MINISTRY OF EDUCATION AND TRAINING VINH UNIVERSITY LE VAN HIEN SOME RESULTS ON METRIC SUBREGULARITY IN VARIATIONAL ANALYSIS AND APPLICATIONS Speciality: Mathematical Analysis Code: 46 01 02 SUMMARY OF MATHEMATICS DOCTORAL THESIS NGHE AN - 2019 Work is completed at Vinh University Supervisors: Dr Nguyen Huy Chieu Assoc Prof Dr Dinh Huy Hoang Reviewer 1: Reviewer 2: Reviewer 3: Thesis will be defended at school-level thesis evaluating council at Vinh University at date month year Thesis can be found at: Nguyen Thuc Hao Library and Information Center - Vinh University Vietnam National Library PREFACE Rationale In order to implement more tools to investigate optimization and related problems, R T Rockafellar and J.-J Moreau proposed and studied the subdifferential for convex functions in the early 1960s In the mid-1970s, F H Clarke and B S Mordukhovich independently introduced the concepts of the subdifferential for possibly non-convex functions Derivatives and coderivatives of set-valued mappings appeared in the early 1980s Besides, many other generalized differential concepts were also presented and examined in the literature In 1998, R T Rockafellar and R J.-B Wets published a monograph book namely “Variational Analysis” based on summarizing, systematizing and complementing basic results in this research direction, marking the birth of Variational analysis Up to now, the first-order variational analysis has been quite perfect, while the second-order variational analysis has been intensively examined and rapidly developed Recently, this field has attracted the attention of many mathematicians The generalization differentiation plays a vital role in variational analysis and its application To any generalized differential structures, there are always two fundamental problems naturally raised: firstly, which feature of the function, mapping or set is reflected by the structure; secondly, how we can calculate or estimate that structure in terms of the initial data In fact, in order to thoroughly address each of these problems, we all need some information about certain regularity of the involved functions, mappings or sets That is why regularity properties are important research objects in variational analysis The metric subregularity is one of the remarkable regularity properties in the firstorder variational analysis Recently, there have been various studies on this property in the second-order variational analysis However, its role in second-order variational analysis is still an interesting and not fully understood issue that requires further investigation With such reasons, we have selected and studied the topic “Some results on metric subregularity in variational analysis and applications” Research Objectives The purpose of the thesis is to establish new research results based on the investigation of the two aforementioned fundamental problems, contributing to clarify the role of the metric subregularity in variational analysis and applications Research Subjects The research subjects of this thesis are regularity properties in variational analysis, subgradient graphical derivative, tilt stability and isolated calmness Research Scopes For the first problem, the thesis focuses on studying the possibility of the subgradient graphical derivative in recognizing tilt stability for unconstrained optimization problems in which the objective function is prox-regular At the same time, the thesis is also interested in nonlinear programs under metric subregularity constraint qualification with the objective and constraint functions being twice continuously differentiable functions For the second one, the thesis focuses on calculating the subgradient graphical derivative for a normal cone mapping under the metric subregularity condition and using this result to investigate the isolated calmness property of the solution mapping for a broad class of generalized equations Research Methodology In this thesis, we use the variational approach and some techniques from functional analysis, convex analysis, set-valued analysis, variational analysis, optimization Scientific and Practical Meaning The thesis contributes to enrich the calculation rules in variational analysis; proposes a new approach to study the tilt stability as well as improves some results of tilt stability for nonlinear programming problems; thereby clarifies the role of metric subregularity in variational analysis and application Moreover, the thesis may be a good reference for those who are interested in variational analysis, optimization and their applications Research Organization 7.1 Research Overview The regularity properties play an important role in variational analysis and its application On the one hand, these properties are used to establish optimality conditions and study stabilities for optimization and related problems On the other hand, they are used to develop calculus rules in variational analysis In addition, they are also utilized to investigate the convergence of algorithms in numerical optimization In variational analysis, mathematicians have proposed and studied many different regularity concepts for sets, extended-real-valued functions and set-valued mappings One of the vital regularity properties in the study of optimal conditions and calculation rules of generalized differentiation is the metric subregularity In 1979, A D Ioffe used this property to define the concept of regular points and set first-order necessary optimality conditions for a