Điều kiện tối ưu và đối ngẫu trong tối ưu không trơn và các vấn đề liên quan

VIETNAM NATIONAL UNIVERSITY - HO CHI MINH CITY UNIVERSITY OF SCIENCE NGUYEN MINH TUNG Optimality conditions and duality in nonsmooth optimization and related problems PhD THESIS IN MATHEMATICS Ho Chi Minh City - 2015 VIETNAM NATIONAL UNIVERSITY HO CHI MINH CITY UNIVERSITY OF SCIENCE NGUYEN MINH TUNG Optimality conditions and duality in nonsmooth optimization and related problems Major: Optimization Theory Code: 62 46 20 01 First examiner: Associate Prof Dr LAM QUOC ANH Second examiner: Dr NGUYEN BA THI Third examiner: Associate Prof Dr NGUYEN NGOC HAI First independent examiner: Associate Prof Dr DO VAN LUU Second independent examiner: Dr LE THANH TUNG SCIENTIFIC SUPERVISOR: Prof DSc PHAN QUOC KHANH Ho Chi Minh City - 2015 Acknowledgements The completion of this doctoral dissertation would not have been possible without the support of several people I would like to express my sincere gratitude to all of them First, I want to express my deepest gratitude to my supervisor, Professor Phan Quoc Khanh, for his valuable guidance, scholarly inputs, and consistent encouragement I received throughout the research work From deciding on the research topic in the beginning to the process of actual writing of the thesis, he offered his unreserved help and guided through every step of my work He provided me with inspiring and insightful guidance, without which this study would never have reached a successful completion Second, I am very pleased to extend my thanks to the reviewers of this thesis Their comments, remarks and questions have truly improved the quality of this manuscript I would like to thank the professors who agreed to be on the jury judging my thesis defence To my colleagues, I would like to express my thankfulness to Dr Nguyen Dinh Tuan who extended his support in a very special way, and I benefited a lot from his personal and scholarly interactions, his suggestions at various points of my research program My thanks also go to the Faculty of Mathematics and Computer Science at the University of Science, Vietnam National University Ho Chi Minh City for their support during the period of preparation of my thesis Finally, I owe a lot to my parents, my brother, my older sister and my younger sister who support, encourage and help me at every stage of my personal and academic life, and long to see this achievement come true They always make sure I am provided with a carefree environment where I can devote myself entirely to my study I am really lucky to be a member of such a fantastic family i Contents Acknowledgements i List of Symbols and Notations iv Preface vi Premilinaries 1.1 Some definitions in set-valued analysis 1.2 First-order and second-order tangent sets 1.3 Set-valued optimization problems Second-order optimality conditions with the envelope-like effect in set-valued optimization 2.1 Introduction 2.2 Approximations as generalized derivatives 2.3 Necessary optimality conditions 12 2.4 Sufficient optimality conditions 19 Second-order KKT optimality conditions with the envelope-like effect in some general set-valued optimization problems 30 3.1 Introduction 30 3.2 Set-valued optimization problems with generalized inequality constraints 33 3.2.1 Second-order necessary optimality conditions 33 3.2.2 Second-order sufficiency optimality conditions 43 Set-valued optimization problems with mixed constraints 47 3.3.1 Second-order necessary optimality conditions 48 3.3.2 Second-order sufficiency optimality conditions 59 Applications 63 3.3 3.4 ii Second-order asymptotic contingent epiderivatives and set-valued optimization problems 66 4.1 Introduction 66 4.2 Second-order asymptotic contingent epiderivatives 67 4.3 Optimality conditions 72 4.4 Duality 79 Optimality conditions and duality for nonsmooth vector equilibrium problems with constraints 82 5.1 Introduction 82 5.2 Optimality conditions 88 5.2.1 Necessary conditions 88 5.2.2 Sufficient conditions 92 Duality 94 5.3.1 Wolfe Duality 94 5.3.2 Mond-Weir duality 97 Applications 99 5.4.1 Constrained vector minimization 99 5.4.2 Cone saddle point problems 101 5.4.3 Variational inequalities 103 Conclusions 105 Expected further works 107 List of the author’s publications related to the thesis 108 List of the author’s conference reports related to the thesis 109 References 109 5.3 5.4 iii List of Symbols and Notations N Q R R = R ∪ {−∞, +∞} Rn Rn+ · X dimX BX BX (x0 , r) X∗ L(X, Y ) B(X, X, Y ) C∗ C ∗i N (D, z0 ) {xn } xn → x clM intM coreM bdM convM