This paper proposes a novel variant of the proximal gradient method for a constrained multiobjective composite optimization problem, in which each objective function is the sum of a smooth function and a non-differentiable one. At each iteration, the descent direction of the differentiable part is calculated by solving a quadratic subproblem, and the non-differentiable part draws on the idea of the multi-objective proximal point method. Under locally Lipschitz continuity assumption, it is proved that the accumulation points of the sequence generated by the algorithm are Pareto critical. Under convexity condition, it is further proved that the sequence generated by the algorithm converges to a weak Pareto efficient point. We also establish the global convergence rates of the proposed approach. More specifically, we present the global convergence rates of O ( 1 k ) for convex case and O ( r k ) with some r ∈ ( 0 , 1 ) for strongly convex case, respectively. In addition, an application to a binary classification problem in supervised machine learning is given to validate the efficiency of the proposed method. Finally, performance experiments suggest that the proposed algorithm can robustly generate Pareto fronts of multiple synthesis test problems compared with existing ones.

Objective: To evaluate the clinical efficacy of liquid-phase concentrated growth factor (LPCGF) alone or in combination with low-level laser therapy (LLLT) under local micro-trauma activation for endogenous healing in the treatment of gingival papilla recession, and to explore the influencing factors. Methods: Patients with aesthetic complaints caused by gingival papilla recession who sought treatment at the Department of Periodontology, Stomatological Hospital of Xi'an Jiaotong University from July 2022 to August 2023 were enrolled in this study. A total of 155 gingival papilla recession sites from 40 patients were included. All sites were numbered after preoperative preparation and randomly divided into two primary groups using a random number table: micro-trauma group (20 patients, 81 sites) and injection-only group (20 patients, 74 sites). Subsequently, each primary group was further randomly subdivided into two subgroups respectively, forming four subgroups in total: Group T1 (micro-trauma combined with LPCGF injection, 10 patients, 37 sites); Group T2 (micro-trauma combined with LPCGF injection and LLLT, 10 patients, 44 sites); Group C1 (LPCGF injection alone, 10 patients, 33 sites); Group C2 (LPCGF injection combined with LLLT, 10 patients, 41 sites). Preoperative cone-beam CT (CBCT) examination and clinical measurements were performed to obtain anatomical parameters, including distance from bone crest to proximal contact point, crown width-length ratio, proximal contact length, interradicular distance and alveolar bone thickness. Meanwhile, gingival biotype, black triangle parameters and Norland & Tarnow classification of gingival papilla recession were recorded. The black triangle height (BTH) and black triangle area (BTA) were measured, and their changes (ΔBTH, ΔBTA) were assessed at 12 months postoperatively. Results: A total of 40 patients were enrolled, including 2 males and 38 females, with age of (30.67±5.20) years, involving 155 sites with gingival papilla recession. At 12 months postoperatively, the reductions in BTH and BTA were (1.21±0.47) mm and 0.75 (0.51, 1.10) mm2 in Group T1, 1.03 (0.61, 1.27) mm and 0.85 (0.55, 1.11) mm2 in Group T2, (0.42±0.65) mm and 0.21 (0.02, 0.94) mm2 in Group C1, and (0.51±0.64) mm and 0.29 (0.02, 0.59) mm2 in Group C2, respectively. Significant improvements in BTH and BTA were observed in all four groups compared with baseline (P<0.001, P<0.001, P<0.01, and P<0.01, respectively). No significant intergroup differences in ΔBTH and ΔBTA were found between Group T1 and Group T2, as well as between Group C1 and Group C2 (both P>0.05). However, visual analogue scale (VAS) assessments of swelling and pain within 7 days postoperatively demonstrated that, compared with the Group T1 and Group C1, the Group T2 and Group C2 presented significantly lower swelling VAS scores from the night of operation to postoperative day 3 (P<0.05), as well as significantly improved pain VAS scores from the night of operation to postoperative day 2 (P<0.05). For thin gingival biotype sites at 3, 6 and 12 months postoperatively, the improvements in ΔBTH and ΔBTA in the micro-trauma groups (Group T1+Group T2) were significantly superior to those in the injection-only groups (Group C1+Group C2) (ΔBTH: Z=3.64, -4.01, 3.46; ΔBTA: Z=-4.18, t=-6.02, t=-7.43; both P<0.001). Canonical correlation analysis revealed a significant correlation between the anatomical variable set and clinical efficacy indicators, with the first canonical correlation coefficient of R=0.486 (P<0.001). According to the standardized canonical weights of variables, alveolar bone thickness presented the strongest positive effect (weight=0.788), while interradicular distance showed a significant negative effect (weight=-0.516). Conclusions: Minimally invasive LPCGF therapy, alone or in combination with LLLT, effectively improves gingival papilla recession. Bone thickness and interproximal root distance are key anatomical determinants of treatment efficacy.