class of optimization problems The term “metric subregularity” was suggested by A L Dontchev and R T Rockafellar in 2004 The metric subregularity of the set-valued mapping is equivalent to calmness of the inverse In 2008, A D Ioffe and J V Outrata established a system of calculation rules for the first-order generalized differentiation in the form of duality using the metric subregularity Recently, researchers have also established many calculation rules for the second-order generalized differentiation structures under the metric subregularity Graphical derivative of a set-valued mapping at a point in its graph is the setvalued mapping whose graph is the tangent cone to the graph of the given set-valued mapping at the point in question This concept was introduced by J -P Aubin in 1981, who called it the contingent derivative The term “graphical derivative” was used in the book “Variational Analysis” by R T Rockafellar and R J -B Wets The graphical derivative is a powerful tool in variational analysis One can use it to investigate the stability and sensitivity of constraint and variational systems, and more general, generalized equations The graphical derivative can also be used to characterize some nice properties of set-valued mappings, such as the metric regularity, the Aubin property, the isolated calmness and the strong metric subregularity In spite of being the key in tackling some important issues in variational analysis, calculation of the graphical derivative of a set-valued mapping is generally a challenging task The problem has been studied by many researchers for a long time, and many interesting results in the direction have been established Consider the set Γ given by the formula Γ := x ∈ Rn | q(x) ∈ Θ , where q : Rn → Rm , q(x) = (q1 (x), q2 (x), , qm (x)) is a twice continuously differentiable mapping and Θ ⊂ Rm is a nonempty closed set Set Mq (x) := q(x) − Θ with x ∈ Rn If Θ = Rm − then Γ is the feasible set of the nonlinear programming problem and, in this case, the Mangasarian–Fromovitz constraint qualification (MFCQ) holds at x¯ ∈ Γ iff the mapping Mq is metrically regular around (¯ x, 0) Moreover, if n adding the assumption qi : R → R, i = 1, 2, , m, are convex functions, the Slater condition holds iff Mq is metrically regular If Θ is a closed convex cone, then Γ is the feasible set of the cone programming and the Robinson constraint qualification (RCQ) is equivalent to the metric regularity of Mq The Slater condition, MFCQ and RCQ are the crucial qualification conditions in optimization theory and its application These conditions are originally the metric regularity of the set-valued mapping Mq Therefore, it is possible to collectively refer these conditions as the metric regularity constraint qualification In 2015, for Γ to be the feasible set of a nonlinear programming problem, H Gfrerer and B S Mordukhovich defined the metric subregularity constraint qualification (MSCQ) as the metric subregularity of Mq The concept has been extended for Θ to be an arbitrary closed set This thesis, we concern the computation the graphical derivative DNΓ of the normal cone mapping NΓ : Rn ⇒ Rn , x → NΓ (x), with Θ being a nonempty polyhedral convex set The first result in this direction was established by A L Dontchev and R T Rockafellar in 1996, where these authors accurately described the graph of DNΓ , with the assumption that Γ is a polyhedral convex set, in terms of the input data of the problem The result was then utilized to calculate the second-order limiting subdifferential of the indicator function of Γ In 2013, combining some calculus rules available in variational analysis, R Henrion et al revealed a nice formula for computing the graphical derivative DNΓ under the metric regularity of the set-valued mapping Mq (x) := q(x) − Θ around the reference point In 2014, H Gfrerer and J V Outrata proved that this formula holds if Θ := Rm − and the metric regularity is replaced by the metric subregularity at the reference point plus a uniform metric regularity around this point Among other things, their important contribution is that they proposed a scheme allowing us to directly prove the formula for calculating the graphical derivative of the normal cone mapping, which paves the way for satisfactorily solving the problem of compuating the graphical derivation of the normal cone mapping In 2015, following this scheme for the case Θ := {0Rm1 } × Rm−m under the metric subregularity constraint qualification, H − Gfrerer and B S Mordukhovich showed that the same result remain to be hold if the uniform metric regularity condition is replaced by the weaker condition, which is the the bounded extreme point property (BEPP) Generally, the result of calculating the graphical derivation by A L Dontchev and R T Rockafellar is independent to the results set later However, basically they all have the assumption under the metric subregularity qualification and a certain additional property This leads to the following natural question: Can we unify the results of the calculating the graphical derivative of the normal cone mapping by removing the additional property? In other words , whether the formulas for calculating the graphical derivative of the normal cone mapping mentioned above are still hold if Mq only assumed to be metric subregular? In Chapter 2, with the assumption that Mq is metric subregular at the reference point and Θ is a polyhedral convex set, removing the additional property, we successfully proved that the