coneM T (M, x0 ) IT (M, x0 ) T (M, x0 , u) A2 (M, x0 , u) IT (M, x0 , u) T (M, x0 , u) A (M, x0 , u) IT (M, x0 , u) f :X→Y F :X⇒Y F −1 : Y ⇒ X domF gphF the natural numbers the rational numbers the real numbers the extended real numbers the n-dimensional Banach space the non-negative orthant of Rn the norm of a space X the dimension of a space X the closed unit ball of a space X the closed ball with the center x0 and the radius r of a space X the dual space of a space X the space of continuous linear mappings from X to Y the space of continuous bilinear mappings from X × X to Y the dual cone of C the quasi-interior dual cone of C the normal cone of D at z0 the sequence with elements xn xn converges to x the closure of a set M the interior of a set M the algebraic interior of a set M the boundary of a set M the convex hull of a set M the cone generated by M the contingent cone of M at x0 the interior tangent cone of M at x0 the second-order contingent set of M at x0 in direction u the second-order adjacent set of M at x0 in direction u the second-order interior set of M at x0 in direction u the second-order asymptotic contingent set of M at x0 in direction u the second-order asymptotic adjacent set of M at x0 in direction u the second-order asymptotic interior set of M at x0 in direction u a single-valued mapping from X to Y a set-valued mapping from X to Y the inverse mapping of F : X ⇒ Y the domain of F the graph of F iv epiF F+ f (x0 ) f (x0 ) df (x0 , u) d2 f (x0 , u)(x) d f (x0 , u)(x) DF (x0 , y0 ) Dl F (x0 , y0 ) D2 F (x0 , y0 , u, v) D (2) F (x0 , y0 , u, v) Dl2 F (x0 , y0 , u, v) D F (x0 , y0 , u, v) D ( ) F (x0 , y0 , u, v) Dl F (x0 , y0 , u, v) EDF (x0 , y0 ) ED2 F (x0 , y0 , u, v) EDc(2) F (x0 , y0 , u, v) ED F (x0 , y0 , u, v) ≤C d(x, M ) u⊥ l.s.c u.s.c t↓0 |·| ·, · [·, ·] (·, ·) (·, ·] and [·, ·) ∅ ∀ ∃ ✷ the epigraph of F the profile/epigraphic map of F the first-order Fr´echet derivative of f at x0 the second-order Fr´echet derivative of f at x0 the Hadamard directional derivative of f at x0 the second-order Hadamard directional derivative of f at x0 in direction (u; x) the asymptotic second-order Hadamard directional derivative of f at x0 in direction (u; x) the contingent derivative of F at (x0 , y0 ) the lower contingent derivative of F at (x0 , y0 ) the second-order contingent derivative of F at (x0 ; y0 ) in direction (u; v) the second-order adjacent derivative of F at (x0 ; y0 ) in direction (u; v) the second-order lower derivative of F at (x0 ; y0 ) in direction (u; v) the asymptotic second-order contingent derivative of F at (x0 ; y0 ) in direction (u; v) the asymptotic second-order adjacent derivative of F at (x0 ; y0 ) in direction (u; v) the asymptotic second-order lower derivative of F at (x0 ; y0 ) in direction (u; v) the contingent epiderivative of F at (x0 , y0 ) the second-order contingent epiderivative of F at (x0 ; y0 ) in direction (u; v) the composed second-order contingent epiderivative of F at (x0 ; y0 ) in direction (u; v) the asymptotic second-order contingent epiderivative of F at (x0 ; y0 ) in direction (u; v) the order relation generated by C the distance from x to M the orthogonal complement of u lower semicontinuous upper semicontinuous t > and t → the absolute value the canonical pairing of any pair of dual spaces the close interval the open interval the half-open intervals the empty set for all there exists the end of a proof v Preface In the intensively developed optimization theory and applications, set-valued vector problems have been taking a remarkably and continuously increasing part in the enormous number of contributions to the literature Vector optimization, including multiobjective problems (as usual the terminology “multiobjective” is used when the number of objectives is finite, and “vector” for the general case), began to be studied more than a century ago, in mathematical applications in economics first, while set-valued optimization began to attract attention from researchers about five decades ago This thesis is devoted to set-valued vector optimization On the other hand, most practical problems are nonsmooth, i.e., the involved maps and functions are not Fr´echet and Gateaux differentiable Hence, in our work we always consider nonsmooth problems, like most important contributions to the literature in recent years Besides derivative-free investigations, the main approach is introducing generalized derivatives to replace the classical ones, and developing their applications There have been numerous notions of