VIETNAM ACADEMY OF SCIENCE AND TECHNOLOGY INSTITUTE OF MATHEMATICS NGUYEN THANH QUI CODERIVATIVES OF NORMAL CONE MAPPINGS AND APPLICATIONS DOCTORAL DISSERTATION IN MATHEMATICS HANOI - 2014 VIETNAM ACADEMY OF SCIENCE AND TECHNOLOGY INSTITUTE OF MATHEMATICS Nguyen Thanh Qui CODERIVATIVES OF NORMAL CONE MAPPINGS AND APPLICATIONS Speciality: Applied Mathematics Speciality code: 62 46 01 12 DOCTORAL DISSERTATION IN MATHEMATICS Supervisors: 1. Prof. Dr. Hab. Nguyen Dong Yen 2. Dr. Bui Trong Kien HANOI - 2014 To my beloved parents and family members Confirmation This dissertation was written on the basis of my research works carried at Institute of Mathematics (VAST, Hanoi) under the supervision of Profes- sor Nguyen Dong Yen and Dr. Bui Trong Kien. All the results presented have never been published by others. Hanoi, January 2014 The author Nguyen Thanh Qui i Acknowledgments I would like to express my deep gratitude to Professor Nguyen Dong Yen and Dr. Bui Trong Kien for introducing me to Variational Analysis and Optimiza- tion Theory. I am thankful to them for their careful and effective supervision. I am grateful to Professor Ha Huy Bang for his advice and kind help. My many thanks are addressed to Professor Hoang Xuan Phu, Professor Ta Duy Phuong, and Dr. Nguyen Huu Tho, for their valuable support. During my long stays in Hanoi, I have had the pleasure of contacting with the nice people in the research group of Professor Nguyen Dong Yen. In particular, I have got several significant comments and suggestions concerning the results of Chapters 2 and 3 from Professor Nguyen Quang Huy. I would like to express my sincere thanks to all the members of the research group. I owe my thanks to Professor Daniel Frohardt who invited me to work at Department of Mathematics, Wayne State University, for one month (Septem- ber 1–30, 2011). I would like to thank Professor Boris Mordukhovich who gave me many interesting ideas in the five seminar meetings at the Wayne State University in 2011 and in the Summer School “Variational Analysis and Applications” at Institute of Mathematics (VAST, Hanoi) and Vietnam Institute Advanced Study in Mathematics in 2012. This dissertation was typeset with LaTeX program. I am grateful to Pro- fessor Donald Knuth who created TeX the program. I am so much thankful to MSc. Le Phuong Quan for his instructions on using LaTeX. I would like to thank the Board of Directors of Institute of Mathematics (VAST, Hanoi) for providing me pleasant working conditions at the Institute. I would like to thank the Steering Committee of Cantho University a lot for constant support and kind help during many years. Financial supports from the Vietnam National Foundation for Science and Technology Development (NAFOSTED), Cantho University, Institute of ii Mathematics (VAST, Hanoi), and the Project “Joint research and training on Variational Analysis and Optimization Theory, with oriented applications in some technological areas” (Vietnam-USA) are gratefully acknowledged. I am so much indebted to my parents, my sisters and brothers, for their love and support. I thank my wife for her love and encouragement. iii Contents Table of Notations vi List of Figures viii Introduction ix Chapter 1. Preliminary 1 1.1 Basic Definitions and Conventions . . . . . . . . . . . . . . . . 1 1.2 Normal and Tangent Cones . . . . . . . . . . . . . . . . . . . 3 1.3 Coderivatives and Subdifferential . . . . . . . . . . . . . . . . 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 Fr´echet 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 Fr´echet Normal Cone to gphF . . . . . . . . 48 3.3 Estimation of the Limiting Normal Cone to gphF . . . . . . . 54 iv 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 Fr´echet Coderivative of N(x, α) . . . . . . . . . . 70 4.2.2 The Mordukhovich Coderivative of N(x, α) . . . . . . 78 4.3 Necessary and Sufficient 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 v Table of Notations IN := {1, 2, . . .} set of positive natural numbers ∅ empty set IR set of real numbers IR ++ set of x ∈ IR with x > 0 IR + set of x ∈ IR with x ≥ 0 IR − set of x ∈ IR with x ≤ 0 IR := IR ∪ {±∞} set of generalized real numbers |x| absolute value of x ∈ IR IR n n-dimensional Euclidean vector space x norm of a vector x IR m×n set of m ×n-real matrices detA determinant of a matrix A A  transposition of a matrix A A norm of a matrix A X ∗ topological dual of a norm space X x ∗ , x canonical pairing x, y canonical inner product  (u, v) angle between two vectors u and v B(x, δ) open ball with centered at x and radius δ ¯ B(x, δ) closed ball with centered at x and radius δ B X open unit ball in a norm space X ¯ B X closed unit ball in a norm space X posΩ convex cone generated by Ω spanΩ linear subspace generated by Ω dist(x; Ω) distance from x to Ω {x k } sequence of vectors x k → x x k converges to x in norm topology x ∗ k w ∗ → x ∗ x ∗ k converges to x ∗ in weak* topology vi ∀x for all x x := y x is defined by y  N(x; Ω) Fr´echet normal cone to Ω at x N(x; Ω) limiting normal cone to Ω at x f : X → Y function from X to Y f  (x), ∇f(x) Fr´echet derivative of f at x ϕ : X → IR extended-real-valued function domϕ effective domain of ϕ epiϕ epigraph of ϕ ∂ϕ(x) limiting subdifferential of ϕ at x ∂ 2 ϕ(x, y) limiting second-order subdifferential of ϕ at x relative to y F : X ⇒ Y multifunction from X to Y domF domain of F rgeF range of F gphF graph of F kerF kernel of F  D ∗ F (x, y) Fr´echet coderivative of F at (x, y) D ∗ F (x, y) Mordukhovich coderivative of F at (x, y) vii [...]