mentioned formula for calculating the graphical derivative of the normal cone mapping is still hold and thus responds affirmatively to the above question To establish this formula, we used the proof scheme of H Gfrerer and J V Outrata combining with an idea of A D Ioffe and J V Outrata Thank to this formula, we obtained formulas for computing the graphical derivative of solution mappings and characterized the isolated calmness of the solution mappings for a generalized equation class Our results incorporate with many related results in this research direction Tilt stability is a property of local minimizers guaranteeing the minimizing point shifts in a Lipschitzian manner under linear perturbations on the objective function of an optimization problem This notion was introduced by R A Poliquin and R T Rockafellar for problems of unconstrained optimization with extended-real-valued objective function Tilt stability is basically equivalent to a uniform second-order growth condition as well as strong metric regularity of the subdifferential The first characterization of tilt stability using second-order generalization differentiation was due to R A Poliquin and R T Rockafellar in 1998 They proved that for an unconstrained optimization problem, under mild assumptions of prox-regularity and subdifferential continuity, a stationary point is a tilt-stable local minimizer if and only if the second-order limiting subdifferential is positive-definite at the point in question Furthermore, using this result together with a formula of A L Dontchev and R T Rockafellar for the second-order limiting subdifferential of the indicator function of a polyhedral convex set, they obtained a second-order characterization of tilt stability for nonlinear programming problems with linear constraints In 2012, by establishing new second-order subdifferential calculi, B S Mordukhovich and R T Rockafellar derived second-order characterizations of tilt-stable minimizers for some classes of constrained optimization problems Among other important things, they showed that for C -smooth nonlinear programming problems, under the linear independence constraint qualification (LICQ), a stationary point is a tilt-stable local minimizer if and only if the strong second-order sufficient condition (SSOSC) holds In the same year, under the validity of both the MFCQ and CRCQ, B S Mordukhovich and J V Outrata proved that SSOSC is a sufficient condition for a stationary point to be a tilt-stable local minimizer in nonlinear programming In 2015, B S Mordukhovich and T T A Nghia showed that SSOSC is indeed not a necessary condition for tilt stability and then introduced the uniform second-order sufficient condition (USOSC) to characterize tilt stability when both MFCQ and CRCQ occur Recently, H Gfrerer and B S Mordukhovich obtained some point-based second-order sufficient conditions for tilt-stable local minimizers under the validity of both the MSCQ and BEPP Furthermore, when supplementing either nondegeneracy in critical directions or the 2-regularity, the point-based second-order characterization of tilt stability were established Instead of using the second-order subdifferential, we mainly use the subgradient graphical derivative of an extended-real-valued function to characterize tilt stability This tilt stability approach has never been applied by other researchers We note that one of the biggest advantages of this approach is the workable computation of the graphical derivative in various important cases under very mild assumptions in initial data Furthermore, several results on tilt stability were established based on the calculation of the subgradient graphical derivative as a mediate step These observations lead us to the following natural questions: Is it possible to use the subgradient graphical derivative to characterize tilt stability of local minimizers for unconstrained optimization problems in which the objective function is prox-regular and subdifferentially continuous? If yes, is such a characterization useful in helping us to improve the knowledge of tilt stability for nonlinear programming problems? Is it possible to remove prox-regular condition? Chapter of the thesis will answer these questions in a sufficient way, as follows: We have established tilt stability characteristics of local minimizer for the unconstrained optimal problem via the subgradient graphical derivative Applying this result to the nonlinear programming problem under MSCQ, we obtained the necessary and sufficient conditions for tilt-stable local minimizer 7.2 Research Organization The contents of this dissertation are divided into three chapters Chapter is devoted to present the preparatory knowledge as a basis for introducing the main results of the thesis in the remaining chapters Chapter focuses on studying the formula for computing the graphical derivative of normal cone mapping in case Θ is a polyhedral convex set with Mq which is metric subregular and its applications Section 2.1, we present the formula for computing the graphical derivative of the normal cone mappings Then, in section 2.2, we show how to use this formula to compute the graphical derivative of solution mappings as well as derive the new results on the isolated calmness for generalized equations and stationary point mappings Chapter presents the results on tilt stability of local minimizer of for optimization problem In section 3.1, we establish a new second-order characterization of tilt-stable local