(Luận văn) đối đạo hàm của ánh xạ nón pháp tuyến và ứng dụng

VIETNAM ACADEMY OF SCIENCE AND TECHNOLOGY INSTITUTE OF MATHEMATICS NGUYEN THANH QUI lu an n va to p ie gh tn CODERIVATIVES OF NORMAL CONE MAPPINGS AND APPLICATIONS d oa nl w nf va an lu z at nh oi lm ul DOCTORAL DISSERTATION IN MATHEMATICS z m co l gm @ an Lu n va HANOI - 2014 ac th si VIETNAM ACADEMY OF SCIENCE AND TECHNOLOGY INSTITUTE OF MATHEMATICS Nguyen Thanh Qui lu an n va CODERIVATIVES OF NORMAL CONE MAPPINGS AND APPLICATIONS p ie gh tn to Speciality code: 62 46 01 12 d oa nl w Speciality: Applied Mathematics nf va an lu DOCTORAL DISSERTATION IN MATHEMATICS z at nh oi lm ul Supervisors: z Prof Dr Hab Nguyen Dong Yen m co l gm @ Dr Bui Trong Kien an Lu n va HANOI - 2014 ac th si lu an n va p ie gh tn to To my beloved parents and family members d oa nl w nf va an lu z at nh oi lm ul z m co l gm @ an Lu n va ac th si Confirmation This dissertation was written on the basis of my research works carried at Institute of Mathematics (VAST, Hanoi) under the supervision of Professor Nguyen Dong Yen and Dr Bui Trong Kien All the results presented have never been published by others lu an n va Hanoi, January 2014 p ie gh tn to The author d oa nl w Nguyen Thanh Qui nf va an lu z at nh oi lm ul z m co l gm @ an Lu n va ac th i si 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 Optimization Theory I am thankful to them for their careful and effective supervision lu 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 an n va p ie gh tn to 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 and from Professor Nguyen Quang Huy I would like to express my sincere thanks to all the members of the research group w d oa nl I owe my thanks to Professor Daniel Frohardt who invited me to work at Department of Mathematics, Wayne State University, for one month (September 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 nf va an lu z at nh oi lm ul z This dissertation was typeset with LaTeX program I am grateful to Professor Donald Knuth who created TeX the program I am so much thankful to MSc Le Phuong Quan for his instructions on using LaTeX @ l gm I would like to thank the Board of Directors of Institute of Mathematics (VAST, Hanoi) for providing me pleasant working conditions at the Institute m co I would like to thank the Steering Committee of Cantho University a lot for constant support and kind help during many years an Lu n va Financial supports from the Vietnam National Foundation for Science and Technology Development (NAFOSTED), Cantho University, Institute of ac th ii si 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 lu an n va p ie gh tn to d oa nl w nf va an lu z at nh oi lm ul z m co l gm @ an Lu n va ac th iii si Contents Table of Notations vi List of Figures viii lu an Introduction ix n va Chapter Preliminary Basic Definitions and Conventions gh Normal and Tangent Cones tn to 1.1 Coderivatives and Subdifferential 1.4 Lipschitzian Properties and Metric Regularity 1.5 Conclusions 11 d oa nl w 1.3 p ie 1.2 lu nf va an Chapter Linear Perturbations of Polyhedral Normal Cone Mappings 12 