, 1995: 69-83.
[59] Ferris M C, Pang J S. Engineering and economic applications of complementarity problems. Discussion Papers in Economics 95-4, Depart went of Economics, University of Colorado, Boulder, Colorado, 1995.
[60] Facchiniei F, Pang J S. Finite-dimensinal variational inequalities and complementarity problems. Volume II, Springer-Verlag, New York, 2003.
[61] Lobo M S, Vandenberghe L, Boyd S, Lebret H. Applications of second-order cone programming. Lin. Algeb. Appl., 1998, 284: 193 - 228.
[62] Monteiro RDC, Tsuchiya T. Polynomial convergence of primal-dual algorithms for the second-order cone programs based on the MZ-family of directions. Math. Program., 2000, 88: 61 - 83.
[63] Tsuchiya T. A convergence analysis of the scaling-invariant primal-dual path-following algorithms for second-order cone programming. Optim. Methods Softw., 1999, 11: 141 - 182.
[64] Bai Y Q, Gehmi M, Roos C. A comparative study of Kernel functions for primal-dual interior-point algorithms in linear optimization. SIAM Journal on Optimization, 2004, 15(1): 101-128,.
[65] Bai Y Q, Roos C. A polynomial-time algorithm for linear optimization based on a new simple kernel function, Optimization Methods and Software. December 2003, 18(6): 631-646.
[66] Bai Y Q, Gehmi M, Roos C. A new efficient large-update primal-dual interior-point method based on a finite barrier. SIAM Journal on Optimization, 13(3): 766-782.
[67] Fukushima M, Luo Z Q, Tseng P. Smoothing functions for second-order-cone complementarity problems. SIAM J. Optim., 2001, 12: 436 - 460.
[68] Chen J S. Merit function and nonsmooth functions for second-order cone complementarity problems. Master thesis, Department of Mathematics, University of Washington, Seattle, April 2001
[69] Chen X D, Sun D, Sun J. Complementarity functions and numerical experiments on some smoothing Newton methods for second-order cone complementarity problems. Computational Optimization and Applications, 2003, 25: 39-56.
[70] Huang N J. Mann and Ishikawa type perturbed iterative algorithms for generalized nonlinear implicit quasi-variational inclusions. Comput. Math. Appl., 1998,35(10): 1-7.
[71] Noor M A, Noor K I, Rassias T M. Set-valued resolvent equatuons and mixed variational inequalities. J. Math. Anal. Appl., 1998, 220: 741-759.
[72] Noor M A. Auxiliary principle for generalized mixed variational-like inequalities. J. Math. Anal. Appl., 1997, 215: 78-85.
[73] Noor M A. A new iterative method for monotone mixed variational inequalities. Math. Comput. Modelling, 1997, 26(7): 29-34.
[74] Shi R Equivalence of variational inequalities with Wiener-Hopf equatuons. Proc. Amer. Math. Soc, 1991,111:339-346.
[75] Bobinson S M. Normal maps induced by linear transformations. Math. Oper. Res., 1992, 17: 691-714.
[76] Ahmad R, Ansari Q H, Irfan S S. Generalized variational inclusions and generalized resolvent equations in banach spaces. Comput. Math. 3 Appl., 2005, 49: 1825 - 1835.
[77] Ding X P. Perturbed proximal point for generalized quasivariational inclusions. J. Math. Anal. Appl., 1997,210: 88-101.
[78] Hassouni A, Moudafi A. A perturbed algorithm for variational-like inclusions. J. Math. Anal. Appl., 1994,185: 706-712.
[79] Lee C H, Ansari Q H, Yao J C. A perturbed algorithm for strongly nonlinear variational-like inclusions. Bull. Aust. Math. Soc, 2000, 62(11): 417 - 426.
[80] Noor M A. Implicit resolvent dynamical systems for quasi variational inclusions. J. Math. Anal. Appl., 2002,269: 216-226.
[81] Qi L, Sun D, Zhou G. A new look at smoothing Newton methods for nonlinear complementarity problems and box constrained variational inequalities. Mathematical Programming, 2000, 87: 1-35.
[82] Sun D, Sun J. Semismooth matrix valued functions. Math. Oper. Res., 2002, 27: 150 - 169.
