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INTERNATIONAL WORKSHOP
Variational Analysis and Related Topics
December 13–15, 2018
Vinh Phuc, Vietnam
PROGRAM&
ABSTRACTS
Hanoi Pedagogical University 2
Aims and Scope
• Variational Analysis is a powerful tool to study optimization problems and equi-librium problems. The advanced techniques based on the notions of tangent andnormal cones to sets, derivative and coderivative of multifuntions, subdifferentials ofextended-real-valued functions, together with the comprehensive calculus rules notonly help to analyze these problems analytically, but also allow ones to constructefficient algorithms to solve them.
• Having a PhD program in Analysis, a master program in Applied Mathematics, anda group of active researchers in Optimization Theory and Equilibrium Theory, HanoiPedagogical University 2 organizes this International Workshop Variational Analysisand Related Topics (December 13-15, 2018) to create a forum for experts and youngresearchers in Variational Analysis, Optimization Theory, and Equilibrium Theoryto exchange new ideas, techniques, and results.
• This Workshop is dedicated to Professor Boris Mordukhovich’s 70th birthday. Hisdistinguished contributions to Variational Analysis, Optimization Theory, and Equi-librium Theory will be highlighted.
Organizing Institutions and Sponsors
• Hanoi Pedagogical University 2
Organizing Committee
• Nguyen Quang Huy (Hanoi Pedagogical University 2) - Chairman
• Tran Van Bang (Hanoi Pedagogical University 2)
• Bui Kien Cuong (Hanoi Pedagogical University 2)
• Nguyen Huy Hung (Hanoi Pedagogical University 2)
• Nguyen Hoang Ngoc (Hanoi Pedagogical University 2)
• Chu Vinh Quyen (Hanoi Pedagogical University 2)
• Phung Gia The (Hanoi Pedagogical University 2)
• Hoang Ngoc Tuan (Hanoi Pedagogical University 2)
• Nguyen Dong Yen (Institute of Mathematics, VAST)
Location
The workshop takes place at Lecture hall 14.8, Hanoi Pedagogical University 2, 32 NguyenVan Linh Street, Xuan Hoa Ward, Phuc Yen City, Vinh Phuc Province, Vietnam
Contact: Dr. Hoang Ngoc Tuan
E-mail: [email protected]
Website: http://hpu2.edu.vn/varanal2018/
Scientific Committee
• Nguyen Quang Huy (Hanoi Pedagogical University 2) - Chairman
• Nguyen Huy Chieu (Vinh University)
• Bui Trong Kien (Institute of Mathematics, VAST)
• Huynh Van Ngai (Quy Nhon University)
• Nguyen Nang Tam (Hanoi Pedagogical University 2)
• Hoang Ngoc Tuan (Hanoi Pedagogical University 2)
• Nguyen Van Tuyen (Hanoi Pedagogical University 2)
• Nguyen Dong Yen (Institute of Mathematics, VAST)
4
PROGRAM
Thursday, December 13, 2018
Morning
08:30 – 09:00 Registration
09:00 – 09:10 Opening Ceremony
Session 1
Chair: Nguyen Quang Huy
09:10 – 09:30 Nguyen Dong Yen (Institute of Mathematics, VAST)Dedication to Professor Boris Mordukhovichon the Occasion of his 70th Birthday
09:30 – 09:50 Coffee Break
Session 2
Chair: Nguyen Dong Yen
09:50 – 10:30 Boris Mordukhovich (Wayne State University)Optimal control of perturbed sweeping processeswith applications
10:30 – 11:00 Michel Thera (Universite de Limoges)Some new developments on the Campanato nearness condition
11:00 – 11:30 Xi-Yin Zheng (Yunnan University)Stability of metric subregularity for multifunctions
11:30 – 12:00 Truong Xuan Duc Ha (Institute of Mathematics, VAST)On Error bounds and weak sharp minima
12:00 – 13:30 Lunch
5
Afternoon
Session 3A
Chair: Jein-Shan Chen
14:00 – 14:30 Ebrahim Sarabi (Miami University)Critical and noncritical Lagrange multipliersfor generalized KKT systems
14:30 – 14:50 Duong Thi Kim Huyen (Institute of Mathematics - VAST)Sensitivity analysis of a stationary point set mapunder Total Perturbations. Part 2: Robinson Stability
14:50 – 15:10 Vu Thi Huong (Institute of Mathematics - VAST)Differentiability properties of a parametric consumer problem
15:10 – 15:30 Vo Thanh Phat (HCMC University of Education)Second-order characterizations of C1-smoothrobustly quasiconvex functions
15:30 – 15:50 Coffee break
Session 3B
Chair: Gue Myung Lee
14:00 – 14:30 Bui Trong Kien (Institute of Mathematics - VAST)Second-order optimality conditions for multi-objectiveoptimal control problems with free right end pointsand mixed pointwise constraints
14:30 – 15:00 Nguyen Huu Tron (Quy Nhon University)Directional Holder metric regularity and applications
15:00 – 15:30 Nguyen Van Tuyen (Hanoi Pedagogical University 2)Existence of efficient and properly efficient solutionsto problems of constrained vector optimization
15:30 – 15:50 Coffee break
6
Session 4A
Chair: Hector Ramırez
15:50 – 16:20 Le Dung Muu (Thang Long University)A fixed point approach for equilibrium problemsdefined by the Nikaido-Isoda inequality
16:20 – 16:50 Nguyen Huy Chieu (Vinh University)Subgradient graphical derivativewith some applications to optimization
