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He (J. Inequal. Appl. 2012:Article ID 162 2012) introduced the proximal point CQ algorithm (PPCQ) for solving the split equilibrium problem (SEP). However, the PPCQ converges weakly to a solution of the SEP and is restricted to monotone bifunctions. In addition, the step-size used in the PPCQ is a fixed constant μ in the interval $(0, \frac{1}{ \| A \|^{2} } )$ ( 0 , 1 ∥ A ∥ 2 ) . This often leads to excessive numerical computation in each iteration, which may affect the applicability of the PPCQ. In order to overcome these intrinsic drawbacks, we propose a robust step-size $\{ \mu _{n} \}_{n=1}^{\infty }$ { μ n } n = 1 ∞ which does not require computation of $\| A \|$ ∥ A ∥ and apply the adaptive step-size rule on $\{ \mu _{n} \}_{n=1}^{\infty }$ { μ n } n = 1 ∞ in such a way that it adjusts itself in accordance with the movement of associated components of the algorithm in each iteration. Then, we introduce a self-adaptive extragradient-CQ algorithm (SECQ) for solving the SEP and prove that our proposed SECQ converges strongly to a solution of the SEP with more general pseudomonotone equilibrium bifunctions. Finally, we present a preliminary numerical test to demonstrate that our SECQ outperforms the PPCQ.

We introduce and investigate a new generalized convexity notion for functions called prox-convexity. The proximity operator of such a function is single-valued and firmly nonexpansive. We provide examples of (strongly) quasiconvex, weakly convex, and DC (difference of convex) functions that are prox-convex, however none of these classes fully contains the one of prox-convex functions or is included into it. We show that the classical proximal point algorithm remains convergent when the convexity of the proper lower semicontinuous function to be minimized is relaxed to prox-convexity.

In this paper, we study modified-type proximal point algorithm for approximating a common solution of a lower semi-continuous mapping and fixed point of total asymptotically nonexpansive mapping in complete CAT(0) spaces. Under suitable conditions, some strong convergence theorems of the proposed algorithms to such a common solution are proved.

In this paper, we propose an accelerated proximal point algorithm for the difference of convex (DC) optimization problem by combining the extrapolation technique with the proximal difference of convex algorithm. By making full use of the special structure of DC decomposition and the information of stepsize, we prove that the proposed algorithm converges at rate of O 1 / k 2 under milder conditions. The given numerical experiments show the superiority of the proposed algorithm to some existing algorithms.

Simple Summary The quality and completeness of colonoscopies might be hampered by suboptimal visualization; often leading to a CT colonography. Colon capsule endoscopy has been suggested as an alternative to CT colonography following incomplete colonoscopy. Colon capsule endoscopy is conducted by the ingestion of a video capsule, which then transmits pictures, allowing complete visualization of the colon and rectum. In our study, we compared the completion rates and diagnostic yields (detection of polyps) of colon capsule endoscopy with those of CT colonography by conducting a systematic review with meta-analysis. We found that the completion rate of CT colonography was slightly higher than that of colon capsule endoscopy, while the diagnostic yield of colon capsule endoscopy was higher, especially when comparing findings of any size polyps. This knowledge might aid clinicians when planning further diagnostics following incomplete colonoscopies. Abstract Following incomplete colonoscopy (IC) patients often undergo computed tomography colonography (CTC), but colon capsule endoscopy (CCE) may be an alternative. We compared the completion rate, sensitivity and diagnostic yield for polyp detection from CCE and CTC following IC. A systematic literature search resulted in twenty-six studies. Extracted data included inter alia, complete/incomplete investigations and polyp findings. Pooled estimates of completion rates of CCE and CTC and complete colonic view rates (CCE reaching the most proximal point of IC) of CCE were calculated. Per patient diagnostic yields of CCE and CTC were calculated stratified by polyp sizes. CCE completion rate and complete colonic view rate were 76% (CI 95% 68–84%) and 90% (CI 95% 83–95%). CTC completion rate was 98% (CI 95% 96–100%). Diagnostic yields of CTC and CCE were 10% (CI 95% 7–15%) and 37% (CI 95% 30–43%) for any size, 13% (CI 95% 9–18%) and 21% (CI 95% 12–32%) for >5-mm and 4% (CI 95% 2–7%) and 9% (CI 95% 3–17%) for >9-mm polyps. No study performed a reference standard follow-up after CCE/CTC in individuals without findings, rendering sensitivity calculations unfeasible. The increased diagnostic yield of CCE could outweigh its slightly lower complete colonic view rate compared to the superior CTC completion rate. Hence, CCE following IC appears feasible for an introduction to clinical practice. Therefore, randomized studies investigating CCE and/or CTC following incomplete colonoscopy with a golden standard reference for the entire population enabling estimates for sensitivity and specificity are needed.