This study aimed to compare the incidence and long-term progression of breast cancer-related lymphedema (BCRL) following autologous or implant-based breast reconstruction, with subgroup analysis based on prepectoral versus subpectoral implant placement. A retrospective review was conducted of 228 patients who underwent mastectomy and immediate breast reconstruction by a single surgeon between 2017 and 2020, using either a deep inferior epigastric perforator (DIEP) flap or a tissue expander. Arm circumference was measured at standardized proximal and distal points. BCRL was diagnosed clinically and confirmed by lymphoscintigraphy demonstrating dermal backflow and/or delayed axillary nodal uptake. Soft-tissue compliance was assessed using ultrasonography to evaluate differences in tissue response with and without compression. The overall incidence of BCRL showed no significant difference among groups (DIEP 8.9%, prepectoral 14.8%, subpectoral 15.6%). However, during 2-year follow-up, the subpectoral group demonstrated faster progression and greater limb volume increase compared with the DIEP and prepectoral groups. Improvements in soft-tissue compliance were greatest in the DIEP group, followed by the prepectoral group, while the subpectoral group showed the least improvement. Autologous reconstruction demonstrated the most favorable lymphatic outcomes, supporting prior literature. Among implant-based approaches, prepectoral placement was associated with a milder course of BCRL than subpectoral placement, possibly due to decreased postoperative pain and shoulder restriction. These findings suggest that reconstruction type and implant placement plane may influence BCRL severity and should be considered when planning surgery to reduce complications and optimize long-term outcomes.

Elastic stable intramedullary nailing (ESIN) is an established treatment for pediatric radius fractures. However, the optimal surgical entry point remains debated due to differing complication profiles. The radial approach carries a risk of superficial branch of the radial nerve (SBRN) irritation, whereas dorsal approaches have been associated with extensor tendon injuries. This study evaluates the neurological complication profile of the radial approach and provides clinically relevant risk factors. A retrospective single-center study was performed including 285 pediatric patients treated with ESIN for radius fractures between 2015 and 2024 using a standardized radial entry technique. Neurological complications were assessed at predefined time points. Associations between patient age, fracture characteristics, entry point location, and neurological outcomes were analyzed. New SBRN lesions occurred in 12 of 285 patients (4.2%) after ESIN implantation and in 3 patients (1.1%) after implant removal. Most deficits were transient, while persistent deficits were observed in 2.1% of cases. A more proximal entry point showed a trend toward increased risk of SBRN injury (p = 0.055), with a potential threshold at approximately 18mm from the distal radial physis. Older patient age was significantly associated with increased risk of neurological complications (p < 0.001). No extensor pollicis longus (EPL) tendon ruptures were observed. Radial entry ESIN is associated with a low rate of predominantly transient neurological complications. Entry point positioning and patient age appear to influence the risk of SBRN injury. Based on our data and the available literature, the radial approach represents a safe and clinically favorable option for ESIN of pediatric radius fractures.

