非光滑拟变分不等式的正则化间隙函数
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摘要
本文将广义拟变分不等式和非光滑拟变分不等式问题转化为关于正则化间隙函数的极小化问题.在最优化问题中,研究目标函数的方向导数及其次微分具有重要意义,如利用目标函数的方向导数和次微分讨论最优性条件,建立基于方向导数或次微分的优化算法.本文讨论了边际函数的方向可微性和一类含参变量凸规划问题的边际函数的Clarke方向导数的上界和下界.利用这些结果,我们研究了广义拟变分不等式问题的正则化间隙函数的方向可微性及极限次微的上界和非光滑拟变分不等式的正则化间隙函数和D-间隙函数的Clarke方向导数.
In this paper, we use regularized gap functions to reformulate quasi-variational inequalities into minimization problems. In optimization prob-lems, it is significant to study the directional derivatives and subdifferentials of objective function, for example, we can use the directional derivatives and sub-differentials of objective function to study the necessary optimality conditions and algorithms for optimization problems. We discuss the directional differentia-bility of marginal function and estimate the upper and lower bounds of a class of marginal functions in parametric convex programs. Then we employ these results to study the directional differentiability and the upper bound of limit-subdifferentials of the regularized gap functions for generalized quasi-variational inequalities and Clarke directional derivatives of the regularized gap functions and D-gap functions for nonsmooth quasi-variational inequalities.
In this paper, we use regularized gap functions to reformulate quasi-variational inequalities into minimization problems. In optimization prob-lems, it is significant to study the directional derivatives and subdifferentials of objective function, for example, we can use the directional derivatives and sub-differentials of objective function to study the necessary optimality conditions and algorithms for optimization problems. We discuss the directional differentia-bility of marginal function and estimate the upper and lower bounds of a class of marginal functions in parametric convex programs. Then we employ these results to study the directional differentiability and the upper bound of limit-subdifferentials of the regularized gap functions for generalized quasi-variational inequalities and Clarke directional derivatives of the regularized gap functions and D-gap functions for nonsmooth quasi-variational inequalities.
引文
[1]P. HARTMAN AND G. STAMPACCHIA On some nonlinear elliptic differential functional equations. Acta Mathematica 115, (1966),153-188.
[2]J.L. LIONS AND G. STAMPACCHIA Variational inequalities. Communications on Pure and Applied Mathematics 20, (1967),493-519.
[3]D. CHAN AND J.S. PANG The generalized quasi-variational inequality prob-lem. Math. Oper. Research 7, (1982),211-222.
[4]F.FACCHI AND J.S.PANG Finit-Dimensional variational inequalities and complementarity problems. Vol.Ⅰ. Springer series in oprerations research, Springer, New York,NY,USA, (2003).
[5]M.FUKUSHIMA Equivalent differentiable optimaization problems and descent methods for asymmetric variational inequality problems. Mathematical Pro-gramming 55,1,(1992),99-110.
[6]J.-M.PENG Equivalence of variational inequality problems to unconstrained minimization. Mathematical Programming 78,3,(1997),347-355.
[7]N.YAMASHITA, K.TAJI AND M.FUKUSHIMA Unconstrained optimization reformulations of asymmetric variational inequality problems. Journal of Op-timization Theory and Applications 92,3, (1997),439-456.
[8]M.FUKUSHIMA A class of gap functions for quasi-variational inequality prob-lems. Journal of Industrial and management Optimization 3,2, (2007),165-171.
[9]K.TAJI On gap functions for quasi-variational inequalities. Abstract and Applied Analysis,Art. ID 531361,7 pp,(2008).
[10]K. KUBOTA AND M.FUKUSHIMA A gap function approach to the general-ized Nash equilibrium problem. Journal of Optimization Theory and Appli-cation 144,3,(2010),511-531.
[11]KUNG Fu NG AND Lu LIN TAN Error bounds of regularized gap functions for nonsmooth variational inequality problems. Mathematical Programming 110,(2007),405-429.