Objective: To validate a computed tomographic (CT) method to measure the femoral trochlear groove depth (FTGD). Study design: Cadaveric study. Sample population: Fifteen dogs, 26 femoral trochleae. Methods: Five points were identified from proximal to distal (proximal point [PP], P25, P50, P75, and distal point [DP]) along the trochlea via three‐dimensional volume‐rendering function on the sagittal plane and measured on multiplanar reconstruction images. Each rater repeated measurements in duplicate, unaware of the identity of the joint. The FTGD was quantitated on the anatomical specimens and statistically compared with CT measurements. Intrarater and interrater agreements were analyzed by using intraclass coefficients. Accuracy was evaluated by using either adjusted R2 coefficients (R2 > 80% was considered acceptable) or Student's t test. The ratio of the patellar and the trochlear width and the ratio of the patellar craniocaudal thickness inside the trochlear groove were calculated at three different patellar locations. Results: Good to excellent intrarater and interrater agreements were observed in four of five trochlear points (P25, P50, P75, and DP), and accuracy was acceptable for these points (R2 > 80%). Computed tomographic measurements differed from the mean anatomical measurements at three of five points (PP, P50, and P75; P

Using proof-theoretical techniques, we analyze a proof by H.-K. Xu regarding a result of strong convergence for the Halpern type proximal point algorithm. We obtain a rate of metastability (in the sense of T. Tao) and also a rate of asymptotic regularity for the iteration. Furthermore, our final quantitative result bypasses the need of the sequential weak compactness argument present in the original proof. This elimination is reflected in the extraction of primitive recursive quantitative information. This work follows from recent results in Proof Mining regarding the removal of sequential weak compactness arguments.

In this work, we introduce a generalized contraction proximal point algorithm and use it to approximate common zeros of maximal monotone operators A and B in a real Hilbert space setting. The algorithm is a two step procedure that alternates the resolvents of these operators and uses general assumptions on the parameters involved. For particular cases, these relaxed parameters improve the convergence rate of the algorithm. A strong convergence result associated with the algorithm is proved under mild conditions on the parameters. Our main result improves and extends several results in the literature.Mathematics Subject Classification (2010): 47J25, 47H05, 47H09.Keywords: Maximal monotone operator, contraction proximal point algorithm, alternating method, nonexpansive map, resolvent operator

Douglas-Rachford Splitting (DRS) methods based on the proximal point algorithms for the Poisson and Gaussian log-likelihood functions are proposed for ptychography and phase retrieval. Fixed point analysis shows that the DRS iterated sequences are always bounded explicitly in terms of the step size and that the fixed points are attracting if and only if the fixed points are regular solutions. This alleviates two major drawbacks of the classical Douglas-Rachford algorithm: slow convergence when the feasibility problem is consistent and divergent behavior when the feasibility problem is inconsistent. Fixed point analysis also leads to a simple, explicit expression for the optimal step size in terms of the spectral gap of an underlying matrix. When applied to the challenging problem of blind ptychography, which seeks to recover both the object and the probe simultaneously, Alternating Minimization with the DRS inner loops, even with a far from optimal step size, converges geometrically under the nearly minimum conditions established in the uniqueness theory., arXiv admin note: text overlap with arXiv:1809.00962

We propose two forward–backward proximal point type algorithms with inertial/memory effects for determining weakly efficient solutions to a vector optimization problem consisting in vector-minimizing with respect to a given closed convex pointed cone the sum of a proper cone-convex vector function with a cone-convex differentiable one, both mapping from a Hilbert space to a Banach one. Inexact versions of the algorithms, more suitable for implementation, are provided as well, while as a byproduct one can also derive a forward–backward method for solving the mentioned problem. Numerical experiments with the proposed methods are carried out in the context of solving a portfolio optimization problem.

The proximal point algorithm is an approximation method for finding a minimizer of a convex function. In this paper, we introduce the resolvent for a convex function in complete geodesic spaces with negative curvature. Using properties of the resolvent, we show the proximal point algorithm in complete geodesic spaces with negative curvature.