16:50 – 17:20 Nguyen Van Luong (Hong Duc University)Weak sharpness and finite convergencefor solutions of mixed variational inequalities
17:20 – 17:50 Lam Quoc Anh (Cantho University)On the stability of solution maps to set optimization problems
Session 4B
Chair: Xiaoqi Yang
15:50 – 16:20 Nguyen Nang Tam (Hanoi Pedagogical University 2)Some perturbation results for the semi-affinevariational inequality problems in Hilbert space
16:20 – 16:50 Wei Ouyang (Yunnan Normal University)Regularity and Newton’s method forGeneral Parametric Variational Systems
16:50 – 17:20 Nguyen Thi Quynh Trang (Vinh University)Tilt stability for quadratic programswith one or two quadratic inequality constraints
17:20 – 17:50 Nguyen Thanh Qui (Can Tho University)Full stability for a class of control problemsof semilinear elliptic partial differential equations
7
Friday, December 14, 2018
Morning
Session 5
Chair: Boris Mordukhovich
08:30 – 09:00 Hong-Kun Xu (Hangzhou Dianzi University)Proximal methods for reweighted lQ-regularizationof sparse signal recovery
09:00 – 09:30 Gue Myung Lee (Pukyong National University)On approximate solutions for robustsemi-infinite optimization problems
09:30 – 10:00 Huynh Van Ngai (Quy Nhon University)Directional metric regularity of multifunctions
10:00 – 10:30 Coffee break
Session 6
Chair: Juan Enrique Martınez-Legaz
10:30 – 11:00 Jane Ye (University of Victoria)Weaker sufficient conditions for metric subregularityand directional quasi/pseudo-normalityn
11:00 – 11:30 Andrew Eberhard (RMIT University)On partial smoothness, tilt stability and the VU–decomposition
11:30 – 12:00 Hector Ramırez (University of Chile)Stability analysis for parameterized conic programs
12:00 – 13:30 Lunch
8
Afternoon
Session 7
Chair: Chong Li
14:00 – 14:30 Juan Enrique Martınez-Legaz (Universitat Autonoma de Barcelona)A quasiconvex asymptotic function with applications in optimization
14:30 – 15:00 Xianfu Wang (University of British Columbia)Global linear convergence of gradient descent methodsin the framework of Bregman distances
15:00 – 15:30 Tan Cao (SUNY Korea)Optimal control for a controlled sweeping processwith applications to the crowd motion model
15:30 – 15:50 Coffee break
Session 8
Chair: Andrew Eberhard
15:50 – 16:20 Chong Li (Zhejiang University)Linearized proximal algorithms with adaptive stepsizesfor convex composite optimization with applications
16:20 – 16:50 Defeng Sun (The Hong Kong Polytechnic University)Sparse semismooth Newton methodsand big data composite optimization
16:50 – 17:20 Xiaoqi Yang (Hong Kong Polytechnic University)On error bound moduli for locally Lipschitz and regular functions
17:20 – 17:50 Jein-Shan Chen (National Taiwan Normal University)Constructions of new complementarity functions for NCP and SOCCP
18:10 – 20:00 Banquet
9
Saturday, December 15, 2018
• Scientific Discussions
• Excursions
ABSTRACTS
13
On the stability of solution maps to set optimizationproblems
Lam Quoc Anh1
Abstract: In this report we study set optimization problems involving set order relations.Some concepts related to convexity, continuity of set-valued maps, and nonlinear scalarizationfunctions for sets are introduced. Employing these notions, we investigate several importantproperties of solutions to such problems, including existence conditions of solutions, stabilityconditions of solution maps, and well-posedness for the reference problems.
1 Department of Mathematics,Teacher College,Cantho UniversityCantho, [email protected]
14
Optimal control for a controlled sweeping processwith applications to the crowd motion model
Tan Cao1 and Boris Mordukhovich2
Abstract: This talk concerns optimal control problems for a new class of dynamics systemsgoverned by the (Moreau) sweeping process which arises in various problems of hysteresis, fer-romagnetism, electric circuits, phase transitions, economics, etc. The dynamics of such systemwas introduced in the 1970s by J. J. Moreau to model quasi-static evolution processes subjectto unilateral constraints, and it can be described by the normal cone mapping to moving poly-hedral convex sets. The main attention is paid to deriving necessary optimality conditions foroptimal control problems using the method of discrete approximations. It should be emphasizedthat the velocity mapping (described by the normal cone) in the differential inclusion is highlynon-Lipschitz and unbounded, which cannot be treated by means of known results in optimalcontrol for differential inclusions. Such challenging issues can be overcome by developing themethod of discrete approximations married with appropriate generalized differential tools ofmodern variational analysis. We also discuss new applications to the controlled crowd motionmodel of traffic equilibria.