minimizers for unconstrained optimization problems in which the objective function is prox-regular and subdifferentially continuous Based on the results obtained in section 3.1, section 2.1 and some other authors’ results, section 3.2 establishes the necessary and sufficient conditions in order that a stationary point of a nonlinear programming problem under MSCQ is a tilt-stable local minimizer CHAPTER PRELIMINARIES In this thesis, all spaces are assumed to be Euclidean spaces with scalar product ·, · and Euclidean norm · 1.1 Basic notions This section recalls some notions and their properties from variational analysis, noted from which are used in the sequel 1.1.1 Definition Mapping F for each x ∈ Rn to one and only one set F (x) ⊂ Rm is called set-valued mapping from Rn to Rm and denoted by F : Rn ⇒ Rm If for every x ∈ Rn set F (x) has only one element, then we say F is a single mapping from Rn to Rm We usually use the standard notation F : Rn → Rm The domain, range and graph to F : Rn ⇒ Rm is defined by domF := x ∈ Rn | F (x) = ∅ , rgeF := y ∈ Rm | ∃x ∈ Rn such that y ∈ F (x) , gphF := (x, y) ∈ Rn × Rm | y ∈ F (x) , respectively The inverse mapping F −1 : Rm ⇒ Rn to F is defined by F −1 (y) = x ∈ Rn | y ∈ F (x) , for all y ∈ Rm 1.1.2 Definition Let Ω be a nonempty subset of Rn (i) The (Bouligand-Severi)tangent/contingent cone to Ω at x¯ ∈ Ω is given by TΩ (¯ x) := v ∈ Rn | there exist tk ↓ 0, vk → v with x¯ + tk vk ∈ Ω for all k ∈ N (ii) The (Fr´echet) regular normal cone to Ω at x¯ ∈ Ω is defined by NΩ (¯ x) := v ∈ Rn | lim sup Ω x→¯ x v, x − x¯ ≤0 , x − x¯ 10 Ω where x → x¯ means that x → x¯ with x ∈ Ω (iii) The (Mordukhovich) limiting/basic normal cone to Ω at x¯ ∈ Ω is defined by NΩ (¯ x) = v ∈ Rn | there exist xk → x¯ and vk ∈ NΩ (xk ) with vk → v If x¯ ∈ Ω, put NΩ (¯ x) = NΩ (¯ x) := ∅ by convention 1.1.4 Definition Consider the set-valued mapping F : Rn ⇒ Rm with domF = ∅ (i) Given a point x¯ ∈ domF, the graphical derivative of F at x¯ for y¯ ∈ F (¯ x) is the n m set-valued mapping DF (¯ x|¯ y ) : R ⇒ R defined by DF (¯ x|¯ y )(v) := w ∈ Rm | (v, w) ∈ TgphF (¯ x, y¯) for all v ∈ Rn , that is, gphDF (¯ x|¯ y ) := TgphF (¯ x, y¯) (ii) The regular coderivative of F at a given point (¯ x, y¯) ∈ gphF is the set-valued mapping D∗ F (¯ x, y¯) : Rm ⇒ Rn defined by D∗ F (¯ x, y¯)(y ∗ ) := x∗ ∈ Rn | (x∗ , −y ∗ ) ∈ NgphF (¯ x, y¯) for all y ∗ ∈ Rm In the case F (¯ x) = {¯ y }, one writes DF (¯ x) and D∗ F (¯ x) for DF (¯ x|¯ y ) and D∗ F (¯ x, y¯), respectively We note that if F : Rn → Rm is a single-valued mapping that is differentiable at x¯, then DF (¯ x) = ∇F (¯ x) and D∗ F (¯ x) = ∇F (¯ x)∗ 1.1.6 Definition Let ϕ : Rn → R := R ∪ {±∞} and x¯ ∈ Rn with y¯ := ϕ(¯ x) finite (i) The regular subdifferential of ϕ at x¯ is defined by ∂ϕ(¯ x) := x∗ ∈ Rn | (x∗ , −1) ∈ Nepiϕ (¯ x, y¯) , where epiϕ := (x, α) ∈ Rn × R | α ≥ ϕ(x) is the epigraph of ϕ (ii) The limiting subdifferential of ϕ at x¯ is defined by ∂ϕ(¯ x) := x∗ ∈ Rn | (x∗ , −1) ∈ Nepiϕ (¯ x, y¯) If |ϕ(¯ x)| = ∞, then put ∂ϕ(¯ x) = ∂ϕ(¯ x) := ∅ by convention Note that ∂ϕ(¯ x) ⊂ ∂ϕ(¯ x) and if ϕ is a convex function, then both ∂ϕ(¯ x) and ∂ϕ(¯ x) coincide with the subdifferential in the sense of convex analysis: ∂ϕ(¯ x) = ∂ϕ(¯ x) = x∗ ∈ Rn | x∗ , x − x¯ ≤ ϕ(x) − ϕ(¯ x) for all x ∈ Rn 1.1.8 Definition Let f : Rn → R be an extended-real-valued function (i) The domain of f is defined by domf := x ∈ Rn | f (x) < ∞ (ii) The function f is said to be proper if domf = ∅ and f (x) > −∞, ∀x ∈ Rn 12 Next, we recall some well-known constraint qualifications in nonlinear programming 1.2.9 Definition Considering Γ is the feasible set of the nonlinear programming Γ := x ∈ Rn | q(x) ∈ Rm − , where q(x) := q1 (x), , qm (x)) with qi : Rn → R is a continuously differentiable mapping, for all i = 1, , m (i) The Mangasarian–Fromovitz constraint qualification (MFCQ) is said to hold at point x¯ ∈ Γ if there exists a vector d ∈ Rn such that ∇qi (¯ x), d < for all i ∈ I(¯ x), where I(¯ x) := i ∈ {1, , m} | qi (¯ x) = is the active index set at x¯ ∈ Γ (ii) The constant rank constraint qualification (CRCQ) is said to hold at x¯ ∈ Γ if there is a neighborhood U of x¯ such that the gradient system {∇qi (x)| i ∈ J} has the same rank in U for any index J ⊂ I(¯ x) (iii) The linear independence constraint qualification (LICQ) is said to hold at x¯ ∈ Γ if the gradient system {∇qi (¯ x), i ∈ I(¯ x)} are linearly independent (iv) The constraint set Γ is said to have the bounded extreme point property (BEPP) at x¯ ∈ Γ if there exist real numbers κ > and r > such that E(x, x∗ ) ⊂ κ x∗ B for all x ∈ Γ ∩ Br (¯ x) and x∗ ∈ Rn , where E(x, x∗ ) denotes the set of extremal points of Λ(x, x∗ ), with Λ(x, x∗ ) denotes the set of multipliers T ∗ Λ(x, x∗ ) := λ ∈ Rm / I(x) + | ∇q(x) λ = x , λi = for i ∈ 13 CHAPTER GRAPHICAL DERIVATIVE OF NORMAL CONE MAPPING UNDER THE METRIC SUBREGULARITY CONDITION This chapter presents the formula for computation the graphical derivative of normal cone mapping under the metric subregularity constraint qualification and its applications 2.1 Computation of graphical derivative for a class of normal cone mappings In this section, we suppose that Γ := {x | q(x) ∈ Θ}, where q : Rn → Rm is a twice continuously differentiable mapping and Θ is a nonempty polyhedral convex set in Rm − ∗ For each x¯ ∈ Γ and x¯ ∈ NΓ (¯ x), put Λ := {λ ∈ NΘ (¯ y ) | ∇q(¯ x)T λ = x¯∗ }, with y¯ := q(¯ x) We denote by Iq (¯ x) := {i = 1, 2, , | bi , y¯ = αi } the active index set of Γ at x¯ and K := TΓ (¯ x) ∩ {¯ x∗ }⊥ the critical cone of Γ at x¯ To proceed, we need the following result, which provides a useful formula for computing the normal cone to the critical cone in terms of the initial data 2.1.1 Lemma Suppose that MSCQ is valid at x¯ and y¯ := q(¯ x) Then, for each v ∈ K and λ ∈ Λ, one has NK (v) = ∇q(¯ x)T µ | µT ∇q(¯ x)v = 0, µ ∈ TNΘ (¯y) (λ) , (2.1) 14 where NΘ (¯ y ) = pos{bi | i ∈ Iq (¯ x) and TNΘ (¯y) (λ) = pos{bi | i ∈ Iq (¯ x) − R+ λ Consequently, for v ∈ K, one has     ⊥ T ∗ (2.2) NK (v) = ti bi ∇q(¯ x) − t0 x¯ | t0 , ti ∈ R+ , i ∈ Iq (¯ x) ∩ v   i∈Iq (¯ x) We now arrive at the main result of this section, which provides a formula for the graphical derivative of the normal cone mapping NΓ in the case where Θ is a nonempty polyhedral convex set under a very weak condition (MSCQ) 2.1.10 Theorem Let MSCQ be satisfied at x¯ ∈ Γ and x¯∗ ∈ NΓ (¯ x) Then, one has TgphNΓ (¯ x, x¯∗ ) = (v, v ∗ ) ∈ Rn × Rn | ∃λ ∈ Λ(v) : v ∗ ∈ ∇2 λT q (¯ x)v + NK (v) (2.3) Therefore, the graphical derivative of the normal cone mapping NΓ (x) is given by DNΓ (¯ x|¯ x∗ )(v) = ∇2 λT q (¯ x)v | λ ∈ Λ(v) + NK (v) (2.4) Here Λ(v) is the optimal solution set of the linear programming LP(v), and the cone NK (v) can be computed by (2.2) For the case where Γ is a feasible set of a nonlinear programming, it may happen that the assumption of Theorem 2.1.10 is fulfilled, while the bounded extreme point property is invalid 2.1.12 Example Let q : R2 ⇒ R2 be given by q(x) := (−x1 , x1 − x21 x22 ), Θ := {(0, 0)}, Γ := x ∈ R2 | q(x) ∈ Θ = {0} × R and x¯ := (0, 0) Then, the assumption of Theorem 2.1.10 is fulfilled, while the bounded extreme point property is invalid at x¯ The next result gives us a formula for computing the regular coderivative of the normal cone mapping, which is a direct consequence of Theorem 2.1.10 2.1.13 Corollary Under the assumption of Theorem 2.1.10, one has D∗ NΓ (¯ x, x¯∗ )(u∗ ) = u | u, v − u∗ , ∇2 λT q (¯ x)v ≤ 0, for all v ∈ K, λ ∈ Λ(v), −u∗ ∈ TK (v) 2.2 Application to generalized equation We first consider the parametric generalized equation of the form: ∈ F (x, y) + NΓ (x), (2.5) 15 where F : Rn × Rs → Rn is a continuous differentiable mapping, x is a variable, y is a parameter, and Γ := {x ∈ Rn | q(x) ∈ Θ}, Θ is a nonempty polyhedral set in Rm , and q : Rn → Rm is a twice continuously differentiable mapping Denote by S the solution mapping to (2.5) given by S(y) := x ∈ Rn | ∈ F (x, y) + NΓ (x) 2.2.2 Theorem Let (¯ y , x¯) ∈ gphS and let Mq be metrically subregular at (¯ x, 0) Then, one has DS(¯ y |¯ x)(z) ⊂ v | − ∇y F (¯ x, y¯)z ∈ ∇x F (¯ x, y¯)v + ∇2 λT q (¯ x)v : λ ∈ Λ(v) + NK (v) , (2.6) for all z ∈ Rs Inclusion (2.6) holds as equality if assume further that ∇y F (¯ x, y¯) is ∗ ⊥ ∗ surjective Here K := TΓ (¯ x) ∩ {¯ x } with x¯ := −F (¯ x, y¯), and Λ(v) is the optimal solution set of LP(v) If q is an affine mapping, then {∇2 λT q (¯ x)v | λ ∈ Λ(v)} = {0} and Mq is automatically metrically subregular Hence, in this case, formula (2.6) can be much more simplified 2.2.3 Corollary Consider the generalized equation (2.5) with q : Rn → Rm being an affine mapping For any (¯ y , x¯) ∈ gphS and x¯∗ := −F (¯ x, y¯), one has DS(¯ y |¯ x)(z) ⊂ v | − ∇y F (¯ x, y¯)z ∈ ∇x F (¯ x, y¯)v + NK (v) , for all z ∈ Rs (2.7) Inclusion (2.7) holds as equality if in addition ∇y F (¯ x, y¯) is surjective 2.2.6 Corollary Consider (2.5) with Θ := {0Rm1 }×Rm−m v`a (¯ y , x¯) ∈ gphS Assume − that CRCQ is fulfilled at x¯ Then, one has x)v + NK (v) , DS(¯ y |¯ x)(z) ⊂ v | − ∇y F (¯ x, y¯)z ∈ ∇x F (¯ x, y¯)v + ∇2 λT q (¯ (2.8) for all z ∈ Rs and λ ∈ Λ Inclusion (2.8) holds as equality if in addition ∇y F (¯ x, y¯) is surjective Next, we consider the so-called isolated calmness of S This property introduced by A L Dontchev, which is an important property in variational analysis 2.2.7 Definition The set-valued mapping F : Rs ⇒ Rn is said to be isolated calm at (¯ y , x¯) ∈ gphF if there exist κ, r > such that F (y) ∩ Br (¯ x) ⊂ {¯ x} + κ y − y¯ BRn , for all y ∈ Br (¯ y ) 16 The following theorem gives a characterization of the isolated calmness of the solution mapping 2.2.9 Theorem Let (¯ y , x¯) ∈ gphS and let Mq be metrically subregular at (¯ x, 0) If the implication ∈ ∇x L(¯ x, y¯, λ)v + NK (v) ⇒ v = (2.9) λ ∈ Λ(v), v ∈ Rn is valid, then S is isolated calm at (¯ y , x¯) The reverse statement also holds if ∇y F (¯ x, y¯) n s m n is surjective Here L : R × R × R → R is given by L(x, y, λ) := F (x, y) + ∇q(x)T λ 2.2.10 Corollary Consider the generalized equation (2.5) with Γ := Θ, n = m and q := In the identity mapping in Rn Let (¯ y , x¯) ∈ gphS and x¯∗ := −F (¯ x, y¯) Then, if (∇x F (¯ x, y¯) + NK )−1 (0) = {0} (2.10) then S is isolated calm at (¯ y , x¯) Moreover, if, in addition, rank∇y F (¯ x, y¯) = n then property (2.10) is necessary and sufficient for S to have the isolated calmness at (¯ y , x¯) Next, we now consider the parametric generalized equation w ∈ F (x, y) + NΓ (x), (2.11) where F : Rn × Rs → Rn is a continuous differentiable mapping, x is the variable, and p := (y, w) represents the parameter, and Γ := {x ∈ Rn | q(x) ∈ Θ} with Θ ⊂ Rm being a polyhedron and q : Rn → Rm being a twice continuously differentiable mapping Let S : Rs × Rn ⇒ Rn be the solution mapping of (2.11), that is, S(p) := x ∈ Rn | w ∈ F (x, y) + NΓ (x) for all p := (y, w) ∈ Rs × Rn (2.12) The following result gives us a characterization of the isolated calmness of the mapping S(p) 2.2.11 Theorem Let (¯ p, x¯) ∈ gphS and let Mq be metrically subregular at (¯ x, 