[12]KUNG FU NG AND LU LIN TAN D-Gap Functions for nonsmooth varia-tional inequality problems. Journal of Optimization Theory and Application 110,(2007),405-429.
[13]LU LIN TAN Regularized gap functions for nonsmooth variational inequal-ity problems. Journal of Mathematical Analysis and Applications 334,2, (2007),1022-1038.
[14]B.S.MORDUKHOVICH Variational Analysis and Generalized Differentiation Ⅰ, Springer-verlag, Berlin, (2006).
[15]J. P. PENOT Differentiability of relations and differential stability of per-turbed optimization problems, SIAM Journal on Control and Optimization 22,(1984),529-551.
[16]F. H. CLARKE Optimization and nonsmooth analysis. Wiley, New York,(1983).
[17]E. M. STERN Singztlar Integrals and Differentiability Properties of Func-tions. Prirzcetorz Matkenzatics Series 30 Princeton University Press. Prince-ton. New Jersey (1970).
[18]L. QI AND J. SUN A nonsmooth version of Newton's method. Mathematical Programming 58,(1993),353-368.
[19]R. MIFFLIN Semismooth and semiconvex functions in constrained optimiza-tion. SIAM Journal on Control and Optimization 15, (1977),957-972.
[20]A. AUSLENDER Differential stability in nonconvex and nondifferentiable programming. Mathematical Programming Study 10, (1979),29-41.153-188.
[21]WILLIAM W. HOGAN Point-to-set maps in mathematical programming. SIAM review 15,3,(1973),591-603.
[22]WILLIAM W. HOGAN Directional derivatives for extremal-value func-tions with applications to the completely convex case. Operations Reseach 21,(1973),188-209.
[23]D. P. BERTSEKAS WITH A. NEDIC AND A. E. OZDAGLAR Convex Analysis and Optimization. Athena Scientific press, ISBN:1-886529-45-0 (2003).
[24]R.T. ROCKAFELLAR Lagrange multipliers and subderivatives of optimal value functions in nonlinear programming. Mathematical Programming Study 17,(1982),28-66.
[25]B.S.MORDUKHOVICH, N.M.NAM AND N.D.YEN Subgradients of marginal functions in parametric mathematical programming Mathematical Program-ming 116,(2009),369-396.
[26]J.B. HIRIART-URRUTYy Approximate First-Order and Second-Order Di- rectional Derivatives of a Marginal Function in Convex Optimization Journal of Optimization Theory and Applications 48,1,(1986),127-140.
[27]L. I. MINCHENKO AND P. P. SAKOLCHIK Holder Behavior of Optimal Solu-tions and Directional Differentiability of Marginal Functions in Nonlinear Pro-gramming Journal of Optimization Theory and Applications 90,3,(1996),555-580.
[28]D.E. WARD AND G.M. LEE Upper Subderivatives and Generalized Gra-dients of the Marginal Function of a Non-Lipschitzian Program Annals of Operations Research 101,3,(2001),299-312.
[2]J.L. LIONS AND G. STAMPACCHIA Variational inequalities. Communications on Pure and Applied Mathematics 20, (1967),493-519.
[3]D. CHAN AND J.S. PANG The generalized quasi-variational inequality prob-lem. Math. Oper. Research 7, (1982),211-222.
[4]F.FACCHI AND J.S.PANG Finit-Dimensional variational inequalities and complementarity problems. Vol.Ⅰ. Springer series in oprerations research, Springer, New York,NY,USA, (2003).
[5]M.FUKUSHIMA Equivalent differentiable optimaization problems and descent methods for asymmetric variational inequality problems. Mathematical Pro-gramming 55,1,(1992),99-110.
[6]J.-M.PENG Equivalence of variational inequality problems to unconstrained minimization. Mathematical Programming 78,3,(1997),347-355.