1 Department of Applied Mathematics and StatisticsSUNY KoreaIncheon 21985, [email protected]
2 Department of Mathematics,Wayne State University,Detroit, MI 48202, [email protected]
15
Constructions of new complementarity functions forNCP and SOCCP
Jein-Shan Chen1
Abstract: It is well known that complementarity functions play an important role in deal-ing with complementarity problems. In this talk, we introduce a few ways to construct newcomplementarity functions for nonlinear complementarity problems and second-order cone com-plementarity problems. The constructions of such new complementarity functions are basedon discrete-type generalization which is a novel idea. Surprisingly, these new complementarityfunctions possess continuous differentiability even though they are discrete-oriented extensions.This is a new discovery to the literature and we believe that such new complementarity functionscan also be used in many other contexts.
1 Department of MathematicsNational Taiwan Normal UniversityTaipei 11677, [email protected]
16
Subgradient graphical derivative with someapplications to optimization
Nguyen Huy Chieu1
Abstract: The subgradient graphical derivative is the graphical derivative of the limiting subdif-fential mapping associated with an extended-real-valued function. This second-order generalizeddifferential concept is attracting considerable interest of many mathematicians. The impetus ofrecent researches is mainly due to the following two facts. The first one is that new calculusrules for subgradient graphical derivative has been established under mild conditions, allowingus to improve and unify various known results in this area. The other is that new applications ofthe subgradient graphical derivative have been shown, especially in numerical optimization, mo-tivating to further study the subgradient graphical derivative. The aim of this talk is to presentour recent findings on the gradient graphical derivative with applications in optimization.
1 Institute of Natural Sciences Education, Vinh University,182 Le Duan, Vinh, Nghe An, [email protected], [email protected]
17
On partial smoothness, tilt stability and theVU–decomposition
Andrew Eberhard1, Yousong Luono2 and shuai Liu3
Abstract: The study of substructure of nonsmooth functions has led to an enrichment of fun-damental theory of nonsmooth functions. Fundamental to this substructure is the presence ofmanifolds along which the restriction of the nonsmooth function exhibits some kind of smooth-ness. In the case of “partially smooth functions” an axiomatic approach is used by A. Lewisto describe the local structure that is observed in a number of important examples. Indeedthe study of tilt stability can be enhanced for the class of partially smooth functions. In thetheory of the “U-Lagrangian” and the associated “VU decomposition” developed by Mifflin andSagastizabal, the existence of a smooth manifold substructure is proven for some special classesof functions. In the extended theory the presence of so called “fast tracks” is assumed and thesealso give rise to similar manifold substructures. In this theory the associated U -Lagrangianis reminiscent of a partial form of “tilt minimisation” and this observation has motivated thisstudy. As fast tracks and related concepts such as “identifiable constraints” are designed to aidthe design of methods for the solution of nonsmooth minimization problems, it seems appropri-ate to ask what additional insight does the existence of a tilt stable local minimum give to thestudy of the VU decomposition? This is the subject of this talk.
Under the assumption of prox-regularity and the presence of a tilt stable local minimum weare able to show that a VU like decomposition gives rise to the existence of a smooth manifoldon which the function in question coincides locally with a smooth function. Moreover a lowerTaylor expansion or second order “subjet ” exists along this manifold. This provides a broadand general class of problems for which these substructure are available for further study.
1,2 Discipline of Mathematical SciencesSchool of SciencesRMIT UniversityMelbourne, Victoria [email protected]
3 Universidade Estadual de CampinasRio de Janeiro, BRAZIL
18
On Error bounds and weak sharp minima
Truong Xuan Duc Ha1
Abstract: The concepts of error bounds and weak sharp minima play an important role inoptimality conditions, subdifferential calculus, stability and convergence of numerical methods.Numerous characterizations of the error bound property have been established in terms of var-ious derivative-like objects either in the primal space (directional derivatives, slopes, etc) orin the dual space (subdifferentials, normal cones). In this talk, we present some results oncharacterization of error bounds and weak sharp minima in vector optimization.
1 Institute of Mathematics, VAST18 Hoang Quoc Viet road, Hanoi, [email protected]
19
Differentiability properties of a parametric consumerproblem
Vu Thi Huong1, Jen-Chih Yao2, and Nguyen Dong Yen3
Abstract: We study the budget map and the indirect utility function of a parametric consumerproblem in a Banach space setting by some advanced tools from set-valued and variationalanalysis. The Lipschitz-likeness and differentiability properties of the budget map, as well asformulas for finding subdifferentials of the infimal nuisance function, which is obtained from theindirect utility function by changing its sign, are established. Our investigation is mainly basedon the paper by Mordukhovich [J. Global Optim. 28 (2004), 347–362] on coderivative analysisof variational systems and the paper of Mordukhovich, Nam, and Yen [Math. Program. 116(2009), 369–396] on subgradients of marginal functions. Economic meanings of the obtainedsubdifferential estimates are explained in details.
1 Graduate Training CenterInstitute of MathematicsVietnam Academy of Science and [email protected], [email protected]
2 Center for General EducationChina Medical [email protected]
3 Department of Numerical Analysis and Scientific ComputingInstitute of MathematicsVietnam Academy of Science and [email protected]
20
Sensitivity analysis of a stationary point set mapunder total perturbations. Part 2: Robinson stability
Duong Thi Kim Huyen1, Jen-Chih Yao2, and Nguyen Dong Yen3
Abstract: In Part 1 of this paper, we have estimated the Frechet coderivative and the Mor-dukhovich coderivative of the stationary point set map of a smooth parametric optimizationproblem with one smooth functional constraint under total perturbations. From these esti-mates, necessary and sufficient conditions for the local Lipschitz-like property of the map havebeen obtained. In this part, we establish sufficient conditions for the Robinson stability of thestationary point set map. This allows us to revisit and extend several stability theorems inindefinite quadratic programming. A comparison of our results with the ones which can beobtained via another approach is also given.