generalized derivatives, each of which is valuable in some classes of problems but none is universal, due to the diversity of practical situations Until now, many derivative-like notions have been proposed and applied to investigate optimality conditions in nonsmooth problems (see [7, 14, 15, 31, 38, 41, 79, 80] for first-order conditions, [4, 43, 56, 57, 72, 73, 95, 101] for higher-order conditions, and the references therein) In the pathbreaking paper [14], Corley employed the contingent and circatangent derivatives to establish a first-order Fritz John necessary optimality condition Jahn and Rauh [41] introduced the notion of contingent epiderivative, which is an extension of directional derivative, but it is a single map and its existence is still an difficult problem For overcoming this difficulty, Chen and Jahn [15] proposed a generalized contingent epiderivative of set-valued maps, and it was applied to obtain first-order optimality conditions in [15, 41] By using contingent epiderivatives, Gotz and Jahn in [31] obtained a first-order Karush-Kuhn-Tucker (KKT) necessary optimality condition With the well-known Dubovitski-Milutin approach, Issac and Khan in [38] established a Lagrange multiplier rule for set-valued optimization with generalized inequality constraints Another fruitful approach in set-valued optimization is the dual space approach initiated by Mordukhovich (see [79, 80]) In our contributions we apply several kinds of generalized derivatives as mathematical tools in order to best serve our purpose Depending on particular research situations, we even propose new concepts of generalized derivatives to suit our problems under consideration Recently, second-order optimality conditions for scalar and vector optimization problems have been intensively developed (for example, we cite [2, 10, 13, vi 26, 34, 50, 53, 54, 61, 62, 75, 82, 83, 91, 93, 100] and the references therein) because they refine the first-order by second-order information which is very helpful for recognizing optimal solutions as well as for designing numerical algorithms for computing them We observe, in most related contributions in the literature, that the core of second-order necessary optimality conditions is a direct extension of the classical result in calculus that the second derivative of the objective map (or the Lagrange map in constrained problems) at minimizers is nonnegative Kawasaki in [50] discovered that the second derivative of the Lagrangian at the minimal solution may be strictly negative in certain critical directions He called this phenomenon the envelope-like effect The Kawasaki result was developed in [13, 46, 83] for C2 scalar programming, in [34, 61] for nonsmooth multiobjective programming, and in [62, 93] for infinite-dimensional nonsmooth optimization However, for set-valued optimization, we observe that there is no publication dealing with the envelope-like effect Let us mention first some papers on second-order optimality conditions for set-valued optimization (not discussing the envelope-like effect) In [19, 20], Durea employed second-order contingent derivatives to establish second-order conditions In [43], Jahn et al proposed a secondorder contingent epiderivative and a generalized second-order contingent epiderivative and applied them to obtain optimality conditions in the primal form for set-valued optimization problems with set constraints For set-valued optimization problem with generalized inequality constraints, following the DubovitskiMilutin approach, in [49], Khan and Tammer proved second-order optimality conditions in terms of second-order asymptotic contingent derivatives; Zhu et al [101] proposed and used the second-order composed contingent derivative to establish dual second-order conditions Higher-order necessary and sufficient optimality conditions for set-valued optimization can be seen in [4, 56, 57, 95] In all the above results concerning second-order necessary optimality conditions, the envelope-like effect was not considered In [91], Studniarski introduced the concept of a higher-order local strict (known also as firm or isolated) minimizer for scalar programming and established necessary conditions and sufficient conditions for set-constrained minimization in infinite-dimensional spaces This notion was extended to vector optimization in [44] and to set-valued vector optimization problems in [23], where conditions for local firm minimizers (of order 1) were obtained by using various generalized derivatives of set-valued maps To our