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    TIẾN SĨ Đối đạo hàm của ánh xạ nón pháp tuyến và ứng dụng

    Nhu Ely Nhu Ely Đang Ngoại tuyến (1771 tài liệu)
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  6. Đối đạo hàm của ánh xạ nón pháp tuyến và ứng dụng

    CODERIVATIVES OF NORMAL CONEMAPPINGS AND APPLICATIONS (Đối đạo hàm của ánh xạ nón pháp tuyến và ứng dụng)
    NĂM 2014

    Table of Notations vi
    List of Figures viii
    Introduction ix
    Chapter 1. Preliminary 1
    1.1 Basic De nitions and Conventions 1
    1.2 Normal and Tangent Cones . 3
    1.3 Coderivatives and Subdi erential 6
    1.4 Lipschitzian Properties and Metric Regularity . 9
    1.5 Conclusions 11

    Chapter 2. Linear Perturbations of Polyhedral Normal Cone Mappings 12
    2.1 The Normal Cone Mapping F(x; b) . 12
    2.2 The Frechet Coderivative of F(x; b) . 16
    2.3 The Mordukhovich Coderivative of F(x; b) . 26
    2.4 AVIs under Linear Perturbations 37
    2.5 Conclusions 42
    Chapter 3. Nonlinear Perturbations of Polyhedral Normal Cone Mappings 43
    3.1 The Normal Cone Mapping F(x; A; b) 43
    3.2 Estimation of the Frechet Normal Cone to gphF 48
    3.3 Estimation of the Limiting Normal Cone to gphF . 54
    3.4 AVIs under Nonlinear Perturbations . 59
    3.5 Conclusions 66

    Chapter 4. A Class of Linear Generalized Equations 67
    4.1 Linear Generalized Equations 67
    4.2 Formulas for Coderivatives 69
    4.2.1 The Frechet Coderivative of N(x; ) 70
    4.2.2 The Mordukhovich Coderivative of N(x; ) 78
    4.3 Necessary and Sucient Conditions for Stability 83
    4.3.1 Coderivatives of the KKT point set map 83
    4.3.2 The Lipschitz-like property 84
    4.4 Conclusions 91
    General Conclusions 92
    List of Author's Related Papers 93
    References 94

    Motivated by solving optimization problems, the concept of derivative was
    rst introduced by Pierre de Fermat. It led to the Fermat stationary princi-
    ple, which plays a crucial role in the development of di erential calculus and
    serves as an e ective tool in various applications. Nevertheless, many fundamental
    objects having no derivatives, no rst-order approximations (de ned
    by certain derivative mappings) occur naturally and frequently in mathematical
    models. The objects include nondi erentiable functions, sets with nonsmooth
    boundaries, and set-valued mappings. Since the classical di erential
    calculus is inadequate for dealing with such functions, sets, and mappings, the
    appearance of generalized di erentiation theories is an indispensable trend.
    In the 1960s, di erential properties of convex sets and convex functions
    have been studied. The fundamental contributions of J.-J. Moreau and
    R. T. Rockafellar have been widely recognized. Their results led to the beautiful
    theory of convex analysis [47]. The derivative-like structure for convex
    functions, called subdi erential, is one of the main concepts in this theory.
    In contrast to the singleton of derivatives, subdi erential is a collection of
    subgradients. Convex programming which is based on convex analysis plays
    a fundamental role in Mathematics and in applied sciences.
    In 1973, F. H. Clarke de ned basic concepts of a generalized di erentiation
    theory, which works for locally Lipschitz functions, in his doctoral dissertation
    under the supervision of R. T. Rockafellar. In Clarke's theory, convexity
    is a key point; for instance, subdi erential in the sense of Clarke is always a
    closed convex set. In the later 1970s, the concepts of Clarke have been developed
    for lower semicontinuous extended-real-valued functions in the works of
    R. T. Rockafellar, J.-B. Hiriart-Urruty, J.-P. Aubin, and others. Although
    the theory of Clarke is beautiful due to the convexity used, as well as to
    the elegant proofs of many fundamental results, the Clarke subdi erential
    and the Clarke normal cone face with the challenge of being too big, so too
    rough, in complicated practical problems where nonconvexity is an inherent
    property. Despite to this, Clarke's theory has opened a new chapter in the
    development of nonlinear analysis and optimization theory (see, e.g., [8], [2]).
    In the mid 1970s, to avoid the above-mentioned convexity limitations of
    the Clarke concepts, B. S. Mordukhovich introduced the notions of limiting
    normal cone and limiting subdi erential which are based entirely on dualspace
    constructions. His dual approach led to a modern theory of generalized
    di erentiation [28] with a variety of applications [29]. Long before the publication
    of these books, Mordukhovich's contributions to Variational Analysis
    had been presented in the well-known monograph of R. T. Rockafellar and
    R. J.-B. Wets [48].
    The limiting subdi erential is generally nonconvex and smaller than the
    Clarke subdi erential. Similarly, the limiting normal cone to a closed set in
    a Banach space is nonconvex in general and usually smaller than the Clarke
    normal cone. Therefore, necessary optimality conditions in nonlinear programming
    and optimal control in terms of the limiting subdi erential and
    limiting normal cone are much tighter than that given by the corresponding
    Clarke's concepts. Furthermore, the Mordukhovich criteria for the Lipchitzlike
    property (that is the pseudo-Lipschitz property in the original terminology
    of J.-P. Aubin [1], or the Aubin continuity as suggested by A. L. Dontchev