0) Then, the following assertions are equivalent (i) The implication ∈ ∇x L(¯ x, p¯, λ)v + NK (v) λ ∈ Λ(v), v ∈ Rn ⇒v=0 is valid (ii) The solution mapping S(p) is isolated calm at (¯ p, x¯) Here L : Rn × Rs × Rn × Rm → Rn is defined by L(x, p, λ) := F (x, y) − w + ∇q(x)T λ with p := (y, w) 17 Finally, we consider the parametric optimization problem g(x, y) − w, x | x ∈ Γ , (2.13) where g : Rn × Rs → R is twice continuously differentiable, the feasible set Γ := {x ∈ Rn | q(x) ∈ Θ}, Θ is a nonempty polyhedral convex set in Rm , q : Rn → Rm is twice continuously differentiable, x is a variable, and y ∈ Rs and w ∈ Rn are parameters Noting that the set-valued mapping XKKT : Rs × Rn ⇒ Rn defined by XKKT (p) := x ∈ Rn | ∈ ∇x g(x, y) − w + NΓ (x) , p := (y, w) ∈ Rs × Rn , is called the stationary point mapping of (2.13) Obviously, the stationary point mapping XKKT (p) is a special case of the setvalued mapping S(p) given by (2.12) So, by Theorem 2.2.11, we get the corresponding characterization of isolated calmness of the stationary point mapping of (2.13) 2.2.12 Corollary.Let (¯ p, x¯) ∈ gphXKKT and let Mq be metrically subregular at (¯ x, 0) Then, the following assertions are equivalent (i) The implication ∈ ∇x L(¯ x, p¯, λ)v + NK (v) λ ∈ Λ(v), v ∈ Rn ⇒v=0 is valid (ii) The mapping XKKT (p) is isolated calm at (¯ p, x¯) Here L : Rn × Rs × Rn × Rm → Rn is defined by L(x, p, λ) := ∇x g(x, y) − w + ∇q(x)T λ with p := (y, w) 18 CHAPTER TILT STABILITY VIA SUBGRADIENT GRAPHICAL DERIVATIVE FOR A CLASS OF OPTIMIZATION PROBLEMS WITH THE PROX-REGULARITY ASSUMPTION In this chapter, we provide a new second-order characterization via the subgradient graphical derivative of tilt-stable local minimizers for unconstrained optimization problems in which the objective function is prox-regular and subdifferentially continuous In the next step, applying the feature set above to nonlinear programming under MSCQ, we obtained a second-order tilt stable characteristic via the relaxed uniform second-order sufficient condition and we continuously obtained the pointbased second-order sufficient condition so that the stationary point of the problem is a tilt stable local minimizer Finally, when applying to the quadratic program with a quadratic inequality constraint, we obtained a simpler feature of the tilt stability 3.1 Second-order characterizations of tilt stability for a class of unconstrained optimization problems First we recall the definition of tilt stability, this concept due to R A Poliquin and R T Rockafellar is defined in 1998 3.1.1 Definition Given f : Rn → R, a point x¯ ∈ dom f is a tilt-stable local minimizer of f with modulus κ > if there is a number γ > such that the mapping Mγ : v → argmin f (x) − v, x x ∈ Bγ (¯ x) is single valued and Lipschitz continuous with constant κ on some neighborhood of ∈ Rn with Mγ (0) = x¯ In this case, we denote tilt (f, x¯) := inf κ| x¯ is a tilt-stable minimizer of f with modulus κ > The following theorem provides the characterizations for tilt stability via the subgradient graphical derivative, which will be the main tool in investigating tilt stability for nonlinear programming problems in section 3.2 19 3.1.3 Theorem Let f : Rn → R be an l.s.c proper function with x¯ ∈ dom f and ∈ ∂f (¯ x) Assume that f is both prox-regular and subdifferentially continuous at x¯ for v¯ = Then the following assertions are equivalent (i) The point x¯ is a tilt-stable local minimizer of f with modulus κ > (ii) There is a constant η > such that for all w ∈ Rn we have w whenever z ∈ D∂f (u, v)(w), (u, v) ∈ gph ∂f ∩ Bη (¯ x, 0) κ Furthermore, we have z, w ≥ tilt (f, x¯) = inf sup η>0 w z ∈ D∂f (u|v)(w), (u, v) ∈ gph ∂f ∩ Bη (¯ x, 0) z, w (3.1) (3.2) with the convention that 0/0 = The following two examples show that the prox-regularity assumption is essential for both (i) ⇒ (ii) and (ii) ⇒ (i) in Theorem 3.1.3 3.1.4 Example Let f : R → R be the function defined by  1  1 if ≤ |x| ≤ ,   ,  + |x| − n+1 n ∗ n n(n + 1) n n ∈ N , f (x) :=   if x = 0,   if |x| > Then x¯ = is a tilt-stable local minimizer and f is subdifferentially continuous but not prox-regular at x¯ = for v¯ = 0, while assertion (ii) of Theorem 3.1.3 is invalid 3.1.5 Example Let f : R2 → R be the function defined by f (x) := x21 + x22 + δΩ (x1 , x2 ), when Ω := {(x1 , x2 ) ∈ R2 | x1 x2 = 0} and x = (x1 , x2 ) Then x¯ = is not a tilt-stable local minimizer and f is subdifferentially continuous but not prox-regular at x¯ = for v¯ = ∈ ∂f (0), while assertion (ii) of Theorem 3.1.3 holds 3.2 Tilt stability in nonlinear programming under the metric subregular condition Consider the nonlinear programming problem g(x) | qi (x) ≤ 0, i = 1, 2, , m , (3.3) where g : Rn → R and qi : Rn → R are twice continuously differentiable functions Let q(x) := q1 (x), q2 (x), , qm (x) for x ∈ Rn and let Γ := {x ∈ Rn | q(x) ∈ Rm − } Based on Definition 3.1.1, people define the tilt stable local minimizer of problem (3.3) as follows 20 3.2.1 Definition We say the point x¯ ∈ Γ is a tilt stable local minimizer of problem (3.3) with modulus κ > if there exists γ > such that the solution mapping ˜ γ (v) := argmin g(x) − v, x | q(x) ∈ Rm M x) − , x ∈ Bγ (¯ is single valued and Lipschitz continuous with constant κ on some neighborhood of ˜ γ (0) = x¯ ∈ Rn with M Thus, x¯ is a tilt stable local minimizer of problem (3.3) if and only if it is a tilt stable local minimizer of the function f := g + δΓ Denote tilt(g, q, x¯) := tilt(f, x¯) For x ∈ Γ, x∗ ∈ NΓ (x), denote I(x) := i ∈ {1, , m} | qi (x) = , T ∗ Λ(x, x∗ ) := λ ∈ Rm / I(x) , + | ∇q(x) λ = x , λi = for i ∈ K(x, x∗ ) := TΓ (x) ∩ {x∗ }⊥ ; I + (λ) := {i = 1, , m | λi > 0} for λ ∈ Rm + Next we introduce a new second-order sufficient condition, which is motivated by the so-called uniform second-order sufficient condition (USOSC) introduced by B S Mordukhovich and T T A Nghia in 2015 3.2.2 Definition We say that the relaxed uniform second-order sufficient condition (RUSOSC) holds at x¯ ∈ Γ with modulus > if there exists η > such that ∇2xx L(x, λ)w, w ≥ w 2, (3.4) whenever (x, v) ∈ gphΨ ∩ Bη (¯ x, 0), here Ψ : Rn ⇒ Rn , Ψ(x) := ∇g(x) + NΓ (x) and λ ∈ Λ x, v − ∇g(x); w with w ∈ Rn satisfying ∇qi (x), w = for i ∈ I + (λ), ∇qi (x), w ≥ for i ∈ I(x)\I + (λ) (3.5) We now arrive at the first result of this section, which gives us a fuzzy characterization of tilt stable local minimizers in terms of RUSOSC and its modification for nonlinear programming problems 3.2.5 Theorem Given a stationary point x¯ ∈ Γ and real numbers κ, γ > 0, suppose that MSCQ is fulfilled at x¯ and γ > subreg Mq (¯ x|0) Then the following assertions are equivalent (i) The point x¯ is a tilt-stable local minimizer of problem (3.3) with modulus κ (ii) The RUSOSC is satisfied at x¯ with modulus := κ−1 (iii) There exists η > such that ∇2xx L(x, λ)w, w ≥ w 2, κ 21 whenever (x, v) ∈ gphΨ ∩ Bη (¯ x, 0) and λ ∈ Λ x, v − ∇g(x); w ∩ γ v − ∇g(x) BRm for w ∈ Rn satisfying ∇qi (x), w = for i ∈ I + (λ) and ∇qi (x), w ≥ for i ∈ I(x)\I + (λ), where Ψ : Rn ⇒ Rn , Ψ(x) := ∇g(x) + NΓ (x) 3.2.6 Corollary Let x¯ be a stationary point of (3.3) at which CRCQ holds Then, the following assertions are equivalent (i) The point x¯ is a tilt stable local minimizer of (3.3) with modulus κ > (ii) There exists η > such that ∇2xx L(x, λ)w, w ≥ w κ whenever (x, v) ∈ gphΨ∩ Bη (¯ x, 0), λ ∈ Λ x, v−∇g(x) , ∇qi (x), w = for i ∈ I + (λ) and ∇qi (x), w ≥ for i ∈ I(x)\I + (λ) We next establish a point-based sufficient condition for tilt stability under MSCQ 3.2.9 Theorem Given a stationary point x¯ ∈ Γ and real numbers κ, γ > 0, suppose that MSCQ is fulfilled at x¯ and γ > subreg Qq (¯ x|0) and that the following second-order condition holds: x, λ)w, w > κ1 w ∇2xx L(¯ whenever w = with ∇qi (¯ x), w = 0, i ∈ I + (λ), and λ ∈ ∆(¯ x), Λ x¯, −∇g(¯ x); v where ∆(¯ x) := (3.6) γ ∇g(¯ x) BRm 0=v∈K x ¯,−∇g(¯ x) Then x¯ is a tilt-stable local minimizer of (3.3) with modulus κ Furthermore, we have the estimation: tilt(g, q, x¯) ≤ sup w | λ ∈ ∆(¯ x), ∇qi (¯ x), w = 0, i ∈ I + (λ) ∇xx L(¯ x, λ)w, w 0, suppose that MSCQ is fulfilled at x¯ and γ > subreg Qq (¯ x|0) and that the following second-order condition holds: w, ∇2xx L(¯ x, λ)w > whenever w = with ∇qi (¯ x), w = 0, i ∈ I + (λ), and λ ∈ ∆(¯ x) := Λ x¯, −∇g(¯ x); v γ ∇g(¯ x) BRm (3.8) 0=v∈K x ¯,−∇g(¯ x) Then, x¯ is a tilt-stable local minimizer for (3.3) 22 3.2.12 Definition We say that the strong second-order sufficient condition (SSOSC) holds at x¯ ∈ Γ if for all λ ∈ Λ x¯, −∇g(¯ x) we have w, ∇2xx L(¯ x, λ)w > (3.9) whenever w = with ∇qi (¯ x), w = 0, i ∈ I + (λ) In 2015, under MFCQ and CRCQ, B S Mordukhovich and J V Outrata proved that the tilt-stability is satisfied under SSOSC In the following corollary we also obtain this property but under condition MSCQ 3.2.13 Corollary Let x¯ be a stationary point of (3.3) at which MSCQ is valid Then, x¯ is a tilt-stable local minimizer of (3.3) provided SSOSC is satisfied at x¯ 3.2.15 Definition The twice differentiable mapping g : Rm → Rs is said to be 2-regular at a given point x¯ ∈ Rm in direction v ∈ Rm if for any p ∈ Rs , the system ∇g(¯ x)u + [∇2 g(¯ x)v, w] = p, ∇g(¯ x)w = admits a solution (u, w) ∈ Rm × Rm , where [∇2 g(¯ x)v, w] denotes the s-vector column with the entrices ∇2 gi (¯ x)v, w , i = 1, , s x), define For each x¯ ∈ Γ and v ∈ TΓlin (¯ I(¯ x, v) := i ∈ I(¯ x) Ξ(¯ x, v) := z ∈ Rn C(¯ x, v) := C ∇qi (¯ x), v = , ∇qi (¯ x), z + v, ∇2 qi (¯ x)v ≤ for i ∈ I(¯ x) , ∇qi (¯ x), z + v, ∇2 qi (¯ x)v = C = i ∈ I(¯ x, v) | with z ∈ Ξ(¯ x, v) 3.2.16 Definition Given a point x¯ ∈ Γ and v ∈ K(¯ x, −∇g(¯ x)) The point x¯ is said to be nondegenerate in the direction v if the set Λ x¯, −∇g(¯ x); v is a singleton The following result provides a second-order necessary condition for tilt-stability, which shows that, under either directional nondegeneracy or 2-regularity, the pointbased second-order sufficient condition established in Theorem 3.2.9 is “not too far” from the necessary one 3.2.20 Theorem Given positive real numbers κ and γ, let x¯ be a tilt-stable local minimizer of (3.3) with modulus κ, and let MSCQ hold at x¯ and γ > subreg Qq (¯ x|0) Suppose that for every v ∈ K x¯, −∇g(¯ x) \{0} one of the following conditions is satisfied: (a) x¯ is nondegenerate in the direction v; (b) for each λ ∈ Λ x¯, −∇g(¯ x); v ∩ γ ∇g(¯ x) BRm there is a maximal element C ∈ C(¯ x, v) with I + (λ) ⊂ C such that the mapping (qi )i∈C is 2-regular at x¯ in the direction v 23 Then, one has w, ∇2xx L(¯ x, λ)w ≥ w κ whenever ∇qi (¯ x), w = 0, i ∈ I + (λ), λ ∈ ∆(¯ x), (3.10) Λ x¯, −∇g(¯ x); v ∩γ ∇g(¯ x) BRm Furthermore, we have where ∆(¯ x) := 0=v∈K x ¯,−∇g(¯ x) tilt(g, q, x¯) = sup w w, ∇2xx L(¯ x, λ)w λ ∈ ∆(¯ x), ∇qi (¯ x), w = 0, i ∈ I + (λ) (3.11) with the convention that 0/0 := in (3.11) By combining Theorem 3.2.9, Theorem 3.2.11 and Theorem 3.2.20, we arrive at the following result, which provides second-order characterizations of tilt-stable local minimizers for (3.3) 3.2.21 Corollary Let x¯ be a stationary point of (3.3) at which MSCQ is valid, and let