[83] Chen X D, Sun D, Sun J. Complementarity functions and numerical experiments on some smoothing Newton methods for second-order-cone complementarity problems. Computational Optimization and Applications, 2003,25: 39-56.
[84] Fischer A. A special Newton-type optimization method. optimization, 1992, 24: 269-284.
[85] Ferris M C, Pang J S, EDS. complimentarity and variational problems: state of the art. SLAM, Philadelphia, 1997.
[86] Pan S, Chen J S. A semismooth Newton method for the SOCCP based on a one-parametric class of SOC complementarity functions. to appear in Computational Optimization and Applications, 2008.
[87] Pang J S, Qi L. Nonsmooth equations: motivation and algorithms. SIAM Journal on Optimization, 1993,3:443-465.
[88] Qi L, Sun J. A nonsmooth version of Newton's method. Mathematical Programming, 1993, 58: 353- 367.
[89] Smale S. Algorithms for solving equations. In Proceedings of the International Congress of Mathematicians, A. M. Gleason, ed., American Mathematical Society, Providence, 1987, 172-195.
[90] Sun D, Sun J. Strong semismoothness of the Fischer-Burmeister SDC and SOC complementarity functions. Source, Mathematical Programming, 2005, 103(3): 575-581.
[91] Tseng P. Merit functions for semidefinite complementarity problems. Mathematical Programming, 1998,83: 159-185.
[92] Fukushima M, Luo Z Q, Tseng P. Smoothing functions for second-order-cone complimentarity problems. SIAM Journal on Optimization, 2002, 12: 436-460.
[93] Faraut U, Koranyi A. Analysis on symmetric cones. Oxford Mathematical Monographs, Oxford University Press, New York, 1994.
[94] Kanzow C, Nagel C. Semidefinite programs: new search directions, smoothing-type methods, and numerical results. SIAM Journal on Optimization, 2002,13: 1-23.
[95] Lobo M S, Vandenberghe L, Boyd S, Lebret H. Applications of second-order cone programming. Linear Algebra Appl., 1998, 284: 193-228.
[96] Monteiro R D C, Tsuchiya T. Polynomial convergence of primal-dual algorithms for the second-order cone programs based on the MZ-family of directions. Math. Programming, 2000, 88: 61-83.
[97] Monteiro R D C, Pang J S. On two interior-point mapping for nonlinear semidefinite complementarity problems. Mathematics of Operations Research, 1998, 23: 39-60.
[98] Monteiro R D C, Pang J S. A potential reduction Newton method for constrained equation. SIAM Journal on Optimization, 1999, 9: 729-754.
[99] Sun J, Sun D, Qi L. A squared smoothing Newton method for nonsmooth matrix equations and its applications in semidefinite optimization problems. SIAM Journal on Optimization, 2004, link. aip.org.
[100] Tsuchiya T. A convergence analysis of the scaling-invariant primal-dual path-following algorithm for second-order-cone programming. Optim. Methods softw., to appear.
[101] Qi L. Convergence analysis of some algorithms for solving nonsmooth equations. Math. Open Res., 1993, 18: 227-244.
[102] Mifflin R. Semismooth and semiconvex functions in constrained optimization. SIAM J. Control Optim., 1977, 15: 957 - 972.
[103] Fang Y P, Huang N J. H-Monotone operator and resolvent operator technique for variational inclusions. Appl. Math. Comput., 2003, 145(8): 795 - 803.
[104] Zeidler E. Nonlinear Functional Analysis and its Applications II: Monotone Operators. Springer- Verlag, Berlin, 1985.
[2]Fichera G.Problemi elastostatici con vincoli unilaterali:il problema di Signorini ambigue condizione al contorno.Attem.Acad.Naz.Lincei.Mem.C1.Sci.Nat.Sez.la,1963/64,7(8):91-140.
[3]Stampacchia G.Forms bilinearires coercitives sur les ensembles convexs.C.R.Acad.Sci.Paris,1964,258:4413-4416.
[4]Lion J L,Stampacchia G.Variational inequalities.Comm.Pure Appl.Math.,1967,20:493-519.
[5]Duvaut G,Lions J L.Inequalities in mechanics and physics.Berlin:Speinger-verlag,1976.