1 Graduate Training CenterInstitute of MathematicsVietnam Academy of Science and TechnologyHanoi, [email protected]
2 Center for General EducationChina Medical UniversityTaichung, [email protected]
3 Nguyen Dong YenInstitute of MathematicsVietnam Academy of Science and TechnologyHanoi, [email protected]
21
Second-order optimality conditions formulti-objective optimal control problems with freeright end points and mixed pointwise constraints
Bui Trong Kien1
Abstract: In this report, we present some results on first-and second-order necessary optimalityconditions of Karush–Kuhn–Tucker-type for H-locally weak solutions in the sense of Paretoof multi-objective optimal control problems with free right end points and control pointwiseconstraint. In order to deal with the problem, we first establish first-and second-order optimalityconditions for a reduction problem with fixed-end time. We then obtain optimality conditionsfor the original problem.
1 Department of Optimization and Control TheoryInstitute of MathematicsVietnam Academy of Science and Technology18 Hoang Quoc Viet road, Hanoi, [email protected]
22
On approximate solutions for robust semi-infiniteoptimization problems
Jae Hyoung Lee1 and Gue Myung Lee2
Abstract: In this talk, we consider semi-infinite optimization problems involving a convexobjective function and infinitely many convex constraint functions with data uncertainty, andgive its robust counterpart (RSIP). Moreover, we consider approximate solutions (ε-solutions)for (RSIP). Using robust optimization approach (worst-case approach), we establish robustnecessary optimality and robust sufficient theorems and give duality results for ε-solutions for(RSIP) under a closed convex cone constraint qualification. Moreover, an example is given toillustrate the obtained duality results.
1,2 Department of Applied Mathematics,Pukyong National University,Busan 48513, [email protected]
23
Linearized proximal algorithms with adaptivestepsizes for convex composite optimization with
applications
Chong Li1, Jinhua Wang2, Xiaoqi Yang3 and Linglingzhi Zhu1
Abstract: In this talk, we continue to study the problem of numerically solving convex compos-ite optimizations. Linearized proximal algorithms (LPA) with adaptive stepsizes for solving theconvex composite optimization problem are proposed. Local and/or global convergence proper-ties of the proposed algorithms are explored, and their superlinear/quadratic convergence resultsare established under the assumptions of local weak sharp minima and the quasi-regularity con-dition. Our proposed algorithms, compared with the LPA with the constant stepsize, have theadvantages of suiting for wider range of problems and of employing higher convergence rates. Weapply the LPA with adaptive stepsizes to solve the wireless sensor network localization problem,and the numerical results show that the LPA with adaptive stepsizes can solve this problemmore efficiently and stable than the LPA with the constant stepsize or other algorithms.
1 School of Mathematical SciencesZhejiang UniversityHangzhou, P.R. [email protected]
2 Department of MathematicsZhejiang University of TechnologyHangzhou, P.R. [email protected]
3 Department of Applied MathematicsThe Hong Kong Polytechnic UniversityKowloon, Hong [email protected]
24
Weak sharpness and finite convergence for solutionsof mixed variational inequalities
Nguyen Van Luong1
Abstract: We introduce and study the weak sharp solutions for mixed variational inequalitiesin Hilbert spaces. We give several characterizations of weak sharpness of solutions of mixedvariational inequalities with and without using dual gap functions. Under certain conditions,we show that sequences generated by algorithms for finding solutions of mixed variational in-equalities terminate after a finite number of iterations provided that the solution set is weaklysharp. Our results extend and improve some known results in the literature.
This talk based on a joint work with Xiaolong Qin and Jen-Chih Yao.
1 Department of Natural SciencesHong Duc UniversityThanh Hoa, [email protected]
25
A quasiconvex asymptotic function with applicationsin optimization
Nicolas Hadjisavvas1, Felipe Lara2 and Juan Enrique Martınez-Legaz3
Abstract: We introduce a new asymptotic function, which is mainly adapted to quasiconvexfunctions. We establish several properties and calculus rules for this concept and compare it toprevious notions of generalized asymptotic functions. Finally, we apply our new definition toquasiconvex optimization problems: we characterize the boundedness of the function, and thenonemptiness and compactness of the set of minimizers. We also provide a sufficient conditionfor the closedness of the image of a nonempty closed and convex set via a vector-valued function.
