Int. J. Med. Sci. 2009, 6 http://www.medsci.org 348IInntteerrnnaattiioonnaall JJoouurrnnaall ooff MMeeddiiccaall SScciieenncceess 2009; 6(6):348-357 © Ivyspring International Publisher. All rights reserved Research Paper Why are some children with early onset of asthma getting better over the years? - Diagnostic failure or salutogenetic factors Eduardo Roel 1, Olle Zetterström 2, Erik Trell 1, Tomas Faresjö 1 1. Department of Medical and Health Sciences/Community Medicine, Faculty of Health Sciences, Linköping University, SE-581 83 Linköping, Sweden. 2. Department of Clinical and Experimental Medicine /Allergy Centre, Faculty of Health Sciences, Linköping University, SE-581 83 Linköping, Sweden. Correspondence to: Tomas Faresjö, Assoc Prof., Department of Medical and Health Sciences / Community Medicine, Faculty of Health Sciences, Linköping University, SE-581 83 Linköping, Sweden. Telephone: +46 13 22 20 00; Fax: +46 13 22 40 20; E-mail: Tomas.Faresjo@liu.se Received: 2009.07.09; Accepted: 2009.11.17; Published: 2009.11.19 Abstract Among children earlier having been identified with a hospital or primary care diagnosis of asthma at least once between 0-7 years of age, almost 40 % of their parents reported in the ISAAC-questionnaire as never having had asthma (NA). These are further analysed and compared with the persisting asthma cases (A) in this study. All these children’s medical records were scrutinized concerning their asthma diagnose retrospectively. The aim of this study was to analyse possible factors related to the outcome in an Asthma diagnosis reassessment by parental questionnaire at the age of ten of the children earlier having been identified with a hospital or primary health care diagnosis of asthma at least once between 0-7 years of age in a total birth-year cohort in a defined Swedish geographical area. A multiple logistic analysis revealed four significant and independent factors associated to the improvement/non-report of asthma at the age of ten. These factors were; not having any past experiences of allergic symptoms (p<0.0001), only having one or two visits at the hos-pital for asthma diagnosis in the 0-7 interval (p=0.001), not living in a flat but a villa at the age of ten (p=0.029) and no previous perception of mist or mould damage in the house (p=0.052). In the early postnatal stage, obstructive and bronchospastic symptoms typical of asthma may be unspecific, and those cases not continuing to persisting disease tend to have identifiable salutogenetic factors of constitutional rather than environmental nature, namely, an overall reduced allergic predisposition. Key words: asthma diagnosis, childhood asthma, diagnose setting, follow-up, salutogenetic fac-tors. Introduction In the last decades, the prevalence of childhood asthma has been increasing in many parts of the world, especially in developed countries (1). Particu-larly in the USA and mainly in urban areas it has al-most reached epidemic levels (2), most marked in low-income urban communities (3). Only recently, this global increase of childhood asthma prevalence has shown signs of levelling out or even in some Western countries reversing (4). Research into the causes of asthma has mostly Int. J. Med. Sci. 2009, 6 http://www.medsci.org 349focused on potential risk factors in the Some Special Cases of Optimizing over the Efficient Set of a Generalized Convex Multiobjective Programming Problem Tran Ngoc Thang Tran Thi Hue School of Applied Mathematics and Informatics Faculty of Management Information System Hanoi University of Science and Technology The Banking Academy of Vietnam Email: thang.tranngoc@hust.edu.vn Email: huett@bav.edu.vn Abstract—Optimizing over the efficient set is a very hard fi , i = 1, , m are linear (resp convex), called a and interesting task in global optimization, in which local linear multiobjective programming problem (resp a con- optima are in general different from global optima At the vex multiobjective programming problem), have received same time, this problem has some important applications in finance, economics, engineering, and other fields In this special attention in the literature (see the survey in [21] article, we investigate some special cases of optimization and references therein) However, to the best of our problems over the efficient set of a generalized convex knowledge, there is little result about numerical methods multiobjective programming problem Preliminary com- in the nonconvex case where fi , i = 1, , m, are putational experiments