[6]Chipot M.Variational inequalities and flow in porous media.New York:Springer-Verlag,1984.
[7]Friedman A.Variational principle and flee-boundary problems.New York:John Wiley,1982.
[8]Glowinski R.Numercial methods for nonlinear variational Problems.New York:Springer-Verlag,1984.
[9]Glowinski R,Lions J L,Tremolieres R.Numerical analysis of variational inequalities.Amsterdam:North-Holland,1981.
[10]Han W,Reddy B D.Plasticity:Mathemaical theory and numercial analysis.New York:Springer-Verlag,1999.
[11]Han W,Sofonea M.Quasistatic contact problems in viscoelasticity and viscoplasticit.American Mathmatical Society and International Press,2002.
[12]Kinderlehrer D,Stampacchia G.An introduction to variational inequalities and their applications.New York:Academic Press,1980.
[13]Panagiotopoulos P D.Inequalities problems in mechanics and applications.Boston:Birkhauser,1985.
[14]Rodrigues J F.Obstacle problems in mathematical physics.Amsterdam:North-Holand,1987.
[15]Huang N J,Deng C X.Auxiliary principle and iterative algorithms for generalized set-valued strongly nonlinear mixed variational-like inequalities.J.Math.Anal.Appl.,2001,256:345-359.
[16]Ding X P.Preditor-corrector iterative algorithms for solving generalized mixed quasi-variational-like inclusion.Journal of Computational and Applied Mathematics,2005,182(1):1-12.
[17]Verma R U.On generalized variational inequalities involving relaxed Lipschitz and relaxed monotone operators.J.Math.Anal.Appl.,1997,213:387-392.
[18]Liu Z,Ume J S,Kang S M.General stongly nonlinear quasi-variational inequalities with relaxed Lipschitz and relaxed monotone mappings.J.Optim.Theory Appl.,2002,114:639-656.
[19]张石生.变分不等式和相补问题理论与应用.上海:上海科学技术文献出版社,1991.
[20] Isac G, Bulavaski V, Klashnikov V. Exceptional families, topological degree and complementarity problems. Journal of Global Optimization, 1997,10(2): 207-225.
[21] Zhao Y B, Han J Y, Qi H D. Exceptional families and existence theorems for variational inequality problems. Journal of Optimization Theory and Applications, 1999, 101(2): 475-495.
[22] Zhou S Z. Bai M R. A new exceptional family of elements for a variational inequality problem on Hilbert space. Appl. Math. Lett., 2004, 17(4):423-428.
[23] Cottle R W, Pang J, Stone R. The linear complementarity problems. Boston: Academic Press, 1992.
[24] Cottle R W. Nonlinear programs with positively bounded Jacobians. Journal of the Society for Industrial and Applied Mathematics, 1966, 14: 147-158.
[25] Habetler G J, Kostreva M M. On a direct algorithm for nonlinear complemen- tarity problems. SIAM Journal on Control and Optimization, 1978, 16: 504-511.
[26] Todd M J. Computation of fixed points and applications. Vol. 124 of Lecture Notes in Economics and Mathematical Systems, Springer Berlag, Heidelberg, 1976.
[27] Kojima M, Mizuno S, Noma T. A new continuation method for complementarity problems with uniform P-functions. Mathematical Programming, 1989,43: 107-113.
[28] Chen B, Harker P T. A non-interior continuation method for linear complementarity problems. SIAM Journal on Matrix Analysis, 1993, 14: 1168-1190.
[29] Kojima M, Shindo S, Hara S, Interior-point methods for the monotone semideffinite linear complementarity problem in symmetric matrices. Research Report on Information Sciences B-282, Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, Tokyo, Japan, 1994.
[30] Wang T, Monteiro R D C, Pang J S. An interior point potential reduction method for constrained equations. Manuscript, Department of Mathematical Sciences, The Johns Hopkins University, Baltimore, Maryland, 1994.
[31] Wright S J. An infeasible-interior-point algorithm for linear complementarity problems. Mathematical Programming, 1994, 67: 29-51.
[32] Lustig I J, Marsten R E, Shanno D F. Interior point methods for linear programming: computational state of the art. ORSA Journal of Computing, 1994, 6(1): 1-38.