    and R. T. Rockafellar [11], [12]) and the metric regularity of multifunctions
    are remarkable tools to study stability of variational inequalities, generalized
    equations, and the Karush-Kuhn-Tucker point sets in parametric optimization
    problems. Note that if one uses Clarke's theory then only sucient
    conditions for stability can be obtained. Meanwhile, Mordukhovich's theory
    provides one with both necessary and sucient conditions for stability. Another
    advantage of the latter theory is that its system of calculus rules is
    much more developed than that of Clarke's theory. So, the wide range of applications
    and bright prospects of Mordukhovich's generalized di erentiation
    theory are understandable.
    In the late 1990s, V. Jeyakumar and D. T. Luc introduced the concepts of
    approximate Jacobian and corresponding generalized subdi erential. It can
    be seen [18] that using the approximate Jacobian one can establish conditions
    for stability, metric regularity, and local Lipschitz-like property of the solution
    maps of parametric inequality systems involving nonsmooth continuous
    functions and closed convex sets. Calculus rules and various applications of
    the approximate Jacobian can be found in the monograph [17]. It is worthy
    to study relationships between the concepts of coderivative and approximate
    Jacobian. In [33], the authors show that the Mordukhovich coderivative and
    the approximate Jacobian have a little in common. These concepts are very
    di erent, and they require di erent methods of study and lead to results in
    di erent forms.
    [1] J.-P. Aubin, Lipschitz behavior of solutions to convex minimization
    problems, Math. Oper. Res., 9 (1984), pp. 87{111.
    [2] J.-P. Aubin and H. Frankowska, Set-Valued Analysis, Birkhauser
    Boston-Basel-Berlin, 1990.
    [3] D. Bartl, A short algebraic proof of the Farkas lemma, SIAM J. Optim.,
    19 (2008), pp. 234{239.
    [4] J. M. Borwein and Q. J. Zhu, Techniques of Variational Analysis,
    CMS Books in Mathematics, Springer, New York, 2005.
    [5] N. H. Chieu, T. D. Chuong, J.-C. Yao, and N. D. Yen, Charac-
    terizing convexity of a function by its Frechet and limiting second-order
    subdi erentials, Set-Valued Var. Anal., 19 (2011), pp. 75{96.
    [6] N. H. Chieu and N. Q. Huy, Second-order subdi erentials and con-
    vexity of real-valued functions, Nonlinear Anal., 74 (2011), pp. 154{160.
    [7] N. H. Chieu and N. T. Q. Trang, Coderivative and monotonicity of
    continuous mappings, Taiwanese J. Math., 16 (2012), pp. 353{365.
    [8] F. H. Clarke, Optimization and Nonsmooth Analysis, CMS Books in
    Mathematics, Wiley, New York, 1983.
    [9] G. Colombo, R. Henrion, N. D. Hoang, and B. S. Mor-
    dukhovich, Optimal control of the sweeping process, Dyn. Contin. Discrete
    Impuls. Syst. Ser. B Appl. Algorithms, 19 (2012), pp. 117{159.
    [10] A. R. Conn, N. I. M. Gould, and P. L. Toint, Trust-Region
    Methods, MPS-SIAM Ser. Optim., Philadelphia, 2000.
    [11] A. L. Dontchev and R. T. Rockafellar, Characterizations of
    strong regularity for variational inequalities over polyhedral convex sets,
    SIAM J. Optim., 6 (1996), pp. 1087{1105.
    [12] A. L. Dontchev and R. T. Rockafellar, Implicit Functions and
    Solution Mappings, Springer, Dordrecht, 2009.
    [13] R. Henrion, B. S. Mordukhovich, and N. M. Nam, Second-order
    analysis of polyhedral systems in nite and in nite dimensions with ap-
    plications to robust stability of variational inequalities, SIAM J. Optim.,
    20 (2010), pp. 2199{2227.
    [14] R. Henrion, J. Outrata, and T. Surowiec, Analysis of M-
    stationary points to an EPEC modeling oligopolistic competition in an
    electricity spot market, ESAIM Control Optim. Calc. Var., 18 (2012),
    pp. 295{317.