are reported and show that the nonconvex (see [4], [16], ) proposed algorithms can work well The main problem in this paper is formulated as AMS Subject Classification: 90 C29; 90 C26 Φ(x) s.t x ∈ XE , (PX ) Keywords: Global optimization, Efficient set, Generalized convexity, Multiobjective programming problem where Φ : X → R is a continuous function and XE is the efficient solution set for problem (GM OP ), i.e I I NTRODUCTION XE = {x0 ∈ X | ∃x ∈ X : f (x0 ) ≥ f (x), f (x0 ) = f (x)} The generalized convex multiobjective programming problem (GM OP ) is given as follows Minf (x) = (f1 (x), , fm (x))T s.t x ∈ X , As usual, the notation y ≥ y , where y , y ∈ Rm , is used to indicate yi1 ≥ yi2 for all i = 1, , m It is well-known that, in general, the set XE is non- where X ⊂ Rn is a nonempty convex compact set convex and given implicitly as the form of a standard and fi , i = 1, , m, are generalized convex functions mathematical programming problem, even in the case on X In the case m = 2, problem (GM OP ) is m = 2, the objective functions f1 , f2 are linear and called a generalized convex biobjective programming the feasible set X is polyhedral Hence, problem (PX ) problem The special cases of problem (GM OP ), where is a global optimization problem and belongs to NP- hard problem class This problem has many applications in economics, finance, engineering, and other fields Recently this problem has a great deal of attention In the case h is differentiable, if h is quasiconvex on X , we have h(x1 ) − h(x2 ) ≤ ⇒ ∇h(x2 ), x1 − x2 ≤ (1) from researchers (for instance, see [1], [5], [6], [7], [8], [12], [15], [16] [17], [19] and references therein) Like problem (GM OP ), there is only few numerical for all x1 , x2 ∈ X , where ∇h(x2 ) is the gradient vector of h at x2 (see Theorem 9.1.4 in [10]) algorithms to solve problem (PX ) in the nonconvex A vector function f (x) = (f1 (x), , fm (x))T is case (see [1], [16], ) In this article, simple convex called convex (resp., pseudoconvex, quasiconvex) on X if programming procedures are proposed for solving three its component functions fi (x), i = 1, , m, are convex special cases of problem (PX ) where XE is the efficient solution set for problem (GM OP ) in the nonconvex (resp., pseudoconvex, quasiconvex) functions on X Recall that a vector function f is called scalarly m i=1 case These special-case procedures require quite little pseudoconvex on X if computational effort in comparison to ones required by X for every λ = (λ1 , , λm ) ≥ (see [16]) algorithms for the general problem (PX ) In Section 2, the theoretical preliminaries are presented to analyze three special cases of optimization over the efficient solution set of problem (GM OP ) Section λi fi is pseudoconvex on By definition, if f is convex then it is scalarly pseudoconvex, and f is scalarly pseudoconvex then it is pseudoconvex Hence, the convex multiobjective programming problem is a special case of problem (GM OP ) proposes the algorithms to solve these cases, including Example II.1 Consider the vector function f (x) over some computational experiments to illustrate the algo- the set X = {x ∈ R2 | Ax ≥ b, x ≥ 0}, where rithms Some conclusions are given in the last section II T HEORETICAL P RELIMINARIES f (x) = and First, recall that a differentiable function h : X → R A= is called pseudoconvex on X if ∇h(x2 ), x1 − x2 ≥ ⇒ h(x1 ) − h(x2 ) ≥ for all x1 , x2 ∈ X For example, by Proposition 5.20 in [2], the fractional function r(x)/l(x), where r : Rn → R is convex on X and l : Rn → R is linear such that x22 + x1 −x21 − 0.6x1 + 0.5x2 , x1 − x2 − x2 − x1 + −1 , b = −2 −1 −10 We have λT f (x) = λ1 (x21 + 0.6x1 − 0.5x2 ) + λ2 (x22 + x1 ) x2 − x1 + ...Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2011, Article ID 475121, 9 pages doi:10.1155/2011/475121 Research Article On the Existence Result for System of Generalized Strong Vector Quasiequilibrium Problems Somyot Plubtieng and Kanokwan Sitthithakerngkiet Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand Correspondence should be addressed to Somyot Plubtieng, somyotp@nu.ac.th Received 3 December 2010; Accepted 12 January 2011 Academic Editor: Qamrul Hasan Ansari Copyright q 2011 S. Plubtieng and K. Sitthithakerngkiet. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We introduce a new type of the system of generalized strong vector quasiequilibrium problems with set-valued mappings in real locally convex Hausdorff topological vector spaces. We establish an existence theorem by using Kakutani-Fan-Glicksberg fixed-point theorem and discuss the closedness of strong solution set for the system of generalized strong vector quasiequilibrium problem. The results presented in the paper improve and extend the main results of Long et al. 2008. 