Several decades ago the Proximal Point Algorithm (PPA) started to gain a long-lasting attraction for both abstract operator theory and numerical optimization communities. Even in modern applications, researchers still use proximal minimization theory to design scalable algorithms that overcome nonsmoothness. Remarkable works established tight relations between the convergence behaviour of PPA and the regularity of the objective function. In this manuscript we derive a nonasymptotic iteration complexity of exact and inexact PPA to minimize convex functions under γ-Holderian growth: O ( log ⁡ ( 1 / ϵ ) ) (for γ ∈ [ 1 , 2 ] ) and O ( 1 / ϵ γ − 2 ) (for 2 $ ]]> γ > 2 ). In particular, we recover well-known results on PPA finite convergence for sharp minima and linear convergence for quadratic growth, even under the presence of deterministic noise. Moreover, when a simple Proximal Subgradient Method is recurrently called as an inner routine for computing each IPPA iterate, novel computational complexity bounds are obtained for Restarting Inexact PPA. Our numerical tests show improvements over existing restarting versions of the Subgradient Method.

This paper introduces a novel framework by merging the concepts of non-Archimedean generalized Menger spaces and (ϰ-ϝ)-weak proximal contractions. Extending the best proximity point concept to a triple of sets, we establish new existence theorems for these contractions without requiring the probabilistic P-property, representing a meaningful advancement beyond prior findings, which is a significant generalization of existing results. The study leverages two control functions (ϰ and ϝ) within the contraction condition to derive optimal approximate solutions to fixed-point equations for non-self mappings. Consequently, our core results not only extend but also unify a range of established theorems within classical probabilistic and G-metric spaces. We present a significant application to theoretical computer science by proving that a self-mapping acting on infinite words possesses a unique fixed point.

Best proximity point theorems in metric spaces with property PC

Abstract In this paper, we study best proximity point results and the existence of solutions for a class of nonlinear functional and integral equations in strictly convex Banach spaces and Banach algebras. By employing measures of weak noncompactness, weak sequential continuity, and Lipschitz-type conditions, we extend classical fixed point theorems to the setting of non-weakly compact operators. We establish general conditions under which proximal condensing operators admit best proximity points, even in non-reflexive Banach spaces. These results are further applied to nonlinear functional equations in $$L^1[0,1]$$ , where the operators involved are Nemytskii-type or integral operators that are not necessarily weakly compact. We also consider product-type operators in Banach algebras satisfying property $$(\textbf{P})$$ , demonstrating the applicability of our main theorems to nonlinear integral equations with an explicit example. The results provide a unified framework for analyzing solvability of nonlinear equations, extending existing approaches based on the Schauder–Tychonoff and O’Regan fixed point theorems, and illustrate the role of weak noncompactness measures in obtaining best proximity solutions.

Venous leg ulcers are the most severe manifestation of chronic venous insufficiency, and their treatment is based on compression therapy, whose effectiveness depends on the magnitude of the pressure and the biomechanical properties of the system. Doubts persist about the actual correlation between interface pressure, bandage stiffness and clinical outcomes in real-world practice. To compare the clinical efficacy and biomechanical behavior of different multicomponent compression systems in venous leg ulcers, analyzing the relationship between interface pressure, static stiffness, edema reduction and variation in the wound area. This is a prospective, observational and multicenter study in six districts/health areas of Andalusia, in adults with active venous ulcers attended by Advanced Practice Nurses in Complex Chronic Wounds. Several multi-component compression systems were applied, and interface pressure was monitored using Tight Alright® at three points on the leg for 96 h, recording final pressure, static stiffness, perimeters and ulcer area. All systems achieved a reduction in leg circumference, more marked at the proximal points, evidencing an overall decongestant effect. The correlation between final pressure and edema reduction was weak, and relevant differences were observed in the reduction in ulcer area, with Urgo K2 and CPK Compress 2 standing out with decreases of more than 50% compared to medium or low yields of other systems with comparable pressures. The static stiffness analysis showed specific patterns according to system and leg size, as well as a heterogeneous longitudinal distribution of pressure. The efficacy of compression in venous ulcers depends on both the interface pressure and the design and biomechanical behavior of the system, with clinically relevant differences between multicomponent dressings. Multipoint pressure and stiffness monitoring provides useful information to optimize system selection and support decisions based on biomechanical parameters and standardized clinical outcomes.

We reformulate equilibrium problems in an unconstrained or less constrained equivalent form, in which the new equilibrium bifunction includes the constraints. Then we associate with the latter an unconstrained or less constrained optimization problem, with which we will design a scheme to build general proximal bundle methods where simpler subproblems are to be solved. Our algorithm works as a pure proximal point algorithm as well as a bundle proximal method, and may generate either a feasible or infeasible sequence. Convergence to a solution of the equilibrium problem is established under various conditions and for equilibrium problems involving cyclically antimonotone bifunctions.