knowledge, there is no publication dealing with second-order multiplier rules in set-valued optimization under inclusion constraints, while such a model is important in applications Furthermore, second-order conditions for firm minimizers in set-valued optimization have not been discussed yet Besides, we observe that major efforts have been focused on kinds of minimization problems, since they are often met in practice and natural generalizations of the classical mathematical programming But, recently more general settings like variational inequalities, equilibrium problems and inclusion or variational relation problems have been intensively developed However, significant results in the literature are mainly on the solution existence, stability and well-posedness, solving algorithms, etc We know only Refs [29, 30, 63, 77] dealing with optimality conditions for equilibrium problems, and [76] for vector variational inequality vii problems In [29] only convex problems were studied Several kinds of epiderivatives were used in [77] to prove optimality conditions for multivalued equilibrium problems In [30] strict derivatives or Mordukhovichs coderivatives were tools for problems involving strictly differentiable mappings or locally Lipschitz mappings First and second-order approximations were employed as generalized derivatives in [77] to establish optimality conditions for equilibrium problems without continuity assumptions As far as we know, there have not been results of duality for equilibrium problems in terms of generalized derivatives Regarding mathematical models, in this thesis we undertake to study the following two most important problems: constrained minimization and equilibrium problems We consider general settings in (infinite-dimensional) normed spaces For each of these problems, we investigate two central issues of optimization theory: optimality conditions and duality When considering optimality conditions, we take into account almost all the important kinds of solution concepts in setvalued vector optimization: weak solutions, Pareto solutions, several types of proper efficiency, firm solutions of various orders, and Q-solutions We consider both local and global solutions We focus on second-order optimality conditions since they are becoming more and more important with a priority over firstand higher-order conditions, because of their efficiency in applications Regarding duality, we study the Wolfe and Mond-Weir duality schemes For each of them, the weak and strong duality properties, as well as the converse duality are investigated Taking advantage of working in favorable conditions of the strong “Group of Optimization in Southern Vietnam”, where interesting and important research issues are discussed based on complete and updated scientific materials, including also manuscripts which are not yet published, we have tried to refer to the most recent important papers in the world’s literature and to manuscripts obtained via a rich international cooperation of the group to ensure our research can keep up with international standards The layout of the thesis is as follows In Chapter 1, we present the basic notions needed in the thesis Here we focus on first- and second-order tangency and corresponding first- and second-order generalized derivatives, since they are very crucial in nonsmooth set-valued optimization Chapter is devoted to secondorder conditions for the above-mentioned kinds of solutions of nonsmooth setvalued optimization Our conditions involve the so-called envelope-like effect, an important phenomenon in second-order optimality conditions which is still not enough well-known Here, we applied second-order approximations as generalized derivatives, a very general notion (including most generalized derivatives) suitable for problems with a high level of nonsmoothness Chapter develops further the topic of second-order optimality conditions with the envelope-like effect studied in Chapter We go deeper in dealing with the critical directions, the cause of the envelope-like effect Instead of using approximations as in the preceding chapter, we employ contingent derivatives and related kinds of derivatives, which reflect, more explicitly and in more details, information about directions Regarding the problem settings of minimization, besides minimization problems with generalized inequality constraints investigated in Chapter 2, we also consider minimization viii 5.4.3 Variational inequalities In this subsection, we apply the results of Subsection 5.3 to obtain