1 Department of Product and Systems Design EngineeringUniversity of the Aegean,Hermoupolis, Syros, [email protected]
2 Departamento de MatematicasUniversidad de Tarapaca,Arica, [email protected]
3 Departament d’Economia i d’Historia EconomicaUniversitat Autonoma de Barcelona,[email protected]
26
Optimal control of perturbed sweeping processeswith applications
Boris Mordukhovich1
Abstract: This talk is devoted to optimal control problems described by a controlled versionof Moreau’s sweeping process governed by convex polyhedra, where measurable control actionsenter additive perturbations. This class of problems, which addresses unbounded discontinuousdifferential inclusions with intrinsic state constraints, is truly challenging and underinvestigatedin control theory while being highly important for various applications. To attack such problemswith constrained measurable controls, we develop a refined method of discrete approximationswith establishing its well-posedness and strong convergence. This approach, married to advancedtools of first-order and second-order variational analysis and generalized differentiations, allowsus to derive adequate collections of necessary optimality conditions for local minimizers, firstin discrete-time problems and then in the original continuous-time controlled sweeping processby passing to the limit. The new results include an appropriate maximum condition and signif-icantly extend the previous ones obtained under essentially more restrictive assumptions. Wecompare them with other versions of the maximum principle for controlled sweeping processesthat have been recently established for global minimizers in problems with smooth sweeping setsby using different techniques. We also discuss new applications of the obtained results to somemodels in robotics and traffic equilibria.
Based on joint work with Giovanni Colombo (University of Padova, Italy) and Dao Nguyen(Wayne State University).
1 Department of Mathematics,Wayne State University,Detroit, MI 48202, [email protected]
27
A fixed point approach for equilibrium problemsdefined by the Nikaido-Isoda inequality
Le Dung Muu1
Abstract: The talk presents a fixed point aproach to the equilibrium problem defined by theNikaido-Isoda-Fan inequality.
1 TIMAS,Thang Long University,Hanoi, [email protected]
28
Directional metric regularity of multifunctions
Huynh Van Ngai1
Abstract: We present the notion of directional metric regularity and some characterizationsof this property by using the strong slope. These slope characterizations allow us to obtain acoderivative type criteration as well as the robustness of the directional metric regularity.
1 Department of MathematicsQuy Nhon UniversityBinh Dinh, [email protected]
29
Regularity and Newton’s method for GeneralParametric Variational Systems
Wei Ouyang1
Abstract: In this paper, our efforts are dedicated to study the regularity properties of generalparametric variational systems. The relationship between the Lipschitz-like and metric regular-ity properties of the solution mapping, the base mapping, and field mapping in the general PVSare established via Dontchev-Hager Fixed Point Theorem. Under the assumption of regularityconditions, we study the application of Newton’s method to such systems and established resultsconcerning parametric properties of the associated sequences of Newton’s iterates.
1 School of MathematicsYunnan Normal UniversityKunming, Yunnan, China [email protected]
30
Second-order characterizations of C1-smooth robustlyquasiconvex functions
Pham Duy Khanh1 and Vo Thanh Phat2
Abstract: This talk investigates the possibility of using the Frechet and Mordukhovich second-order subdifferentials to characterize the robust quasiconvexity of C1-smooth functions. We setup a necessary condition for the robust quasiconvexity of C1,1-smooth functions and univariateC1-smooth ones. We also show that the established necessary condition is indeed a sufficient onefor the robust quasiconvexity of C1-smooth functions.
1 Center for Mathematical Modeling,University of Chile,Santiago, [email protected], [email protected]
2 Department of Mathematics,HCMC University of Education,Ho Chi Minh, [email protected]
31
Full stability for a class of control problems ofsemilinear elliptic partial differential equations*
Nguyen Thanh Qui1 and Daniel Wachsmuth2
Abstract: We investigate full Lipschitzian and full Holderian stability for a class of control prob-lems governed by semilinear elliptic partial differential equations, where all the cost functional,the state equation, and the admissible control set of the control problems undergo perturbations.We establish explicit characterizations of both Lipschitzian and Holderian full stability for theclass of control problems. We show that for this class of control problems the two full stabilityproperties are equivalent. In particular, the two properties are always equivalent in general whenthe admissible control set is an arbitrary fixed nonempty, closed, and convex set.
* This research was supported by the Alexander von Humboldt Foundation.
1 College of Information and Communication TechnologyCan Tho UniversityCampus II, 3/2 Street, Can Tho, [email protected] fur MathematikUniversitat WurzburgEmil-Fischer-Str. 30, 97074 Wurzburg, [email protected]
2 Institut fur MathematikUniversitat WurzburgEmil-Fischer-Str. 30, 97074 Wurzburg, [email protected]
32
Stability analysis for parameterized conic programs
Hector Ramırez1
Abstract: In this talk we first visit several results characterizing well-known stability proper-ties (such as Aubin property, isolated calmness, etc.) for critical maps of parameterized conicprograms. These characterizations are typically carried out via the computation of second ordergeneralized derivatives, and we need the constraint set is defined over a convex cone satisfyinga reducibility assumption and is (strongly) qualified. Then, we present an ongoing work, whichaim is to prove/extend those results under weaker qualification constraints conditions, estab-lishing some connections between second order derivatives and well-known conic tools, such asthe sigma term. Our approach covers seminal examples such as (nonlinear) SDP and SOCP.
1 Center for Mathematical Modeling,University of Chile,Santiago, [email protected]
33
Critical and noncritical Lagrange multipliers forgeneralized KKT systems
Ebrahim Sarabi1
Abstract: This talk focuses mainly on providing a second-order variational analysis of criticaland noncritical Lagrange multipliers for generalized KKT systems. Furthermore, we discussa new characterization of uniqueness of Lagrange multipliers via a dual condition. Finally anapplication to the sequential quadratic programming algorithm for constrained optimizationproblems will be presented.
The talk is based on a joint work with Boris Mordukhovich.