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 z at nh oi lm ul 2.1 z gm @ 42 m co l Chapter Nonlinear Perturbations of Polyhedral Normal Cone Mappings 43 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 an Lu 3.1 n va ac th iv si 3.4 AVIs under Nonlinear Perturbations 59 3.5 Conclusions 66 Chapter 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 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 Conclusions 91 4.3 lu an 4.4 n va General Conclusions 92 tn to 93 p ie gh List of Author’s Related Papers 94 d oa nl w References nf va an lu z at nh oi lm ul z m co l gm @ an Lu n va ac th v si Table of Notations lu an n va p ie gh tn to d oa nl w nf va an lu z at nh oi z m co l gm @ an Lu w∗ x∗k → x∗ set of positive natural numbers empty set set of real numbers set of x ∈ IR with x > set of x ∈ IR with x ≥ set of x ∈ IR with x ≤ set of generalized real numbers absolute value of x ∈ IR n-dimensional Euclidean vector space norm of a vector x set of m × n-real matrices determinant of a matrix A transposition of a matrix A norm of a matrix A topological dual of a norm space X canonical pairing canonical inner product angle between two vectors u and v open ball with centered at x and radius δ closed ball with centered at x and radius δ open unit ball in a norm space X closed unit ball in a norm space X convex cone generated by Ω linear subspace generated by Ω distance from x to Ω sequence of vectors xk converges to x in norm topology lm ul IN := {1, 2, } ∅ IR IR++ IR+ IR− IR := IR ∪ {±∞} |x| IRn kxk IRm×n detA A> kAk X∗ hx∗ , xi hx, yi \ (u, v) B(x, δ) ¯ δ) B(x, BX ¯X B posΩ spanΩ dist(x; Ω) {xk } xk → x x∗k converges to x∗ in weak* topology n va ac th vi si ∀x x := y b (x; Ω) N N (x; Ω) f :X→Y f (x), ∇f (x) ϕ : X → IR domϕ epiϕ ∂ϕ(x) ∂ ϕ(x, y) lu an n va F :X⇒Y domF rgeF gphF kerF b ∗ F (x, y) D D∗ F (x, y) p ie gh tn to for all x x is defined by y Fr´echet normal cone to Ω at x limiting normal cone to Ω at x function from X to Y Fr´echet derivative of f at x extended-real-valued function effective domain of ϕ epigraph of ϕ limiting subdifferential of ϕ at x limiting second-order subdifferential of ϕ at x relative to y multifunction from X to Y domain of F range of F graph of F kernel of F Fr´echet coderivative of F at (x, y) Mordukhovich coderivative of F at (x, y) d oa nl w nf va an lu z at nh oi lm ul z m co l gm @ an Lu n va ac th vii si with b∗I 6≤ 0, i.e., there exists i ∈ I = I(¯ x, ¯b) such that b∗i > Note that {¯ x∗ }⊥ = {(0, α)}⊥ = IR × {0}, T (¯ x; Θ(¯b)) = (−∞, 0] × (−∞, 0], T (¯ x; Θ(¯b)) ∩ {¯ x∗ }⊥ = (−∞, 0] × {0},
∗ T (¯ x; Θ(¯b)) ∩ {¯ x∗ }⊥ = [0, +∞) × IR ¯ = {1}, where λ ¯ = (0, α) We have x¯∗ = (0, α) = 0a∗1 + αa∗2 Hence, I0 (λ) ¯ Choose Observe that {a∗1 , a∗2 } are linearly independent, thus Ξ(¯ x, ¯b, x¯∗ ) = {λ} ∗ x∗ = (1, −1) ∈ T (¯ x; Θ(¯b)) ∩ {¯ x∗ }⊥ ,
b∗ = (b∗1 , b∗2 ) with b∗1 = −1 ≤ 0, b∗2 = 1, and lu v = (γ, 0) ∈ T (¯ x; Θ(¯b)) ∩ {¯ x∗ }⊥ , an n va p ie gh tn to where γ ≤ By the choice of (x∗ , b∗ , v) above, we have the representation x∗ = −(b∗1 a∗1 + b∗2 a∗2 ) and b∗I0 (λ) ¯ ≤ According to Theorem 2.1, we can infer