1. Introduction The equilibrium problem is a generalization of classical variational inequalities. This problem contains many important problems as special cases, for instance, optimization, Nash equilibrium, complementarity, and fixed-point problems see 1–3 and the references therein. Recently, there has been an increasing interest in the study of vector equilibrium problems. Many results on existence of solutions for vector variational inequalities and vector equilibrium problems have been established see, e.g., 4–16. Let X and Z be real locally convex Hausdorff space, K ⊂ X anonemptysubsetand C ⊂ Z be a closed convex pointed cone. Let F : K × K → 2 Z be a given set-valued mapping. Ansari et al. 17 introduced the following set-valued vector equilibrium problems VEPs to find x ∈ K such that F x, y / ⊆−int C ∀y ∈ K, 1.1 2 Fixed Point Theory and Applications or to find x ∈ K such that F x, y ⊂ C ∀y ∈ K. 1.2 If int C is nonempty, and x satisfies 1.1,thenwecallx aweakefficient solution for VEP,andifx satisfies 1.2,thenwecallx a strong solution for VEP. Moreover, they also proved an existence theorem for a strong vector equilibrium problem 1.2see 17. In 2000, Ansari et al. 5 introduced the system of vector equilibrium problems SVEPs, that is, a family of equilibrium problems for vector-valued bifunctions defined on a product set, with applications in vector optimization problems and Nash equilibrium problem 11 for vector-valued functions. The SVEP contains system of equilibrium problems, systems of vector variational inequalities, system of vector variational-like inequalities, system of optimization problems and the Nash equilibrium problem for vector- valued functions as special cases. But, by using SVEP, we cannot establish the existence of a solution to the Debreu type equilibrium problem 7 Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2010, Article ID 513956, 12 pages doi:10.1155/2010/513956 Research Article Some Krasnonsel’ski ˘ ı-Mann Algorithms and the Multiple-Set Split Feasibility Problem Huimin He, 1 Sanyang Liu, 1 and Muhammad Aslam Noor 2, 3 1 Department of Mathematics, Xidian University, Xi’an 710071, China 2 Mathematics Department, COMSATS Institute of Information Technology, Islamabad, Pakistan 3 Mathematics Department, College of Science, King Saud University, Riyadh 11451, Saudi Arabia Correspondence should be addressed to Huimin He, huiminhe@126.com Received 3 April 2010; Revised 7 July 2010; Accepted 13 July 2010 Academic Editor: S. Reich Copyright q 2010 Huimin He et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Some variable Krasnonsel’ski ˘ ı-Mann iteration algorithms generate some sequences {x n }, {y n }, and {z n }, respectively, via the formula x n1 1 − α n x n α n T N ···T 2 T 1 x n , y n1 1 − β n y n β n N i1 λ i T i y n , z n1 1 − γ n1 z n γ n1 T n1 z n ,whereT n T n modN and the mod function takes values in {1, 2, ,N}, {α n }, {β n },and{γ n } are sequences in 0, 1, and {T 1 ,T 2 , ,T N } are sequences of nonexpansive mappings. We will show, in a fairly general Banach space, that the sequence {x n }, {y n }, {z n } generated by the above formulas converge weakly to the common fixed point of {T 1 ,T 2 , ,T N }, respectively. These results are used to solve the multiple-set split feasibility problem recently introduced by Censor et al. 2005. The purpose of this paper is to introduce convergence theorems of some variable Krasnonsel’ski ˘ ı-Mann iteration algorithms in Banach space and their applications which solve the multiple-set split feasibility problem. 1. Introduction The Krasnonsel’ski ˘ ı-Mann K-M iteration algorithm 1, 2 is used to solve a fixed point equation Tx x, 1.1 where T is a self-mapping of closed convex subset C of a Banach space X. The K-M algorithm generates a sequence {x n } according to the recursive formula x n1 1 − α n x n α n Tx n , 1.2 2 Fixed Point Theory and Applications where {α n } is a sequence in the interval 0, 1 and the initial guess x 0 ∈ C is chosen arbitrarily. It is known 3 that if X is a uniformly convex Banach space with a Frechet differentiable norm in particular, a Hilbert space,ifT : C → C is nonexpansive, that is, T satisfies the property Tx − Ty ≤ x − y ∀x, y ∈ C 1.3 and if T has a fixed point, then the sequence {x n } generated by the K-M algorithm 1.2 converges weakly to a fixed point of T provided that {α n } fulfils the condition ∞ n0 α n 1 − α n ∞. 