... a closed convex cone, the problems of computing the Fr´chet coderivative (respectively, the Mordukhovich coderivative) of the e Fr´chet normal cone mapping (x, w2 ) → N (x; Θ(w2 )) (respectively, of the e limiting normal cone mapping (x, w2 ) → N (x; Θ(w2 ))), are interesting, but very difficult All the above-mentioned normal cone mappings are special cases of the last two normal cone mappings It will... (¯; Ω) coincides with the notion of x tangent cone in the sense of Convex Analysis This means that T (¯; Ω) is x the topological closure of the cone {λ(x − x)| x ∈ Ω, λ ≥ 0} ¯ In contrast to the limiting normal cone, the Fr´chet normal cone can be e dual of a tangent cone to a set in the primal space Relationships between the Fr´chet normal cone and the contingent cones are described as follows e 5... Q Trang [50] proposed some developments and refinements of the results of [32] G M Lee and N D Yen [23] computed the Fr´chet coderivatives of the e normal cone mappings to a perturbed Euclidean balls and derived from the results a stability criterion for the Karush-Kuhn-Tucker point set mapping of parametric trust-region subproblems As concerning normal cone mappings to nonlinearly perturbed polyhedra,... homogeneous if 0 ∈ F (0) and F (αx) ⊃ αF (x) for all x ∈ X and α > 0 The latter is equivalent to saying that the graph of F is a cone in X × Y The norm of a positively homogeneous multifunction F is defined by F := sup y y ∈ F (x) with x ≤ 1 2 1.2 Normal and Tangent Cones In this section, we recall the concepts of normals and tangents to sets in Banach spaces and discuss their properties and relationships... upper estimates for the Fr´chet normal cone and the limiting normal cone to the graph of the normal e cone mapping of a system of linear inequalities under linear perturbations The results in [52] are applied to stability analysis of parametric variational inequalities, whose constraint sets are linearly perturbed polyhedra [53] Further developments of the studies [11], [52], and [53] can be seen in [13],... computed coderivatives of the normal cone to a rotating closed half-space The normal cone mapping considered in [23] is a special case of the normal cone mapping to the solution set Θ(w2 ) = Θ(p) := {x ∈ X| ψ(x, p) ≤ 0} where ψ : X × P → I is a C 2 -smooth function defined on the product space R of Banach spaces X and P More generally, for the solution map Θ(w2 ) = Θ(p) := {x ∈ X| Ψ(x, p) ∈ K} of a parametric... with the well-known dual characterizations of the two fundamental properties of multifunctions: the local Lipschitz-like property and the metric regularity 11 Chapter 2 Linear Perturbations of Polyhedral Normal Cone Mappings Generalized differentiability properties of the normal cone mappings allow us to get useful information about solution sensitivity/stability of variational inequalities with polyhedral... polyhedral normal cone mapping to the perturbed polyhedron Θ(b) (or, the normal cone mapping F(·), for short) Here, the set N (x; Θ(b)) =   x∗ ∈ X ∗ x∗ , u − x ≤ 0, ∀u ∈ Θ(b) , if x ∈ Θ(b) ∅, if x ∈ Θ(b) denotes the normal cone to Θ(b) at x in the sense of convex analysis It is well-known that the problem of computing the Fr´chet coderivative e and the Mordukhovich coderivative [28] of the normal cone. .. x The Fr´chet normal cone has a tight connection with the concepts of cone tingent tangent cone and of weak contingent cone Definition 1.3 (See [28, Definition 1.8]) Let Ω be a subset of a Banach space X and x ∈ Ω ¯ (i) The set T (¯; Ω) ⊂ X defined by x T (¯; Ω) := Limsup x t↓0 Ω−x ¯ , t (1.7) where the “ Limsup ” is taken with respect to the norm topology of X, is called the contingent cone to Ω at x... 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 differentiation theory are understandable In the late 1990s, V Jeyakumar and D T Luc introduced the concepts of approximate Jacobian and corresponding generalized subdifferential . ACADEMY OF SCIENCE AND TECHNOLOGY INSTITUTE OF MATHEMATICS NGUYEN THANH QUI CODERIVATIVES OF NORMAL CONE MAPPINGS AND APPLICATIONS DOCTORAL DISSERTATION IN MATHEMATICS HANOI - 2014 VIETNAM ACADEMY OF. Mordukhovich and N. M. Nam [13] to compute coderivatives of the normal cone mappings to a fixed polyhedral convex set in Banach space. N. M. Nam [32] showed that the results of [52], [53] on normal cone mappings. some developments and refinements of the results of [32]. G. M. Lee and N. D. Yen [23] computed the Fr´echet coderivatives of the normal cone mappings to a perturbed Euclidean balls and derived from