[33] Cryer C W. The methods of Christopherson for solving free boundary problems for infinite journal bearings by means of finite differences. Mathematics of computation, 1971,25: 435-443.
[34] Mangasarian O L. Solution of symmetric linear complementarity problems by iterative methods. Journal of Optimization Theory and Applications, 1977,22: 465-485.
[35] Ahn B H. Solution of nonsymmetric linear complementarity problems by iterative methods. Journal of Optimization Theory and Applications, 1981, 33: 175-185.
[36] Mangasarian O L, De Leone R. Parallel successive overrelaxation methods for the symmetric linear complementaaity problems by iterative methods. Journal of Optimization Theory and Applications, 1987,54: 437-446.
[37] De Leone R, Mangasarian O L. Asynchronous parallel successive overrelaxation for symmetric linear complementarity problem. Mathematics Programming, 1988, 42: 347-361.
[38] Oleaty D, White R. Multisplittings of matrices and parallel solution of linear systems. SLAM Journal of Algebraic Discrete Methods, 1985, 6: 630-640.
[39] Frommer A, Mayer G. On the theory and practice of multisplitting methods in parallel computation. Computing, 1992,49: 63-74.
[40] Frommer A, Szyld D B. H-splittings and two-stage iterative methods. Numer. Math., 1992,63: 345-356.
[41] Frommer A, Schwandt H. A unified representation and theory of algebraic additive Schwarz and multi- splitting methods. SIAM J. Martrix Anal. Appl., 1997, 18: 893-912.
[42] Frommer A, Schwandt H, Szyld D B. Asynchronous weighted additive Schwarz methods. Elec. Traus. Numer. Anal., 1997, 5: 48-61.
[43] Frommer A, Szyld D B. Weighted max norms, splitting, and overlapping additive Schwarz iterations. Numer. Math., 1999, 83: 259-278.
[44] Benassi M, White R. Parallel numerical solution of variational inequalities. SLAM Journal of Numerical Mathematic and Analysis, 1994, 31: 813-830.
[45] Bai Z Z, Evans D J. Matrix multisplitting relaxation methods for linear complementarity problems. International Journal of Computer Mathematics, 1997, 63:309-326.
[46] Bai Z Z. On the convergence of the multisplitting methods for the linear complementarity problem. SIAM Journal of Matrix Analysis and Application, 1999, 21(1): 67-78.
[47] Bai Z Z, Evans D J. Matrix multisplitting methods with applications to linear complementarity problems: Parallel synchronous and chaotic methods. Calculatears Pasalleles, 2001, 13(1): 125-154.
[48] Bai Z Z, Evans D J. Matrix multisplitting methods with applications to linear complementaaity problems: Parallel asynchronous methods. International Journal of Computer Mathematics, 2002, 79(2): 205-232.
[49] Bai Z Z, Huang Y G. A class of asynchronous parallel multisplitting relaxation methods for the large sparse linear complementarity problems. Journal of Compu- tational Mathematics, 2003, 21(6), 773- 790.
[50] Mangasarian O L. Equivalence of the complementarity problem to a system of nonlinear equations. SIAM Journal of Applied Mathematics, 1976, 31: 89-92.
[51] Watson L T. Solving the nonlinear complementarity problem by a homotopy method. SIAM Journal on Control and Optimization, 1979,17: 36-46.
[52] Chen X, Qi L, and Sun D. Global and superlinear convergence of the smoothing Newton method and its application to general box constrained variational inequalities. Mathematics of Computation, 1998, 67(222): 519-540.
[53] Chen C. Smoothing methods in mathematical programming. PhD thesis, Computer Sciences Department, University of Wisconsin, Madison, Wisconsin, 1995.
[54] Kanzow C. Some equation-based methods for the nonlinear complementarity problem. Optimization Methods and Software, 1994, 3: 327-340.
[55] Pang J S. Newton's method for B-differentiable equations. Mathematics of Oper- ations Research, 1990, 15: 311-341.
[56] Pang J S. A B-differentiable equation based on globally and locally quadratically convergent algorithm for nonlinear programs, complementarity and variational in- equality problems. Mathematical Programming, 1991, 51: 101-132.