Nowadays, analysing data from different classes or over a temporal grid has attracted a great deal of interest. As a result, various multiple graphical models for learning a collection of graphical models simultaneously have been derived by introducing sparsity in graphs and similarity across multiple graphs. This paper focuses on the fused multiple graphical Lasso model which encourages not only shared pattern of sparsity, but also shared values of edges across different graphs. For solving this model, we develop an efficient regularized proximal point algorithm, where the subproblem in each iteration of the algorithm is solved by a superlinearly convergent semismooth Newton method. To implement the semismooth Newton method, we derive an explicit expression for the generalized Jacobian of the proximal mapping of the fused multiple graphical Lasso regularizer. Unlike those widely used first order methods, our approach has heavily exploited the underlying second order information through the semismooth Newton method. This can not only accelerate the convergence of the algorithm, but also improve its robustness. The efficiency and robustness of our proposed algorithm are demonstrated by comparing with some state-of-the-art methods on both synthetic and real data sets. Supplementary materials for this article are available online.

In one-leader one-follower two-stage games, also called Stackelberg games, multiplicity of subgame perfect Nash equilibria (henceforth SPNEs) arises when the best reply correspondence of the follower is not a single-valued map. This paper concerns a new method to approach SPNEs which makes use of a sequence of SPNEs of perturbed games where the best reply correspondence of the follower is single-valued. The sequence is generated by a learning method where the payoff functions of both players are modified subtracting a term that represents a physical and behavioral cost to move and which relies on the proximal point methods linked to the Moreau–Yosida regularization. Existence results of SPNEs approached via this method are provided under mild assumptions on the data, together with numerical examples and connections with other methods to construct SPNEs.

In this paper, we develop a parameterized proximal point algorithm (P-PPA) for solving a class of separable convex programming problems subject to linear and convex constraints. The proposed algorithm is provable to be globally convergent with a worst-case O(1/t) convergence rate, wheret denotes the iteration number. By properly choosing the algorithm parameters, numerical experiments on solving a sparse optimization problem arising from statistical learning show that our P-PPA could perform significantly better than other state-of-the-art methods, such as the alternating direction method of multipliers and the relaxed proximal point algorithm.

In this article, we study the stability properties of a Gauss-type proximal point algorithm for solving the inclusion y ϵ T (x), where T is a set-valued mapping acting on a Banach space X with locally closed graph that is not necessarily monotone and y is a parameter. Consider a sequence of bounded constants {λk} which are away from zero. Under this consideration, we present the semi-local and local convergence of the sequence generated by an iterative method in the sense that it is stable under small variation in perturbation parameter y whenever the set-valued mapping T is metrically regular at a given point. As a result, the uniform convergence of the Gauss-type proximal point method will be established. A numerical experiment is given which illustrates the theoretical result.

OBJECTIVE: To establish the radiographic distances from posterior cruciate ligament (PCL) tibial insertions centers to the lateral and medial tibial cortex in the anteroposterior view, and from these centers to the PCL facet most proximal point on the lateral view, in order to guide anatomical tunnels drilling in PCL reconstruction and for tunnel positioning postoperative analysis.STUDY DESIGN: Controlled laboratory study.METHODS: Twenty cadaver knees were evaluated. The PCL's bundles tibial insertions were identified and marked out using metal tags, and the knees were radiographed. On these radiographs, the bundles insertion sites center location relative to the tibial mediolateral measure, and the distances from the most proximal PCL facet point to the bundle's insertion were determined. All measures were calculated using the ImageJ software.RESULTS: On the anteroposterior radiographs, the mean distance from the anterolateral (AL) bundle insertion center to the medial tibial edge was 40.68 ± 4.10 mm; the mean distance from the posteromedial (PM) bundle insertion center to the medial tibial edge was 38.74 ± 4.40 mm. On the lateral radiographs, the mean distances from the PCL facet most proximal point to AL and PM bundles insertion centers were 5.49 ± 1.29 mm and 10.53 ± 2.17 mm respectively.CONCLUSIONS: It was possible to establish a radiographic pattern for PCL tibial bundles insertions, which may be useful for intraoperative tunnels locations control and for postoperative tunnels positions analysis. OBJETIVO: Estabelecer as distâncias radiográficas dos centros das inserções tibiais do ligamento cruzado posterior (LCP) até as corticais medial e lateral da tíbia, nas radiografias em anteroposterior, e desses centros até o ponto mais proximal da faceta do LCP, nas radiografias em perfil, para que possam orientar a criação de túneis anatômicos na reconstrução do LCP e para análise pós-operatória do posicionamento dos túneis.DESENHO: Estudo laboratorial controlado.MÉTODOS: Vinte joelhos de cadáveres foram avaliados. As inserções tibiais das bandas do LCP foram identificadas e demarcadas com marcadores metálicos e os joelhos foram radiografados. Nas radiografias determinamos a localização do centro de inserção das bandas relativa à medida mediolateral da tíbia e as distâncias do ponto mais proximal da faceta do LCP até o centro de inserção das bandas. Todas as medidas foram calculadas com o software ImageJ.RESULTADOS: Nas radiografias em anteroposterior a distância média entre o centro da inserção da banda anterolateral (AL) até a borda medial da tíbia foi de 40,68 ± 4,10 mm; a distância entre o centro de inserção da banda posteromedial (PM) até a borda medial da tíbia foi, em média, de 38,74 ± 4,40 mm. Nas radiografias em perfil as distâncias médias entre o ponto mais proximal da faceta do LCP e os centros de inserção das bandas AL e PM foram de 5,49 ± 1,29 mm e 10,53 ± 2,17 mm, respectivamente.CONCLUSÕES: Foi possível estabelecer um padrão radiográfico das inserções tibiais das bandas do LCP que pode ser útil para o controle intraoperatório da localização dos túneis e para análise pós-operatória da posição dos túneis.