1 Department of MathematicsMiami UniversityOxford, OH, [email protected]
34
Sparse semismooth Newton methods and big datacomposite optimization
Defeng Sun1
Abstract: Big data optimization problems provide many challenges as well as plenty of op-portunities for algorithm developers. Concerned with inherent huge computational burdensof the interior point methods (IPMs) for solving optimization problems of large scales, manyresearchers and practitioners tend to the first order methods (FOMs) such as the acceleratedproximal gradient methods and the alternating direction methods of multipliers for the rescue.While these FOMs have indeed been very successful in a number of interesting applications,they also encounter enormous numerical difficulties in dealing with many real data optimizationproblems of big scales even with a low or moderate solution quality. New ideas for solving theseproblems are highly sought both in practice and in academic research. In this talk, we shallexplain why sparse semismooth Newton methods can fully exploit the second order sparsity/lowrank information exhibited in big composite optimization models. Consequently, scalable andefficient algorithms can often be designed to overcome the mentioned numerical difficulties eitherin IPMs or in FOMs. A highly efficient software called LassoNAL for solving the well-knownLasso problem will be used to demonstrate that solving big data optimization models even withhigher accuracy are becoming realistic.
1 Department of Applied Mathematics,The Hong Kong Polytechnic University,Hong Kong,[email protected]
35
Some perturbation results for the semi-affinevariational inequality problems in Hilbert space
Nguyen Nang Tam1
Abstract: In this talk, we present some perturbation results for the semi-affine variationalinequality problems in Hilbert space. Our results complement and generalize some existingresults.
1 Hanoi Pedagogical University 2Xuan Hoa, Phuc Yen, Vinh Phuc, [email protected]
36
Some new developments on the Campanato nearnesscondition
Michel Thera1
Key words: Elliptic PDEs, Birkoff-James orthogonality, Campanato’s nearness.
Abstract: I propose to survey the theory of nearness between operators acting on normedspaces and developed by S. Campanato at the end of the eighties in a series of papers (see e.g.[1]). The aim of S. Campanato was to study existence and regularity results for some differentialelliptic equations. Given X a set with at least two elements, and (Y, ‖ · ‖) a real normed space,he said that the function a : X → Y is near the function b : X → Y if the inequality
‖ (b(x2)− αa(x2))− (b(x1)− αa(x1)) ‖ ≤ κ ‖b(x2)− b(x1)‖ ∀x1, x2 ∈ X (1)
holds for some positive constant α, and some real number κ such that 0 < κ < 1.
Obviously nearness is a reflexive relation. The first part of the talk adresses the naturalquestion of the symmetry of the nearness relation, as developed recently in [2]. We observethat when (Y is an inner product space and a is near b for the constants α and κ, then b is
near a, but for the different constants 1−κ2α and κ. When the dimension of Y is greater or equal
to three, then the three following properties are equivalent : Y is an inner product space, theBirkhoff-James orthogonality is symmetric, and the Campagnato nearness is symmetric.
In a second part of the talk, I will propose an extension of the nearness property to mul-tifunctions, as developed in [3], and investigate which properties of set-valued mappings arepreserved by nearness.
References
[1] S. Campanato, On the condition of nearness between operators, Ann. Mat. Pura Appl. (4),vol. 167, pp. 243–256, (1994).
[2] A. Barbagallo, E. Ernst, M. Thera, Symmetry of Campagnato’s nearness condition and themeasure of right angles in a normed space, Preprint (2018).
[3] A. Barbagallo, E. Ernst, M. Thera, Campanato’s nearness for multifunctions, Preprint(2018).
1 Universite de LimogesLimoges, [email protected]
37
Tilt stability for quadratic programs with one or twoquadratic inequality constraints
Nguyen Huy Chieu1, Le Van Hien2 and Nguyen Thi Quynh Trang3
Abstract: This work examines the tilt stability for quadratic programs with one or twoquadratic inequality constraints. Exploiting specific features of these problems and using someknown results on tilt stability in nonlinear programming, we establish quite simple charac-terizations of tilt-stable local minimizer for quadratic programs with one quadratic inequalityconstraint under the metric subregularity constraint qualification. By the same way, we alsoderive various tilt stability conditions for quadratic programs with two quadratic inequality con-straints and satisfying certain suitable assumptions. Especially, the obtained results show thatsome tilt stability conditions only known to be sufficient in nonlinear programming become thenecessary ones when the considered problems are quadratic programs with one or two quadraticinequality constraints.
1 Institute of Natural Sciences Education, Vinh University,182 Le Duan, Vinh, Nghe An, VietnamEmail: [email protected], [email protected]
2 Department of Natural Science Teachers, Ha Tinh University,Ha Tinh, VietnamEmail: [email protected]
3 Institute of Natural Sciences Education, Vinh University,182 Le Duan, Vinh, Nghe An, VietnamEmail: [email protected]
38
Directional Holder metric regularity and applications
Nguyen Huu Tron1
Abstract: This talk sheds new light on regularity of multifunctions through various characteri-zations of directional Holder/Lipschitz metric regularity, which are based on the concepts of slopeand coderivative. By using these characterizations, we show that directional Holder/Lipschitzmetric regularity is stable, when the multifunction under consideration is perturbed suitably.Applications of directional Holder/Lipschitz metric regularity to investigate the stability andthe sensitivity analysis of parameterized optimization problems are also discussed.