[57] Pang J S, Qi L. Nonsmooth equations: Motivation and algorithms. SIAM Journal on Optimization, 1993, 3: 443-465.
[58] Facchinei F, Soares J. Testing a new class of algorithms for nonlinear complemen tarity problems. Variational Inequalities and Network Equilibrium Problems, Plenum Press, New York, 1995: 69-83.
[59] Ferris M C, Pang J S. Engineering and economic applications of complementarity problems. Discussion Papers in Economics 95-4, Depart went of Economics, University of Colorado, Boulder, Colorado, 1995.
[60] Facchiniei F, Pang J S. Finite-dimensinal variational inequalities and complementarity problems. Volume II, Springer-Verlag, New York, 2003.
[61] Lobo M S, Vandenberghe L, Boyd S, Lebret H. Applications of second-order cone programming. Lin. Algeb. Appl., 1998, 284: 193 - 228.
[62] Monteiro RDC, Tsuchiya T. Polynomial convergence of primal-dual algorithms for the second-order cone programs based on the MZ-family of directions. Math. Program., 2000, 88: 61 - 83.
[63] Tsuchiya T. A convergence analysis of the scaling-invariant primal-dual path-following algorithms for second-order cone programming. Optim. Methods Softw., 1999, 11: 141 - 182.
[64] Bai Y Q, Gehmi M, Roos C. A comparative study of Kernel functions for primal-dual interior-point algorithms in linear optimization. SIAM Journal on Optimization, 2004, 15(1): 101-128,.
[65] Bai Y Q, Roos C. A polynomial-time algorithm for linear optimization based on a new simple kernel function, Optimization Methods and Software. December 2003, 18(6): 631-646.
[66] Bai Y Q, Gehmi M, Roos C. A new efficient large-update primal-dual interior-point method based on a finite barrier. SIAM Journal on Optimization, 13(3): 766-782.
[67] Fukushima M, Luo Z Q, Tseng P. Smoothing functions for second-order-cone complementarity problems. SIAM J. Optim., 2001, 12: 436 - 460.
[68] Chen J S. Merit function and nonsmooth functions for second-order cone complementarity problems. Master thesis, Department of Mathematics, University of Washington, Seattle, April 2001
[69] Chen X D, Sun D, Sun J. Complementarity functions and numerical experiments on some smoothing Newton methods for second-order cone complementarity problems. Computational Optimization and Applications, 2003, 25: 39-56.
[70] Huang N J. Mann and Ishikawa type perturbed iterative algorithms for generalized nonlinear implicit quasi-variational inclusions. Comput. Math. Appl., 1998,35(10): 1-7.
[71] Noor M A, Noor K I, Rassias T M. Set-valued resolvent equatuons and mixed variational inequalities. J. Math. Anal. Appl., 1998, 220: 741-759.
[72] Noor M A. Auxiliary principle for generalized mixed variational-like inequalities. J. Math. Anal. Appl., 1997, 215: 78-85.
[73] Noor M A. A new iterative method for monotone mixed variational inequalities. Math. Comput. Modelling, 1997, 26(7): 29-34.
[74] Shi R Equivalence of variational inequalities with Wiener-Hopf equatuons. Proc. Amer. Math. Soc, 1991,111:339-346.
[75] Bobinson S M. Normal maps induced by linear transformations. Math. Oper. Res., 1992, 17: 691-714.
[76] Ahmad R, Ansari Q H, Irfan S S. Generalized variational inclusions and generalized resolvent equations in banach spaces. Comput. Math. 3 Appl., 2005, 49: 1825 - 1835.
[77] Ding X P. Perturbed proximal point for generalized quasivariational inclusions. J. Math. Anal. Appl., 1997,210: 88-101.
[78] Hassouni A, Moudafi A. A perturbed algorithm for variational-like inclusions. J. Math. Anal. Appl., 1994,185: 706-712.
[79] Lee C H, Ansari Q H, Yao J C. A perturbed algorithm for strongly nonlinear variational-like inclusions. Bull. Aust. Math. Soc, 2000, 62(11): 417 - 426.
[80] Noor M A. Implicit resolvent dynamical systems for quasi variational inclusions. J. Math. Anal. Appl., 2002,269: 216-226.