Advances in optimization theory have been made systematically by the desire to solve more and more complicated geometric structures that are realised in contemporary applications. This is a rigorous investigation of monotone vector fields and proximal algorithms in the deep geometrical setting of generalized metric spaces (G-metric spaces). Our study fills a general deficiency in the literature by generalizing classical monotonicity principles and proximal point algorithms to support the complex three-point distance structure of G-metric spaces.In this way, by conductinga strict theoretical study, we prove the existence and uniqueness of solutions in the concept of monotone inclusion, are able to develop effective proximal algorithms with guaranteed convergence rates, and illustrate their successful application in different areas of practice. Theoretical contributions that we have made include: (1) the extension of monotonicity theory in all its forms to G-metric spaces with complete characterizations, (2) the construction of strongly convergent proximalpoint algorithms that are explicit in rate of convergence, and (3) its application to variational inequalities and multi-objective optimization problems in non-standard geometries, where the old metric structures are no longer applicable. Our findings create new opportunities to deal with optimization problems in complex networks, social systems, and the present-day machine learning paradigms. Keywords: G-metric spaces, monotone operators, proximal algorithms,

Increased Xanthine Oxidoreductase Activity in Patients With Systemic Sclerosis-Associated Pulmonary Arterial Hypertension

To evaluate agreement between the tooth landmark localization patterns of artificial intelligence (AI) and those from human examiners. Three-dimensional (3D) digital dental model images were obtained from 284 participants. On a total of 5583 permanent teeth, six landmarks per tooth were manually identified and annotated using custom-made 3D annotation software. To develop an AI model capable of automatically identifying tooth landmarks, a deep-learning algorithm was applied to a training dataset consisting of 4519 teeth. To select the optimal AI model, datasets of 556 and 508 teeth were used as validation and test datasets, respectively. For intraexaminer and interexaminer reliability tests, 280 teeth from 10 participants were randomly selected, and two human examiners identified the same six landmarks on two separate occasions. The mean error in tooth landmark localization of the AI model ranged from 0.01 mm to 1.68 mm. The intraclass correlation coefficient between the AI model and human examiner for all landmarks was excellent, ranging from 0.97 to 1.0. The landmark localization error from the AI model was smaller than human interexaminer differences for mesial and distal proximal points. However, errors for the cusp tip and facial axis points were greater in the AI model than the interexaminer differences. AI exhibited localization accuracy for tooth landmarks comparable with that of human examiners for specific measurements related to tooth size. Nonetheless, its accuracy still needs improvement to match that of orthodontic clinicians in identifying cusp tips and facial axis points.

Quasi-Newton L-BFGS based inexact primal–dual proximal point algorithms to solve nonsmooth convex composite programs for sparse signal recovery applications

In this work, we establish the conditions for ensuring the existence and uniqueness of common best proximity points for non-self-mappings defined on the general metric spaces. A unified theoretical framework is formulated to cover a broad class of contraction mappings. We describe the required conditions on the real-valued functions (&amp;alefsym;, &amp;Phi;) : [0,&amp;infin;) &amp;rarr; R and verify that these secure the existence of common best proximity points for (&amp;alefsym;, &amp;Phi;)&amp;minus;interpolative contractions in complete metric spaces. The study further extends this concept by examining multiple forms of interpolative proximal-type contractions, such as proximal, Ćirić &amp;mdash;Reich&amp;mdash;Rus, Kannan, and Hardy-Rogers variants, through the use of the auxiliary functions (&amp;alefsym;, &amp;Phi;). Several illustrated examples are included to demonstrate the applicability of our findings. Finally, we conclude with an application involving a nonlinear fractional differential equation, showing that it fully satisfies the assumption of our main result.