[7]N.YAMASHITA, K.TAJI AND M.FUKUSHIMA Unconstrained optimization reformulations of asymmetric variational inequality problems. Journal of Op-timization Theory and Applications 92,3, (1997),439-456.
[8]M.FUKUSHIMA A class of gap functions for quasi-variational inequality prob-lems. Journal of Industrial and management Optimization 3,2, (2007),165-171.
[9]K.TAJI On gap functions for quasi-variational inequalities. Abstract and Applied Analysis,Art. ID 531361,7 pp,(2008).
[10]K. KUBOTA AND M.FUKUSHIMA A gap function approach to the general-ized Nash equilibrium problem. Journal of Optimization Theory and Appli-cation 144,3,(2010),511-531.
[11]KUNG Fu NG AND Lu LIN TAN Error bounds of regularized gap functions for nonsmooth variational inequality problems. Mathematical Programming 110,(2007),405-429.
[12]KUNG FU NG AND LU LIN TAN D-Gap Functions for nonsmooth varia-tional inequality problems. Journal of Optimization Theory and Application 110,(2007),405-429.
[13]LU LIN TAN Regularized gap functions for nonsmooth variational inequal-ity problems. Journal of Mathematical Analysis and Applications 334,2, (2007),1022-1038.
[14]B.S.MORDUKHOVICH Variational Analysis and Generalized Differentiation Ⅰ, Springer-verlag, Berlin, (2006).
[15]J. P. PENOT Differentiability of relations and differential stability of per-turbed optimization problems, SIAM Journal on Control and Optimization 22,(1984),529-551.
[16]F. H. CLARKE Optimization and nonsmooth analysis. Wiley, New York,(1983).
[17]E. M. STERN Singztlar Integrals and Differentiability Properties of Func-tions. Prirzcetorz Matkenzatics Series 30 Princeton University Press. Prince-ton. New Jersey (1970).
[18]L. QI AND J. SUN A nonsmooth version of Newton's method. Mathematical Programming 58,(1993),353-368.
[19]R. MIFFLIN Semismooth and semiconvex functions in constrained optimiza-tion. SIAM Journal on Control and Optimization 15, (1977),957-972.
[20]A. AUSLENDER Differential stability in nonconvex and nondifferentiable programming. Mathematical Programming Study 10, (1979),29-41.153-188.
[21]WILLIAM W. HOGAN Point-to-set maps in mathematical programming. SIAM review 15,3,(1973),591-603.
[22]WILLIAM W. HOGAN Directional derivatives for extremal-value func-tions with applications to the completely convex case. Operations Reseach 21,(1973),188-209.
[23]D. P. BERTSEKAS WITH A. NEDIC AND A. E. OZDAGLAR Convex Analysis and Optimization. Athena Scientific press, ISBN:1-886529-45-0 (2003).
[24]R.T. ROCKAFELLAR Lagrange multipliers and subderivatives of optimal value functions in nonlinear programming. Mathematical Programming Study 17,(1982),28-66.
[25]B.S.MORDUKHOVICH, N.M.NAM AND N.D.YEN Subgradients of marginal functions in parametric mathematical programming Mathematical Program-ming 116,(2009),369-396.
[26]J.B. HIRIART-URRUTYy Approximate First-Order and Second-Order Di- rectional Derivatives of a Marginal Function in Convex Optimization Journal of Optimization Theory and Applications 48,1,(1986),127-140.
[27]L. I. MINCHENKO AND P. P. SAKOLCHIK Holder Behavior of Optimal Solu-tions and Directional Differentiability of Marginal Functions in Nonlinear Pro-gramming Journal of Optimization Theory and Applications 90,3,(1996),555-580.
[28]D.E. WARD AND G.M. LEE Upper Subderivatives and Generalized Gra-dients of the Marginal Function of a Non-Lipschitzian Program Annals of Operations Research 101,3,(2001),299-312.