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In this paper, four extragradient-type algorithms are presented for solving bilevel variational inequalities of pseudomonotone and non-Lipschitz continuous operators in real Hilbert spaces. The proposed iterative schemes employ two Armijo-type linesearch methods making them work adaptively. Strong convergence theorems of the suggested algorithms are established under some mild conditions. Finally, some numerical experiments and applications are given to verify the advantages and efficiency of the proposed algorithms over some previously known ones. [ABSTRACT FROM AUTHOR]

Quasi-Newton method is one of the most popular methods for solving unconstrained single and multiobjective optimization problems. In a quasi-Newton method, the search direction is computed based on a quadratic model of the objective function, where some approximations replace the true Hessian at each iteration. Several Hessian approximation schemes with an adequate line search technique provided higher-order accuracies in approximating the curvature and made the methods more effective in terms of convergence to solution. Considering the aforementioned reasons, we write a survey on quasi-Newton methods for multiobjective optimization problems. We discuss the development of all the variants of the quasi-Newton method for multiobjective optimization problems, along with some of the advantages and disadvantages of the existing methods. We give a brief discussion about the line search for these variants too. Two cases have been considered for BFGS, Huang BFGS, and self-scaling BFGS multiobjective versions of quasi-Newton methods: one is in the presence of the Armijo line search, and the other is in the absence of any line search. Subsequently, a nonmonotone line search version is also explained for multiobjective optimization problems. Commentary is given on the convergence properties of these methods. The rate of convergence of all these methods is highlighted. To prove the convergence of every method, it is reported that every sequence of points generated from the method converges to a critical point of the multiobjective optimization problem under some mild assumptions. Based on the available numerical data, we provide an unbiased opinion on the effectiveness of quasi-Newton methods for multiobjective optimization problems. [ABSTRACT FROM AUTHOR]

In a real Hilbert space, let the GSVI and CFPP represent a general system of variational inclusions and a common fixed point problem of countable nonexpansive mappings and an asymptotically nonexpansive mapping, respectively. In this paper, via a new inertial subgradient extragradient rule we introduce and analyze two iterative algorithms for solving the monotone bilevel equilibrium problem (MBEP) with the GSVI and CFPP as constraints. Some strong convergence theorems for the proposed algorithms are established under some mild assumptions. Our results improve and extend some corresponding results in the earlier and very recent literature. [ABSTRACT FROM AUTHOR]

Linear fractional vector optimization problems are special non-convex vector optimization problems. They were introduced and first studied by E.U. Choo and D.R. Atkins in the period 1982–1984. This paper investigates the properness in the sense of Geoffrion of the efficient solutions of linear fractional vector optimization problems with unbounded constraint sets. Sufficient conditions for an efficient solution to be Geoffrion's properly efficient solution are obtained via Benson's characterization [An improved definition of proper efficiency for vector maximization with respect to cones. J Math Anal Appl. 1979;71:232–241] of Geoffrion's proper efficiency. [ABSTRACT FROM AUTHOR]