1 Department of MathematicsQuy Nhon UniversityBinh Dinh, [email protected]
39
Existence of efficient and properly efficient solutionsto problems of constrained vector optimization
Nguyen Van Tuyen1
Abstract: The paper is devoted to the existence of global optimal solutions for a general classof nonsmooth problems of constrained vector optimization without boundedness assumptionson constraint sets. The main attention is paid to the two major notions of optimality in vectorproblems: Pareto efficiency and proper efficiency in the sense of Geoffrion. Employing adequatetools of variational analysis and generalized differentiation, we first establish relationships be-tween the notions of properness, M -tameness, and the Palais–Smale conditions formulated forthe restriction of the vector cost mapping on the constraint set. These results are instrumental toderive verifiable necessary and sufficient conditions for the existence of Pareto efficient solutionsin vector optimization. Furthermore, the developed approach allows us to obtain new sufficientconditions for the existence of Geoffrion-properly efficient solutions to such constrained vectorproblems.
The talk is based on a joint work with Do Sang Kim, Boris Mordukhovich and Pham TienSon.
1 Department of MathematicsHanoi Pedagogical University 2Xuan Hoa, Phuc Yen, Vinh Phuc, [email protected]
40
Global linear convergence of gradient descentmethods in the framework of Bregman distances
Xianfu Wang1
Abstract: We study the linear convergence of a generalized gradient descent method (GGDM)in the framework of Bregman distances. The GGDM can solve nonconvex and nonsmoothminimization problems. Introducing the LC-condition, Bregman-Polyak- Lojasiewicz condition,and a lower control function, we establish the decreasing property and global linear convergenceof the GGDM. Our work extends the results by Polyak, and recent results by Karimi, Nutiniand Schmidt and by Lu, Freund and Nesterov. Examples are given to illustrate the advantagesof our results. Joint work with H. Bauschke, J. Bolte, J. Chen and M. Teboulle
1 Department of MathematicsUniversity of British Columbia OkanaganKelowna, BC, [email protected]
41
Proximal methods for reweighted lQ-regularization ofsparse signal recovery
Hong-Kun Xu1
Abstract: To recover a sparse signal from a noised linear measurement system Ax = b + e,convex lp regularization methods (i.e., 1 ≤ p < 2, in particular, p = 1) are commonly usedunder certain conditions. Recently, however, more attentions have been paid to nonconvex lqregularization methods (i.e., 0 < q < 1, in particular, q = 1/2) for recovering a sparse signal. Inthis paper, we use proximal methods to discuss both convex and nonconvex reweighted lQ reg-ularization for recovering a sparse signal. Convex lQ regularization is introduced by S. Voroninand I. Daubechies (An iteratively reweighted least squares algorithm for sparse regularization,arXiv:1511.08970v3). We extend it to the nonconvex case and our results therefore supplementthose of Voronin and Daubechies. We also study Nesterov’s acceleration method for the noncon-vex case. Our numerical experiments show that nonconvex lQ regularization can more effectivelyrecover sparse signals.
1 Department of MathematicsHangzhou Dianzi UniversityHangzhou 310018, [email protected]
42
On error bound moduli for locally Lipschitz andregular functions
Minghua Li,1 Kaiwen Meng2 and Xiaoqi Yang, 3
Abstract: In this paper we study local error bound moduli for a locally Lipschitz and regularfunction via its outer limiting sub differential set. We show that the distance from 0 to the outerlimiting subdierential of the support function of the subdierential set, which is essentially thedistance from 0 to the end set of the subdierential set, is an upper estimate of the local errorbound modulus. This upper estimate becomes tight for a convex function under some regularityconditions. We show that the distance from 0 to the outer limiting subdierential set of a lowerC1 function is equal to the local error bound modulus.
1 School of Mathematics and Finance,Chongqing University of Arts and Sciences,Yongchuan, Chongqing, 402160, [email protected]
2 School of Economics and Management,Southwest Jiaotong University,Chengdu 610031, [email protected]
1 Department of MathematicsHong Kong Polytechnic UniversityHong [email protected]
43
Weaker sufficient conditions for metric subregularityand directional quasi/pseudo-normality
Kuang Bai1, Jane Ye2 and Jin Zhang3
Abstract: In this paper we study sufficient conditions for metric subregularity of a set-valuedmap which is the sum of a single-valued continuous map and a locally closed subset. First wederive a sufficient condition for metric subregularity which is weaker than the so-called first-order sufficient condition for metric subregularity (FOSCMS) by adding an extra sequentialcondition. Then we introduce a directional version of the quasi-normality and the pseudo-normality which is weaker than the classical quasi-normality and pseudo-normality respectively.The directional quasi/pseudo-normality is stronger than the new weaker sufficient conditionfor metric subregularity but easier to verify. Moreover we introduce a nonsmooth version ofthe second-order sufficient condition for metric subregularity (SOSCMS) and show that it is asufficient condition for the new sufficient condition for metric subregularity to hold. An exampleis used to illustrate that the directional pseduo-normality can be weaker than FOSCMS. For theclass of set-valued maps where the single-valued mapping is affine and the abstract set is theunion of finitely many convex polyhedral sets, we show that the directional pseudo-normalityholds automatically at each point of the graph.