Building upon the works proposed in [1] and [2], we introduce an advanced version of regularized proximal point methods to solve nonlinear complementarity problems (NCP). Our contribution is characterized by two key innovations. Firstly, we introduce an innovative square root quadratic term as part of the regularized subproblem framework, replacing the commonly used logarithmic quadratic term. Secondly, we implement the conjugate gradient algorithm in two stages: the intermediate step and the correction step. This dual approach employs two optimal descent directions with two step lengths to achieve multiplicative progress in each iteration, significantly accelerating convergence. We establish the global convergence of our innovative algorithm, under the condition that F exhibits monotonicity. Initial numerical experiments are presented to confirm the algorithm’s practical effectiveness.

Fiber-optic shape sensing is being actively developed for navigation in minimally invasive procedures, but reconstruction accuracy remains limited by both physical error sources and the reconstruction procedure itself. The four commonly used algorithms can achieve shape reconstruction including the Frenet-Serret frame (FS), the rotation minimizing frame (RMF), the helical extension method (HEM), and the transformation matrix method (TM). However, all four algorithms are based on the centerline by integrating local curvature and orientation along the fiber, in which small errors near the proximal end can accumulate and lead to larger deviations at the distal tip. In order to achieve high accuracy of the distal tip, we have theoretically compared the reconstruction accuracy and analyzed the noise robustness of the four reconstruction algorithms. In noise-free studies, TM algorithm can produce the smallest reconstruction errors compared with the FS, RMF, and HEM algorithms, showing great potential in the reliability of tip localization in clinical procedures. But when the strain noise and angular noise exist, the reconstruction accuracy of TM degrades, especially at high spatial resolution. To enhance the reconstruction accuracy, we have proposed Kabsch-algorithm-based pose correction that the principal axis direction of the shape reconstruction is corrected based on the minimum deviation of the proximal coordinate points. As a result, the relative accuracy enhancement of TM exceeds 35.7% under strain-spatial-resolution coupling noise and exceeds 37.8% and 42.7% under angular noises with 0.24° and 0.45° standard deviation, respectively. We have further investigated the reconstruction accuracy of complex path in minimally invasive procedures, e.g. brachial-artery-to-heart path. By utilization of TM algorithm with pose correction, the mathematical expectation of the mean shape reconstruction error in the presence of noise can be reduced from 1.25 mm to 0.90 mm, indicating the accuracy can be enhanced by 28%. Our obtained results provide a general guidance for selecting noise-robust reconstruction methods and show that Kabsch-algorithm-based pose correction offers a simple way to improve reconstruction accuracy for clinically relevant catheter paths.

In this article, the affiliation details for author Xiaopeng Luo were incorrectly given as "Xiaopeng Luo 1 " but should have been "Xiaopeng Luo 1,2 ". The original article has been corrected.

We implement proximal point type algorithms for finding an efficient point for nonconvex multiobjective optimization problems in which the objective functions are quasiconvex and satisfy a prox-convexity assumption. Our proposed algorithms combine the proximal point method for the minimization problem with the infeasible projection method for variational inequalities and their generate iterative sequences that converges to efficient solution points of the multiobjective optimization problem under mild assumptions. Furthermore, we also propose accelerated versions of the proposed algorithms by adding an inertial term and we established the nonasymptotic O ( 1 k ) convergence rate, too. Numerical illustrations shows the practical usability of the proposed algorithms.

Abstract In this manuscript, we introduce a new class of proximal contractions, termed a modified proximal contraction, to establish coincidence and best proximity point results supported by illustrative examples. These results are then specialized to recover existing best proximity point theorems and further reduced to well-known common fixed point and fixed point results, thereby highlighting the unifying power and flexibility of the proximal contraction framework. In particular, we show how classical results of Banach and Reich arise naturally as special cases, demonstrating that common fixed point theorems can be reduced to classical fixed point theorems under suitable conditions. As an application, we investigate the fourth-order Euler-Bernoulli beam equation using Green’s function techniques. Several graphical representations of the derived Green’s function, including surface plots, heatmaps, and slice profiles, are provided to visualize symmetry, boundary effects, and maximum midspan deflection, thereby connecting the abstract theory to the physical behavior of beams under point loads. Finally, the existence and uniqueness of solutions to the Euler-Bernoulli boundary value problem are established through the developed fixed-point results.