1.4 See 4, 5 for details on the fixed point theory for nonexpansive mappings. Many problems can be formulated as a fi xed point equation 1.1 with a nonexpansive T and thus K-M algorithm 1.2 applies. For instance, the split feasibility problem SFP introduced in 6–8, which is to find a point x ∈ C such that Ax ∈ Q, 1.5 where C and Q are closed convex subsets of Hilbert spaces H 1 and H 2 , respectively, and A is a linear bounded operator from H 1 to H 2 . This problem plays an important role in the study of signal processing and image reconstruction. Assuming that the SFP 1.5 is consistent i.e., 1.5 has a solution, it is not hard to see that x ∈ C solves 1.5 if and Karush-Kuhn-Tucker necessary conditions for local Pareto minima of constrained multiobjective programming problems DO VAN LUU Department of Mathematics and Informatics Thang Long University Abstract Under the constraint qualification of Abadie type we establish necessary efficiency conditions described by inconsistent inequalities, Karush-Kuhn-Tucker necessary conditions and strong Karush-Kuhn-Tucker necessary conditions for lo- cal Pareto minima of nonsmooth multiobjective programming problems involving inequality, equality and set constraints in Banach spaces in terms of convexificators. Note that all the constraint functions involving in the considering problem are not necessarily continuous, except inactive constraints. 1 Introduction The Karush-Kuhn-Tucker conditions of nonsmooth multiobjective programming problems is a significal topic in optimization. If Lagrange multipliers corresspond- ing to all the components of the objective function are positive, they are called strong Karush-Kuhn-Tucker conditions. The Lagrange multiplier rules in terms of different subdifferentials for nonsmooth optimization problems have been studied by many authors (see, e. g., [4], [6 - 16], and references therein).The notion of convex and compact convexificator was first introduced by Demyanov [2]. This is a generalization of the notions of upper convex and lower concave approximations in [3]. The notions of nonconvex closed convexificator and approximate Jacobian were introduced by Jeyakumar and Luc in [6] and [7], respectively. They have provided good calculus rules for establishing necessary optimality conditions in nonsmooth optimization. The notion of convexificator is a generalization of some notions of known subdifferentials such as the subdifferentials of Clarke [1], Michel-Penot [15], Mordukhovich [16]. In this paper, under the constraint qualifications of Abadie type we establish necessary efficiency conditions described by inconsistent inequalities, Karush-Kuhn- Tucker necessary conditions and strong Karush-Kuhn-Tucker necessary conditions for local Pareto minima of nonsmooth multiobjective programming problems involv- ing inequality, equality and set constraints in Banach spaces in terms of convexifi- cators .\WXF{QJWUuQKNKRDKeF3K?Q, 7UuQJ9LKeF7KQJ/RQJ 7UuQJ9LKeF7KQJ/RQJ 52 2 Notions and definitions Let X be a Banach space, X ∗ topological dual of X and f a extended-real-valued function defined on X. The lower (resp. upper) Dini directional derivatives of f at x ∈ X in a direction v ∈ X are defined, respectively, by f − (x; v) = lim inf t↓0 f(x + tv) − f (x) t , resp. f + (x; v) = lim sup t↓0 f(x + tv) − f (x) t . In case f + (x; v) = f − (x; v), we denote their common value by f (x; v), which is called Dini derivative of f at x in the direction v. The function f is Dini differentiable at x if its Dini derivative at x exists in all directions. Recall [6] that the function f is said to have an upper (resp. lower) convexificator ∂ ∗ f(x) (resp. ∂ ∗ f(x)) at x if ∂ ∗ f(x) (resp. ∂ ∗ f(x)) ⊆ X ∗ is weak ∗ closed, and f − (x; v) sup ξ∈∂ ∗ f(x) ξ, v (∀v ∈ X), resp. f + (x; v) inf ξ∈∂ ∗ f(x) ξ, v (∀v ∈ X) . A weak ∗ closed set ∂ ∗ f(x) ⊆ X ∗ is said to be a convexificator of f at x if it is both upper and lower convexificator of f at x. Note that upper and lower convexificators are not unique. For a locally Lipschitz function, the Clarke subdifferential and the Michel-Penot subdifferential are convexificators of f at x (see [6]). The function f is said to have an upper semi-regular convexificator ∂ ∗ f(x) at x if ∂ ∗ f(x) ⊆ X ∗ is weak ∗ closed, and f + (x; v) sup ξ∈∂ ∗ f(x) ξ, v (∀v ∈ X). (1) Following [6], if equality holds in (1) then ∂ ∗ f(x) is called an upper regular convex- ificator. For a locally