    [15] N. Q. Huy and J.-C. Yao, Exact formulae for coderivatives of normal
    cone mappings to perturbed polyhedral convex sets, J. Optim. Theory
    Appl., 157 (2013), pp. 25{43.
    [16] A. D. Ioffe and V. M. Tihomirov, Theory of Extremal Problems,
    North-Holland Publishing Company, Amsterdam-New York-Oxford,
    [17] V. Jeyakumar and D. T. Luc, Nonsmooth Vector Functions and
    Continuous Optimization, Optimization and Its Applications, Springer,
    New York, 2008.
    [18] V. Jeyakumar and N. D. Yen, Solution stability of nonsmooth contin-
    uous systems with applications to cone-constrained optimization, SIAM
    J. Optim., 14 (2004), pp. 1106{1127.
    [19] H. A. Le Thi, T. Pham Dinh, and N. D. Yen, Properties of two
    DC algorithms in quadratic programming, J. Global Optim., 49 (2011),
    pp. 481{495.
    [20] G. M. Lee, N. N. Tam, and N. D. Yen, Quadratic Programming and
    Ane Variational Inequalities: A Qualitative Study, Springer-Verlag,
    New York, 2005.
    [21] G. M. Lee, N. N. Tam, and N. D. Yen, Stability of linear-quadratic
    minimization over Euclidean balls, SIAM J. Optim., 22 (2012), pp. 936{
    [22] G. M. Lee and N. D. Yen, Frechet and normal coderivatives of im-
    plicit multifunctions, Appl. Anal., 90 (2011), pp. 1011{1027.
    [23] G. M. Lee and N. D. Yen, Coderivatives of a Karush-Kuhn-Tucker
    point set map and applications, Nonlinear Anal., 95 (2014), pp. 191{201.
    [24] A. B. Levy and B. S. Mordukhovich, Coderivatives in parametric
    optimization, Math. Program., 99 (2004), pp. 311{327.
    [25] S. Lu and S. M. Robinson, Variational inequalities over perturbed
    polyhedral convex sets, Math. Oper. Res., 33 (2008), pp. 689{711.
    [26] J. M. Martinez, Local minimizers of quadratic functions on Euclidean
    balls and spheres, SIAM J. Optim., 4 (1994), pp. 159{176.
    [27] B. S. Mordukhovich, Coderivative analysis of variational systems, J.
    Global Optim., 28 (2004), pp. 347{362.
    [28] B. S. Mordukhovich, Variational Analysis and Generalized Di eren-
    tiation, vol. I: Basic Theory, Springer-Verlag, Berlin, 2006.
    [29] B. S. Mordukhovich, Variational Analysis and Generalized Di eren-
    tiation, vol. II: Applications, Springer-Verlag, Berlin, 2006.
    [30] B. S. Mordukhovich and J. V. Outrata, On second-order subdif-
    ferentials and their applications, SIAM J. Optim., 12 (2001), pp. 139{
    [31] B. S. Mordukhovich and R. T. Rockafellar, Second-order subdif-
    ferential calculus with applications to tilt stability in optimization, SIAM
    J. Optim., 22 (2012), pp. 953{986.
    [32] N. M. Nam, Coderivatives of normal cone mappings and Lipschitzian
    stability of parametric variational inequalities, Nonlinear Anal., 73
    (2010), pp. 2271{2282.
    [33] N. M. Nam and N. D. Yen, Relationships between approximate Jaco-
    bians and coderivatives, J. Nonlinear Convex Anal., 8 (2007), pp. 121{
    [34] T. Pham Dinh and H. A. Le Thi, A d.c. optimization algorithm for
    solving the trust-region subproblem, SIAM J. Optim., 8 (1998), pp. 476{
    [35] R. R. Phelps, Convex Functions, Monotone Operators and Di eren-
    tiability, Springer-Verlag, Berlin, 1993.
    [36] H. T. Phung, On the locally uniform openness of polyhedral sets, Acta
    Math. Vietnam., 25 (2000), pp. 273{284.
    [37] R. A. Poliquin and R. T. Rockafellar, Tilt stability of a local
    minimum, SIAM J. Optim., 8 (1998), pp. 287{299.
    [38] N. T. Qui, Linearly perturbed polyhedral normal cone mappings and
    applications, Nonlinear Anal., 74 (2011), pp. 1676{1689.
    [39] N. T. Qui, New results on linearly perturbed polyhedral normal cone
    mappings, J. Math. Anal. Appl., 381 (2011), pp. 352{364.
    [40] N. T. Qui, Upper and lower estimates for a Frechet normal cone, Acta

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