Introduction Juvenile Idiopathic Arthritis (JIA) is a group of conditions characterized by chronic inflammatory joint disease of unknown aetiology before the age of 16. This joint damage is complicated by joint deformities, amplitude limitations, muscle imbalances causing a reduction in the functional capacities of the child. The delayed diagnosis, the inaccessibility of biotherapies give rise to deformations which hinder the development of the physical capacities and the social, educational and then professional integration of the child. Objective Sensitize practitioners about early diagnosis and multidisciplinary management of JIA, in order to avoid severe deformities. Observation A 17-year-old girl, out of school, was admitted to a rheumatology unit for the evolution of chronic polyarthralgia with transient skin eruption evolving by inflammatory attack with partial remission since the age of 8 years. There was no notable story. On physical examination, the affected joints were painful. Deformities were observed such as valgus of the knees and forefeet, ulnar deviation of the wrists and nonspecific of the fingers (Figs. 1 and 2). The biological inflammatory syndrome was present with (CRP 90 mg/l, VS 48 mm/h) and a blood count (Hb 12.5 g/dl, Plq 205 000/μL, GB 14. 109/l). Serum ferritin was elevated at 197.7 μg/ml. The other assessments in particular (blood cultures, FAN, FR, blood smears) came back negative. The diagnosis of AJI in systemic form was retained. The X-ray showed osteoarticular destruction with fixed exostoses, Larsen's stage 4 (Figs. 3 and 4) and Steinbrocker's stage 3 with minimal functional capacity. The treatment included an NSAID (ibuprofen at 40 mg/kg/day), followed by corticosteroid therapy (Prednisolone at a dose of 1 mg/kg/day) and as background treatment methotrexate (10 mg/week). evolution at 3 months was favorable on the pain. The management of the deformities was the subject of a multidisciplinary consultation meeting (physiotherapists, rheumatologists, orthopedic surgeons and paediatricians) very expensive for the parents. Thus it benefited a physical treatment made up of joint maintenance physiotherapy and equipment for joints that are still mobile. E59 Figures 1 and 2 Open in new tab Download slide E59 Figures 3 and 4 Open in new tab Download slide Discussion In Africa, osteoarticular pathology in children most often evokes diagnoses of sickle cell disease and acute articular rheumatism. From the first symptoms, children are referred to traditional healers, especially in rural areas where there is no specialist, then to pediatrics [1]. These reasons could explain the long delay in diagnosis as in the patient whose delay was 9 years. The diagnostic age of our patient was 17 years, higher than elsewhere [2] but consistent with the results of Diomandé in Abidjan with an average delay of 10 years [3]. Most studies find a female predominance with an age of onset between 4 and 8 years [4, 5]. In our patient, the involvement was polyarticular from the outset and the FRs were negative, as in most studies [1, 3]. The presence of these deformities was a source of significant handicap in our patient, which led to her dropping out of school, as stipulated in the literature. In first intention, we used NSAIDs, however unsatisfactory. Faced with the persistence of pain, she received prednisolone at 1 mg/kg/d favorable. As basic treatment she received methotrexate, biotherapies being inaccessible in our context because of their high cost. The management of deformities was the subject of a multidisciplinary consultation meeting (physiotherapists, rheumatologists, orthopedic surgeons and paediatricians) whose costs were beyond the reach of indigent parents. Conclusion The diagnosis and the multidisciplinary treatment must be early and continuous of the disease in order to avoid the deformations, which once installed are irreversible. References 1. V. Weber, L. Goffin, A. Kolivras, C. Adler and G. Casimir. An atypical case of systemic juvenile idiopathic arthritis: Rev Med Brux 2017; 38: 511–4 2. Diomandé M, Coulibaly AK, Kouakou ESCL, Yao JC. Profile of Juvenile Idiopathic Arthritis Observed in Abidjan (Côte d'Ivoire): A Report about 17 Cases. British Journal of Medicine & Medical Research. 2016; 16 (11): 1–6 3. Chkirate B, Jabouril F, Aïtouamar H, Rouiti A et al. Juvenile idiopathic arthritis (about 70 cases). Medical hope. 2001; 8: 256–260 4. Juvenile idiopathic arthritis—Focus on Disability/Orphanet Disability Encyclopedia www.orpha.net/data/patho/Han/Int/fr/ArthriteJuvenileIdiopathique%5fFR%5ffr%5fHAN%5fORPHA92.pdf | January 2018 5. F. Beltramo, Claude Gavillot, Irène Lemelle, Danièle Sommelet. Physical Medicine and Rehabilitation in Juvenile Idiopathic Arthritis: Rheumatism Review 70 (2003) 507–510 Conflict of interest None [ABSTRACT FROM AUTHOR]

In this paper, we introduce and study a split minimization problem with multiple output sets. We propose a new iterative method, which employs the inertial Halpern approximation technique, for a common solution of the split minimization problem and the fixed point problem with a finite family of Bregman relatively nonexpansive mappings in the framework of p-uniformly convex and uniformly smooth Banach spaces. Our iterative method uses the step sizes which do not require prior knowledge of the operators norm, and we prove a strong convergence result under some mild conditions. Moreover, we present some applications of our result and further demonstrate the efficiency and applicability of our algorithm with some numerical examples. The results presented in this paper unify and complement several existing results in the literature. [ABSTRACT FROM AUTHOR]