Wolfe duality results for variational inequalities Let X, Y, Z, S, C, D and Q be as in Subsection 5.1, g : S → Z, and T : X → L(X, Y ), L(X, Y ) being the space of all continuous linear mappings from X to Y The variational inequality with constraints under our consideration is of (VI) finding x0 ∈ S such that T x0 , x − x0 ∈ / −Q for all x ∈ Ω, where the constraint set is Ω := {x ∈ S| g(x) ∈ −D} (recall that T x0 , x − x0 denotes the value of the linear mapping T x0 at x − x0 ) (VI) becomes a special case of (VEP) by setting F (x1 , x2 ) = { T x1 , x2 − x1 } and G(x1 ) = {g(x1 )} for x1 , x2 ∈ S Remark 5.4.11 (i) If S is η-arcwise-connected, then Fx0 is clQ-η-arcwiseconnected on S (ii) y ∈ Fx0 (x) means that y = T x0 , x − x0 , and z ∈ G(x) ∩ (−D), i.e., z = g(x) ∈ −D (iii) Direct computations give (Fx0 , G)∦ (x, (y, z), η) = {( T x0 , u(x,x ) , g ∦ (x, η(x,x ) (t)))|u(x,x ) = limt→0+ η(x, x , t), ∀x ∈ Ω}, where η(x,x ) (t) = tη(x, x , t) for x, x ∈ Ω We define the Wolfe dual problem (DVI)W as follows maximize ψ(q ∗ , d∗ , x) = q ∗ , T x0 , x − x0 s.t q ∗ , T x0 , u(x,x ) + d∗ , g(x) + d∗ , g ∦ (x, η(x,x ) (t)) ≥ 0, u(x,x ) = limt→0+ η(x, x , t), q ∗ ∈ Q∗ \ {0}, d∗ ∈ D∗ , x, x ∈ Ω (5.4.1) (5.4.2) An element (q ∗ , d∗ , x) is a feasible element of (DVI)W wrt x0 iff (q ∗ , d∗ , x) satisfies (5.4.1) and (5.4.2) Corollary 5.4.12 (strong duality) Assume that g is D-η-arcwise-connected on the η-arcwise-connected set S, (q ∗ , d∗ , x) is a feasible element of (DVI)W wrt x0 , and x¯ ∈ Ω If T x0 , x¯ − x0 ∈ clQ and q ∗ , T x0 , x − x0 + d∗ , g(x) = q ∗ , T x0 , x¯ − x0 , then x0 is a global solution of (VI), and (q ∗ , d∗ , x) is a global optimal solution of (DVI)W Proof Since g is D-η-arcwise-connected on the η-arcwise-connected set S, the connectedness assumption of Theorem 5.3.2 is satisfied Since (q ∗ , d∗ , x) is a feasible element of (DVI)W and x¯ ∈ Ω, then (q ∗ , d∗ , x, T x0 , x − x0 , g(x)) is a 103 feasible element of (DVEP)W and (¯ x, T x0 , x¯ − x0 , g(¯ x)) is a feasible triple of (VEP) wrt x0 As q ∗ , T x0 , x − x0 + d∗ , g(x) = q ∗ , T x0 , x¯ − x0 , Theorem 5.3.2 yields that x0 is a global Q-solution of (VEP), and (q ∗ , d∗ , x, T x0 , x −x0 , g(x)) is a global optimal solution of (DVEP)W This implies that x0 is a global solution of (VI) and (q ∗ , d∗ , x) is a global optimal solution of (DVI)W Corollary 5.4.13 Assume that g is D-η-arcwise-connected on η-arcwise-connected set S (i) (direct duality) If x0 is a global solution of (VI) and there exists x¯ ∈ Ω such that T x0 , x¯ − x0 ∈ −clQ and ∈ int(g(¯ x) + g ∦ (¯ x) + D), then there exist ∗ ∗ ∗ ∗ ∗ ∗ q ∈ Q \ {0} and d ∈ D such that (q , d , x¯) is a global optimal solution to (DVI)W (ii) (converse duality) If there exist q ∗ ∈ Q∗ \ {0}, d∗ ∈ D∗ , x¯ ∈ Ω such that (q ∗ , d∗ , x¯) is a global optimal solution of (DVI)W , then x0 is a global solution of (VI) Proof (i) Similarly to Corollary 5.4.12, we can see that (¯ x, T x0 , x¯ − x0 , g(¯ x)) is a feasible triple of (VEP) Since T x0 , x¯ − x0 ∈ −clQ, ∈ int(g(¯ x) + g ∦ (¯ x) + D) and ∈ T x0 , Ω − x0 , according to Theorem 5.3.4, there exist q ∗ ∈ Q∗ \ {0} and d∗ ∈ D∗ such that (q ∗ , d∗ , x, T x0 , x − x0 , g(x)) is a global optimal solution of (DVEP)W Therefore, (q ∗ , d∗ , x¯) is a global optimal solution of (DVI)W (ii) Similarly to (i), applying Theorem 5.3.6, we are done 104 Conclusions In this thesis, we have obtained a series of second-order optimality conditions and duality theorems for set-valued optimization problems It includes the following main results • By developing and using approximations as generalized derivatives of setvalued mappings, we establish both Fritz John necessary conditions and sufficient conditions of order one and two for various kinds of solutions of a multivalued vector optimization problem with general inequality constraints Our results are with a high level of nonsmoothness An emphasis was given to the important (but still not widely known) so-called envelope-like effect • Karush-Kuhn-Tucker second-order optimality conditions for set-valued optimization with attention to the envelope-like effect are obtained To analyze deeper than in the results just above-mentioned the critical feasible directions, which produce this effect, we use contingent derivatives, adjacent derivatives and the corresponding asymptotic derivatives, since directions are explicitly involved in these kinds of derivatives To pursue strong multiplier rules, we impose coneAubin conditions to deal with the objective and constraint maps separately In this way, we