b (¯ ¯ we that (x∗ , b∗ , v) ∈ N x, ¯b, x¯∗ ); gphF Since b∗2 = > and ∈ I\I0 (λ),
b (¯ have shown that the inclusion (x∗ , b∗ , v) ∈ N x, ¯b, x¯∗ ); gphF does not imply b∗I ≤ The proof is complete nl w The following fact plays an important role in our subsequent investigations d oa Theorem 2.3 (See [38, Theorem 3.2]) Let (¯ x, ¯b, x¯∗ ) ∈ gphF and I = I(¯ x, ¯b) If the vectors {a∗i | i ∈ I} are positively linearly independent, then an lu b (¯ N x, ¯b, x¯∗ ); gphF = H(¯ x, ¯b, x¯∗ ), nf va
(2.21) lm ul where H(¯ x, ¯b, x¯∗ ) is defined by (2.10) z at nh oi Let us prove that (2.21) holds without any additional assumption Thus, the positive linear independence assumption on the vectors {a∗i | i ∈ I} can be removed from the formulation of Theorem 2.3 z gm @ Theorem 2.4 For any (¯ x, ¯b, x¯∗ ) ∈ gphF, the equality (2.21) is valid \ λ∈Ξ(¯ x,¯b,¯ x∗ ) Eλ (¯ x, ¯b, x¯∗ ) an Lu
b (¯ N x, ¯b, x¯∗ ); gphF ⊂ m co l Proof First, we prove the inclusion “ ⊂ ” in (2.21) For every λ ∈ Ξ(¯ x, ¯b, x¯∗ ),
b (¯ by Theorem 2.1 we deduce that N x, ¯b, x¯∗ ); gphF ⊂ Eλ (¯ x, ¯b, x¯∗ ) Hence, n va
b (¯ From this and Lemma 2.3 we get N x, ¯b, x¯∗ ); gphF ⊂ H(¯ x, ¯b, x¯∗ ) ac th 22 si To justify the opposite inclusion in (2.21), fix any (x∗ , b∗ , v) ∈ H(¯ x, ¯b, x¯∗ )
b (¯ We have to show that (x∗ , b∗ , v) ∈ N x, ¯b, x¯∗ ); gphF We suppose to the
b (¯ contrary that (x∗ , b∗ , v) 6∈ N x, ¯b, x¯∗ ); gphF Then, by the definition of the
b (¯ Fr´echet normal cone N x, ¯b, x¯∗ ); gphF , there exist δ > and a sequence gphF (xk , bk , x∗ ) −→ (¯ x, ¯b, x¯∗ ) such that k hx∗ , xk − x¯i + hb∗ , bk − ¯bi + hv, x∗k − x¯∗ i ≥ δ > 0, ∀k ∈ IN kxk − x¯k + kbk − ¯bk + kx∗k − x¯∗ k (2.22) Since I(xk , bk ) ⊂ T for all k ∈ IN and (xk , bk ) → (¯ x, ¯b), we may assume that I(xk , bk ) = Ie ⊂ I for all k ∈ IN This implies that x∗k ∈ N (xk , Θ(bk )) ⊂ N (¯ x, Θ(¯b)), ∀k ∈ IN lu Hence, by the inclusion (x∗ , b∗ , v) ∈ H(¯ x, ¯b, x¯∗ ) and by (2.10), an n va hv, x∗k − x¯∗ i = hv, x∗k i ≤ tn to Besides, the relations x∗ = − P i∈I (2.23) b∗i a∗i and b∗I¯ = imply that p ie gh hx∗ , xk − x¯i + hb∗ , bk − ¯bi b∗i  ha∗i , x¯i −  ha∗i , xk i i∈I w = X = X oa nl = X + X b∗i  (bk )i − ¯bi  i∈I  (2.24)  b∗i (bk )i − ha∗i , xk i d i∈I  (bk )i − ha∗i , xk i  + X b∗i  (bk )i −  ha∗i , xk i i∈I\I1 nf va an lu i∈I1 b∗i X  (bk )i −  ha∗i , xk i z at nh oi i∈I1 b∗i lm ul Since b∗i ≤ and ha∗i , xk i ≤ (bk )i for all i ∈ I1 , we have ≤ (2.25) z e Let us show that x¯∗ ∈ pos{a∗i | i ∈ I} If x¯∗ = then it is obvious that e Suppose now that x x¯∗ ∈ pos{a∗i | i ∈ I} ¯∗ 6= Since x∗k → x¯∗ , there exists k0 > such that x∗k 6= for all k ≥ k0 For every k ∈ IN , the equality I(xk , bk ) = Ie together with inclusion x∗k ∈ F(xk , bk ) = N (xk ; Θ(bk )) and e By Lemma 2.1, one can find Jk ⊂ Ie formula (2.5) yield x∗k ∈ pos{a∗i | i ∈ I} such that {a∗i | i ∈ Jk } are linearly independent and x∗k ∈ pos{a∗i | i ∈ Jk } Since Ie is finite, the set below is finite m co l gm @ an Lu n o n va Γ := J ⊂ Ie a∗i , i ∈ J, are linearly independent ac th 23 si Consequently, there must exist Je ∈ Γ and a subsequence {k` } of {k} such e for all ` ∈ IN This