1 Department of Mathematics and StatisticsUniversity of VictoriaVictoria, British [email protected]
2 Department of Mathematics and StatisticsUniversity of VictoriaVictoria, British [email protected]
3 Department of MathematicsHong Kong Baptist UniversityHong [email protected]
44
Stability of metric subregularity for multifunctions
Xi Yin Zheng1
Abstract: It is known that the metric regularity of a closed multifunction F always has thestability when the objective F undergoes “small Lipschitz perturbations”. However the metricsubregularity of F is not stable even when F is convex and undergoes “small smooth perturba-tions”. On the other hand, metric subregularity, as extensions of two fundamental notions oferror bound and weak sharp minima in optimization, is more useful than metric regularity insome aspects of variational analysis and optimization. This paper devotes to consider the stabil-ity of metric subregularity when the objective undergoes either “small Lipschitz perturbations”or “small calm perturbations”. Some results obtained in this paper improve and generalize thecorresponding issues for error bounds in the literature.
1 Department of MathematicsYunnan UniversityKunming, Yunnan, China [email protected]
LIST OF PARTICIPANTS
48
Lam Quoc Anh 6, 13Cantho [email protected]
Trang Van Bang 2Hanoi Pedagogical University [email protected]
Tan Cao 8, 14SUNY [email protected]
Jein-Shan Chen 5, 8, 15National Taiwan Normal [email protected]
Nguyen Huy Chieu 3, 6, 16Vinh [email protected]
Mooi Lee ChooiHanoi Pedagogical University [email protected]
Bui Kien CuongHanoi Pedagogical University [email protected]
Dang Van CuongDuy Tan [email protected]
Nguyen Phuong DongHanoi Pedagogical University [email protected]
Nguyen Trung DzungHanoi Pedagogical University [email protected]
Andrew Eberhard 8, 7, 17RMIT University, [email protected]
Truong Xuan Duc Ha 4, 18Institute of Mathematics, [email protected]
Vu Trung HieuPhuong Dong [email protected]
Pham Thi HuongHanoi Pedagogical University [email protected]
Vu Thi Huong 5, 19Institute of Mathematics, [email protected]; [email protected]
Nguyen Quang Huy 2, 3Hanoi Pedagogical University [email protected]
Duong Thi Kim Huyen 5, 20Institute of Mathematics, [email protected]
Bui Trong Kien 3, 5, 21Institute of Mathematics, [email protected]
Poom KumamKMUTT University, [email protected]
Gue Myung Lee 5, 7, 22Pukyong National [email protected]
Chong Li 8, 8, 23Zhejiang [email protected]
Nguyen Ngoc LuanHanoi National University of [email protected]
Nguyen Van Luong 6, 24Hong Duc [email protected]
Juan Enrique Martınez-Legaz 7, 8, 25Universitat Autonoma de Barcelona, [email protected]
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Boris Mordukhovich 4, 7, 26Wayne state [email protected]
Le Dung Muu 6, 27TIMAS, Thang Long [email protected]
Nguyen Thi Kieu NgaHanoi Pedagogical University [email protected]
Le Thanh NgaHanoi Pedagogical University [email protected]
Khuat Van NinhHanoi Pedagogical University [email protected]
Huynh Van Ngai 3, 7, 28Quy Nhon [email protected]
Wei Ouyang 6, 29Yunnan Normal [email protected]
Vo Thanh Phat 5, 30HCMC University of [email protected]
Nguyen Thanh Qui 6, 31Can Tho [email protected]
Hector Ramırez 7, 6, 32University of [email protected]
Ebrahim Sarabi 5, 33Miami [email protected]
Doan Thai SonInstitute of Mathematics, [email protected]
Nguyen Hai SonHanoi University of Science & [email protected]
Defeng Sun 8, 34The Hong Kong Polytechnic [email protected]
Nguyen Nang Tam 3, 6, 35Hanoi Pedagogical University [email protected]
Vo Thanh TaiAn Giang [email protected]
Le Chi ThanhHanoi University Of [email protected]
Le Xuan ThanhInstitute of Mathematics, [email protected]
Michel Thera 4, 36Universite de [email protected]
Tran Thi ThuHanoi Pedagogical University [email protected]
Le Quang ThuyHanoi University of Science & [email protected]
Nguyen Thi ToanHanoi University of Science & [email protected]
Nguyen Thi Quynh Trang 6, 37Vinh [email protected]
Nguyen Huu Tron 5, 38Quy Nhon [email protected]
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Hoang Ngoc Tuan 2, 3Hanoi Pedagogical University [email protected]
Nguyen Quoc TuanHanoi Pedagogical University [email protected]
Tran Van TuanHanoi Pedagogical University [email protected]
Pham Thanh TuanHanoi Pedagogical University [email protected]
Nguyen Van Tuyen 3, 5, 39Hanoi Pedagogical University [email protected]
Pham Thi VuiNaresuan University,[email protected]
Xianfu Wang 8, 40University of British Columbia Okanagan
Rabian WangkeereeNaresuan University,[email protected]
Hong-Kun Xu 7, 41Hangzhou Dianzi [email protected]
Xiaoqi Yang 8, 6, 42Hangzhou Dianzi [email protected]
Jane Ye 7, 43University of [email protected]
Nguyen Dong Yen 3, 4, 4Institute of Mathematics, [email protected]
Xi Yin Zheng 4, 44Yunnan [email protected]