γ > subreg Qq (¯ x|0) Suppose that for every = v ∈ K x¯, −∇g(¯ x) one of the conditions (a) and (b) given in Theorem 3.2.20 is satisfied Then, the following assertions hold: (i) Given κ > 0, the point x¯ is a tilt-stable local minimizer of (3.3) with any modulus κ > κ if and only if the second-order condition (3.10) is valid; (ii) The point x¯ is a tilt-stable minimizer of (3.3) if and only if the positivedefiniteness condition (3.8) is valid Finally, we consider the quadratic program of the form: g(x) | q(x) ≤ , x∈Rn (3.12) where g(x) := 21 xT Ax + aT x, q(x) = q0 (x) := 12 xT B0 x + bT0 x + β0 , with A, B0 ∈ S n , a, b0 ∈ Rn and β0 ∈ R By using some known results on tilt stability in nonlinear programming, we establish quite simple characterizations of tilt-stable local minimizer for (3.12) under the metric subregularity constraint qualification 3.2.23 Theorem Let x¯ be a stationary point of (3.12) with q(¯ x) = Then the following assertions hold: (i) When ∇q(¯ x) = and ∇g(¯ x) = 0, x¯ is a tilt stable local minimizer if and only if A is positively definite (ii) When ∇q(¯ x) = and ∇g(¯ x) = 0, x¯ is a tilt-stable local minimizer if and only if w, B0 x¯ + b0 A + A¯ x + a B0 w > 0, for all w ∈ Rn \{0} with B0 x¯ + b0 , w = (iii) When ∇q(¯ x) = and MSCQ is valid at x¯, x¯ is a tilt-stable local minimizer if and only if A is positively definite while −B0 is positively semidefinite 24 GENERAL CONCLUSIONS AND RECOMMENDATIONS General conclusions This thesis is intended to study the metric subregularity and its applications The main results of the thesis include: - Establishing a formula for exactly computing the graphical derivative of the normal cone mapping under the metric subregular constraint qualification At the same time, we exhibit formulas for computing the graphical derivative of solution mappings and present characterizations of the isolated calmness for a broad class of generalized equations Our results incorporate many important results in this research direction - Setting up the characterization of the tilt-stable local minimizers for a class of unconstrained optimization problems with the objective function is prox-regular and subdifferentially continuous via the uniform positive-definiteness of the subgradient graphical derivative of objective function Instead of using the second-order subdifferential, here we used the subgradient graphical derivative to examine tilt stability This is a new, unprecedented approach used by previous authors Moreover, we proved that the prox-regularity of the objective function is essential not only for the necessary implication but also for the sufficient one - Obtaining some second-order necessary and sufficient conditions for tilt stability in nonlinear programming under the metric subregularity constraint qualification to be a tilt-stable local minimizer In particular, we show that each stationary point of a nonlinear programming problem satisfying MSCQ is a tilt-stable local minimizer if strong second-order sufficient condition is satisfied In addition, the quadratic program with one quadratic inequality constraint satisfies the metric subregular constraint qualification, by exploiting the specificity of the problem, we have come up with a simple and more explicit characterization of tilt-stable local minimizers 25 Recommendations We find that the topic of this thesis is still able to continuously develop in the following directions: - Using the approach to tilt stability via graphical derivative, examining the tiltstability for the nonpolyhedral conic programs Recently, Benko et al obtained some results for the second-order cone programs with this approach For other cone programs classes, this issue needs further research - Investigating whether it is possible to study the full stability according to LevyPoliquin-Rockafellar by using subgradient graphical derivative Currently, no actual results have been set in this reseacrh direction except from some characterization of full stability via the second-order subdifferential 26 LIST OF PUBLICATIONS RELATED TO THE THESIS N H Chieu and L V Hien (2017), Computation of graphical derivative for a class of normal cone mappings under a very weak condition, SIAM J Optim., 27, 190–204 N H Chieu, L V Hien and T T A Nghia (2018), Characterization of tilt stability via subgradient graphical derivative with applications to nonlinear programming, SIAM J Optim., 28, 2246–2273 N H Chieu, L V Hien and N T Q Trang (2018), Tilt stability for quadratic programs with one or two quadratic inequality constraints, submitted The results of the thesis are reported at: - The seminar of the Analysis Department, Institute of Natural Pedagogy - Vinh University, 2014 - 2019 - The 15th Work shop on Optimization and Scientific Computing, Ba Vi, April 20-22, 2017 - The 16th Work shop on Optimization and Scientific Computing, Ba Vi, April 19-21, 2018 ... variational analysis has been intensively examined and rapidly developed Recently, this field has attracted the attention of many mathematicians The generalization differentiation plays a vital role in... set-valued mappings, such as the metric regularity, the Aubin property, the isolated calmness and the strong metric subregularity In spite of being the key in tackling some important issues in variational... function Tilt stability is basically equivalent to a uniform second-order growth condition as well as strong metric regularity of the subdifferential The first characterization of tilt stability using