means that that x∗k` ∈ pos{a∗i | i ∈ J} x∗k` = X λik` a∗i e for some λki ` ≥ 0, i ∈ J (2.26) i∈Je e we infer that Combining (2.26) with the linear independence of {a∗i | i ∈ J} ∗ x¯ = lim `→∞ x∗k` X = lim `→∞ λki ` a∗i X = lim `→∞ i∈Je λki `  a∗i X = i∈Je λi a∗i , i∈Je e Thus, where λi := lim`→∞ λki ` ≥ for all i ∈ J e ⊂ pos{a∗ | i ∈ I} e x¯∗ ∈ pos{a∗i | i ∈ J} i lu e Indeed, since Ie ⊂ I, it holds I\I e ⊂ I\I1 Conversely, We have I\I1 = I\I by the inclusion Je ⊂ Ie we have an n va X ∗ tn to x¯ = λi a∗i = i∈Je X λi a∗i (2.27) i∈Ie gh p ie e J e By (2.27) and definition of provided that we put λi = for all i ∈ I\ I1 = I1 (¯ x, ¯b, x¯∗ ) in (2.9), we see that I\Ie ⊂ I1 Since Ie ⊂ I, this implies e = I e Hence I\I1 ⊂ I\I e The equality I\I1 = I\I e that I\I1 ⊂ I\(I\I) has been proved Furthermore, we have ha∗i , xk i = (bk )i for any k ∈ IN and e Thus, i ∈ I(xk , bk ) = I d oa nl w   an (bk )i − ha∗i , xk i nf va i∈I\I1 b∗i lu X = X b∗i  (bk )i −  ha∗i , xk i = (2.28) e i∈I\I lm ul From (2.24), (2.25) and (2.28), it follows that z at nh oi hx∗ , xk − x¯i + hb∗ , bk − ¯bi ≤ Combining the latter with (2.23), we obtain z l gm @ hx∗ , xk − x¯i + hb∗ , bk − ¯bi + hv, x∗k − x¯∗ i ≤ 0, ∀k ∈ IN kxk − x¯k + kbk − ¯bk + kx∗k − x¯∗ k m co
b (¯ This contradicts (2.22) Therefore, we have (x∗ , b∗ , v) ∈ N x, ¯b, x¯∗ ); gphF The equality (2.21) has been established an Lu n va Let us consider an example to see how Theorem 2.4 can be used for prac
b (¯ tical computation of the Fr´echet normal cone N x, ¯b, x¯∗ ); gphF ac th 24 si Example 2.1 Let X = IR2 and let {a∗i | i ∈ T } ⊂ X ∗ where T = {1, 2, 3}, and a∗1 = (1, 0), a∗2 = (0, 1), a∗3 = (1, 2) For ¯b = (0, 0, 0) ∈ IR3 , x¯ = (0, 0) ∈ X, we have Θ(¯b) = x ∈ IR2 | ha∗i , xi ≤ 0, i ∈ T = (−∞, 0] × (−∞, 0],  I(¯ x, ¯b) = i | ha∗i , x¯i = ¯bi = {1, 2, 3}, F(¯ x; ¯b) = N (¯ x; Θ(¯b)) = pos{a∗ , a∗ , a∗ } = [0, +∞) × [0, +∞)  For α > 0, one has x¯∗ = (0, α) ∈ F(¯ x; ¯b), thus (¯ x, ¯b, x¯∗ ) ∈ gphF We observe that lu an n va p ie gh tn to {¯ x∗ }⊥ = {(0, α)}⊥ = IR × {0},
∗ T (¯ x; Θ(¯b)) = N (¯ x; Θ(¯b)) = (−∞, 0] × (−∞, 0], T (¯ x; Θ(¯b)) ∩ {¯ x∗ }⊥ = (−∞, 0] × {0},
∗ T (¯ x; Θ(¯b)) ∩ {¯ x∗ }⊥ = [0, +∞) × IR oa nl w Since I := I(¯ x, ¯b) = {1, 2, 3}, it holds I¯ = T \I = ∅ There is only one way to P represent x¯∗ = i∈I λi a∗i where λi ≥ for i ∈ I as follows d x¯∗ = (0, α) = 0(1, 0) + α(0, 1) + 0(1, 2) = 0a∗1 + αa∗2 + 0a∗3 an lu  (x , b , v) x∗ ∈ [0, +∞) × IR, v ∈ (−∞, 0] × {0}, ∗ ∗ ∗ x = z at nh oi =
lm ul b (¯ N x, ¯b, x¯∗ ); gphF nf va Hence, I1 := I1 (¯ x, ¯b, x¯∗ ) = {1, 3} By (2.21) and (2.10), we obtain (−b∗1 − b∗3 , −b∗2 − 2b∗3 ), b∗1 ≤ 0, b∗3  ≤0 z @   (−β1 − β3 , −β2 − 2β3 ), (β1 , β2 , β3 ), (γ, 0) β1 , β3 , γ ∈ IR− co l gm =  m The next statement is immediate from Theorem 2.4 and the definition of the Fr´echet coderivative an Lu n va Theorem 2.5 For any (¯ x, ¯b, x¯∗ ) ∈ gphF, the Fr´echet coderivative of F(·) at ac th 25 si the point (¯ x, ¯b, x¯∗ ) is computed by the formula b ∗ F(¯ D x, ¯b, x¯∗ ) : X ∗∗ ⇒ X ∗ × IRm , b ∗ F(¯ D x, ¯b, x¯∗ )(v)  = 
∗ ∗ ∗ b (x , b ) ∈ X × IR (x , b , −v) ∈ N (¯ x, ¯b, x¯ ); gphF ∗ ∗ ∗ m