can invoke constraint qualifications of the Kurcyusz-Robinson-Zowe type • We propose a notion of a second-order asymptotic contingent epiderivative for set-valued mappings, study its properties and relations to some earlier existing notions second-order contingent epiderivatives Two existence results for second-order asymptotic contingent epiderivatives are established Then, by using this kind of epiderivatives, we investigate set-valued optimization problems with generalized inequality constraints Both second-order necessary and sufficient optimality conditions are established in terms of Karush-Kuhn-Tucker multipliers For duality, the Wolfe and Mond-Weir schemes are investigated • We consider optimality conditions and duality for a general nonsmooth setvalued vector equilibrium problem with inequality constraints We focus on the Q-solution, which contains most other concepts, and the firm solution, which is hardly expressed as a Q-solution To face high-level nonsmoothness, we employ several notions of contingent variations as generalized derivatives As relaxed convexity assumptions, some types of arcwise connectedness conditions are imposed Both necessary and sufficient optimality conditions for firm solutions of order m ≥ are investigated for global Q-solutions and local firm solutions, with consequences for Henig- and Benson-proper solutions For duality, the Wolfe and Mond-Weir schemes are dealt with, using first-order contingent variations We 105 discuss weak, strong, direct and converse duality As illustrative applications, we choose three optimization-related models: a vector minimization problem with inequality constraints, a cone saddle point problem, and a variational inequality The results obtained in the thesis are new, improve, or include many known ones in the literature 106 Expected further works (1) We are in the course of development of some of our results to higher orders In this direction, we can expect some good results because the generalized derivatives we use in this thesis allow natural extensions to higher orders Of course, higherorder considerations are meaningful since they may provide stronger, more general and flexible results to meet the diversity of applications (2) Realizing that Lagrange duality schemes are more interesting and important than the Wolfe and Mond-Weir ones, we are planning to develop results which may be similar to those obtained in this thesis, also in terms of generalized derivatives (involved in Lagrange functions) (3) The kinds of generalized derivatives used in this thesis may be used to consider sensitivity analysis of nonsmooth optimization-related problems In fact, contingent derivatives were already involved in several contributions related to such quantitative stability studies In a near future, we expect to develop some results on applying generalized derivatives familiar to us in sensitivity analysis of optimization models (4) The model of equilibria includes most optimization problems, including all the types of constrained minimization problems Hence, any significant extension of results for minimization problems to this general model is useful In the recent literature, we can see a significant amount of contributions in such extensions for existence considerations and stability or well-posedness issues, but only a small number of papers are concerned with such extensions for optimality conditions and duality We think that we can try to take in to account some extensions in this direction 107 List of the author’s publications related to the thesis (1) Khanh, P.Q., Tung, N.M.: Optimality conditions and duality for nonsmooth vector equilibrium problems with constraints Optimization, 64, 15471575 (2015) (2) Khanh, P.Q., Tung, N.M.: Second-order optimality conditions with the envelope-like effect for set-valued optimization J Optim Theory Appl., 167, 68-90 (2015) (3) Khanh, P.Q., Tung, N.M.: First and second-order optimality conditions without differentiability in multivalued vector optimization Positivity, Onlinefirst, DOI 10.1007/s11117-015-0330-z (2015) (4) Khanh, P.Q., Tung, N.M.: Second-order optimality conditions for Q minimizers and firm minimizers in set-valued 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optimality conditions by imposing assumptions of the cone Aubin property to separate derivatives... (x) − y0 ) ∩ (−intK) = ∅, ∀x ∈ Ω ∩ U (iv) Supposing C has a base B, a pair (x0 , y0 ) is a local strong Henig-proper minimizer of (P), denoted by (x0 , y0 ) ∈ LsHMin(P), iff there exist a neighborhood