[81] Qi L, Sun D, Zhou G. A new look at smoothing Newton methods for nonlinear complementarity problems and box constrained variational inequalities. Mathematical Programming, 2000, 87: 1-35.
[82] Sun D, Sun J. Semismooth matrix valued functions. Math. Oper. Res., 2002, 27: 150 - 169.
[83] Chen X D, Sun D, Sun J. Complementarity functions and numerical experiments on some smoothing Newton methods for second-order-cone complementarity problems. Computational Optimization and Applications, 2003,25: 39-56.
[84] Fischer A. A special Newton-type optimization method. optimization, 1992, 24: 269-284.
[85] Ferris M C, Pang J S, EDS. complimentarity and variational problems: state of the art. SLAM, Philadelphia, 1997.
[86] Pan S, Chen J S. A semismooth Newton method for the SOCCP based on a one-parametric class of SOC complementarity functions. to appear in Computational Optimization and Applications, 2008.
[87] Pang J S, Qi L. Nonsmooth equations: motivation and algorithms. SIAM Journal on Optimization, 1993,3:443-465.
[88] Qi L, Sun J. A nonsmooth version of Newton's method. Mathematical Programming, 1993, 58: 353- 367.
[89] Smale S. Algorithms for solving equations. In Proceedings of the International Congress of Mathematicians, A. M. Gleason, ed., American Mathematical Society, Providence, 1987, 172-195.
[90] Sun D, Sun J. Strong semismoothness of the Fischer-Burmeister SDC and SOC complementarity functions. Source, Mathematical Programming, 2005, 103(3): 575-581.
[91] Tseng P. Merit functions for semidefinite complementarity problems. Mathematical Programming, 1998,83: 159-185.
[92] Fukushima M, Luo Z Q, Tseng P. Smoothing functions for second-order-cone complimentarity problems. SIAM Journal on Optimization, 2002, 12: 436-460.
[93] Faraut U, Koranyi A. Analysis on symmetric cones. Oxford Mathematical Monographs, Oxford University Press, New York, 1994.
[94] Kanzow C, Nagel C. Semidefinite programs: new search directions, smoothing-type methods, and numerical results. SIAM Journal on Optimization, 2002,13: 1-23.
[95] Lobo M S, Vandenberghe L, Boyd S, Lebret H. Applications of second-order cone programming. Linear Algebra Appl., 1998, 284: 193-228.
[96] Monteiro R D C, Tsuchiya T. Polynomial convergence of primal-dual algorithms for the second-order cone programs based on the MZ-family of directions. Math. Programming, 2000, 88: 61-83.
[97] Monteiro R D C, Pang J S. On two interior-point mapping for nonlinear semidefinite complementarity problems. Mathematics of Operations Research, 1998, 23: 39-60.
[98] Monteiro R D C, Pang J S. A potential reduction Newton method for constrained equation. SIAM Journal on Optimization, 1999, 9: 729-754.
[99] Sun J, Sun D, Qi L. A squared smoothing Newton method for nonsmooth matrix equations and its applications in semidefinite optimization problems. SIAM Journal on Optimization, 2004, link. aip.org.
[100] Tsuchiya T. A convergence analysis of the scaling-invariant primal-dual path-following algorithm for second-order-cone programming. Optim. Methods softw., to appear.
[101] Qi L. Convergence analysis of some algorithms for solving nonsmooth equations. Math. Open Res., 1993, 18: 227-244.
[102] Mifflin R. Semismooth and semiconvex functions in constrained optimization. SIAM J. Control Optim., 1977, 15: 957 - 972.
[103] Fang Y P, Huang N J. H-Monotone operator and resolvent operator technique for variational inclusions. Appl. Math. Comput., 2003, 145(8): 795 - 803.
[104] Zeidler E. Nonlinear Functional Analysis and its Applications II: Monotone Operators. Springer- Verlag, Berlin, 1985.
© 2004-2018 中国地质图书馆版权所有 京ICP备05064691号 京公网安备11010802017129号 地址：北京市海淀区学院路29号 邮编：100083 电话：办公室：(+86 10)66554848；文献借阅、咨询服务、科技查新：66554700 |