Lipschitz function and regular in the sense of Clarke [1], the Clarke subdifferential is an upper regular convexificator (see Kỷ yếu công trình khoa học 2014 – Phần I Trường Đại học Thăng Long 26 OPTIMIZING OVER THE EFFICIENT SET OF A CONVEX MULTIPLE OBJECTIVE PROBLEM: TWO SPECIAL CASES 1 Assoc. Prof. Nguyen Thi Bach Kim 1 and M.Sc.Tran Ngoc Thang 2 , 1,2 School of Applied Mathematics and Informatics Hanoi University of Science and Technology 1 Email: kim.nguyenthibach@hust.edu.vn 2 Email: thang.tranngoc@hust.edu.vn Abstract. Optimizing over the efficient set is a very hard and interesting task in multiple objective optimization. Besides, this problem has some important applications in finance, economics, engineering, and other fields. In this article, we propose convex programming procedures for solving the problem of minimizing a real function over the efficient set of a convex multiple objective programming problem in the two special cases. Preliminary computational experiments show that these procedures can work well. AMS Subject Classification: 2000 Mathematics Subject Classification. Primary: 90 C29; Secondary: 90 C26 Key words: Global optimization, Optimization over the efficient set, Outcome set, Convex programming. 1. Introduction Let X be a nonempty, compact and convex set in n R . Let ( ), =1, , i f x i k , 2k , be convex functions defined on a suitable open set containing X . Then the convex multiple objective programming problem may be written as 1 Min ( ) = ( ( ), , ( )) s.t. . T k f x f x f x x X (CMP) When = 2,k problem ()CMP is called a bicriteria convex programming problem. Such problem are not uncommon and have received special attention in the literature (see, for instance, H.P.Benson and D.Lee [2], J.Fulop and L.D.Muu [3], N.T.B.Kim and T.N.Thang [6], H.X.Phu [10], B.Schandle, K.Klamroth and M.M.Wiecek [11], Y.Yamamoto [14] and references therein). Let h be a real-valued function on n R . The central problem of interest in this paper is the following problem min ( ) s.t. , X h x x E (OP 0 ) where X E is the efficient decision set for problem ()CMP and defined as follows: 0 0 0 ={ | such that ( ) ( ) and ( ) ( )}. X E x X x X f x f x f x f x 1 THIS RESEARCH IS FUNDED BY VIETNAM NATIONAL FOUNDATION FOR SCIENCE AND TECHNOLOGY DEVELOPMENT (NAFOSTED) UNDER GRANT NUMBER "101.01-2013.19" Kỷ yếu công trình khoa học 2014 – Phần I Trường Đại học Thăng Long 27 As usual, we use the notation 12 yy , where 12 , k yyR , to indicate 12 ii yy for all =1, ,ik . It is well-known that, the set X E is generally neither convex set nor given explicitly as the form of a standard mathematical programming problems, even in the case of linear multiple objective programming problem when the component functions 1 ,, k ff of f are linear and X is a polyhedral convex set. Therefore, problem 0 ()OP is one of hard global programming problems. This problem has applications in finance, economics, engineering, and other fields. Recently this problem has been attracted a great deal of attention from researcher (see e.g. [1, 2, 3, 4, 5, 6, 8, 9, 12, 14] and references therein). In this article, simple convex programming procedures are proposed for solving two special cases of problem 0 ()OP . These special-case procedures require quite little computational effort in comparison to ones required by algorithms for the general problem 0 ()OP . 2 Preliminaries Let Q be a nonempty subset in k R . The set of all efficient points of Q is denoted by ep Q and given by 0 0 0 ={ | such that and }. ep Q q Q q Q q q q q It is clear that a point 0 ep qQ if 00 ( ) ={ } k Q q q R , where ={ | 0, =1, , } kk i y y i k RR . Let ={ | = ( ) for some }. k Y y y f x x XR As usual, the set Y is said to be the outcome set for problem ()CMP ; see, for instance, [2, 6, 12, 15]. Since the functions , =1, , i f i k are continuous and n X R is a nonempty, compact set, the outcome set Y is also compact set in k R . ... three special cases of problem (PX ) under consideration Case The feasible set XE is the efficient solution set of problem (GM OP ), the ideal point y I belongs (3) (4) The relationship between the. .. pseudoconvex on By definition, if f is convex then it is scalarly pseudoconvex, and f is scalarly pseudoconvex then it is pseudoconvex Hence, the convex multiobjective programming problem is a special. .. pseudoconvex Let Y = {y ∈ Rm |y = f (x) for some x ∈ X } As usual, the set Y is said to be the outcome set for to the outcome set Y and the objective function Φ(x) of problem (GM OP ) problem