Regularized iterative reconstruction methods in computed tomography can be effective when reconstructing from mildly inaccurate undersampled measurements. These approaches will fail, however, when more prominent dataerrors, or outliers, are present. These outliers are associated with various inaccuracies of the acquisition process: defective pixels or miscalibrated camerasensors, scattering, missing angles, etc. To account for such large outliers, robust data misfit functions, such as the generalized Huber function, have beenapplied successfully in the past. In conjunction with regularization techniques, these methods can overcome problems with both limited data and outliers. Thispaper proposes a novel reconstruction approach using a robust data fitting term which is based on the Student’s t distribution. This misfit promises to beeven more robust than the Huber misfit as it assigns a smaller penalty to large outliers. We include the total variation regularization term and automaticestimation of a scaling parameter that appears in the Student’s t function. We demonstrate the effectiveness of the technique by using a realistic synthetic phantom and also apply it to a real neutron dataset.

Nonrecurrent laryngeal nerve (non-RLN) is an anatomical variation increasing the risk of vocal cord palsy. Prediction and early identification of non-RLN may minimize such a risk of injury. This study assessed the effect of intraoperative neuromonitoring (IONM) on the detection of non-RLN. A total of 462 (236 right) nerves in 272 patients were identified and totally exposed, and all intraoperative steps of IONM were sequentially applied on the vagus nerve (VN) and RLN. Right predissection VN stimulation at a distal point did not create a sound signal in three cases (3/236; 1.27%). Proximal dissection of the right VN under IONM guidance established a proximal point, creating a positive signal. The separation point of non-RLN from VN was discovered in all three patients. Non-RLNs were exposed from separation to laryngeal entry. Positive IONM signals were obtained after resection of thyroid lobes, and postoperative period was uneventful in patients with non-RLN. Absence of distal VN signal is a precise predictor of the non-RLN. IONM-guided proximal dissection of the right VN leads to identification of the non-RLN. The prediction of non-RLN by the absence of the VN signal at an early stage of surgery may prevent or minimize the risk of nerve injury.

In this paper, we investigate and analyze a proximal point algorithm via viscosity approximation method with error. This algorithm is introduced for finding a common zero point for a countable family of inverse strongly accretive operators and a countable family of nonexpansive mappings in Banach spaces. Our result can be extended to some well known results from a Hilbert space to a uniformly convex and 2−uniformly smooth Banach space. Finally, we establish the strong convergence theorems for the proximal point algorithm. Also, some illustrative numerical examples are presented.

In this paper we propose a linear scalarization proximal point algorithm for solving arbitrary lower semicontinuous quasiconvex multiobjective minimization problems. Under some natural assumptions and using the condition that the proximal parameters are bounded we prove the convergence of the sequence generated by the algorithm and when the objective functions are continuous, we prove the convergence to a generalized critical point. Furthermore, if each iteration minimize the proximal regularized function and the proximal parameters converges to zero we prove the convergence to a weak Pareto solution. In the continuously differentiable case, it is proved the global convergence of the sequence to a Pareto critical point and we introduce an inexact algorithm with the same convergence properties. We also analyze particular cases of the algorithm obtained finite convergence to a Pareto optimal point when the objective functions are convex and a sharp minimum condition is satisfied., The main motivation to study this problem are the consumer demand theory in economy, where the quasiconvexity of the objective vector function is a natural condition associated to diversication of the consumption,. arXiv admin note: text overlap with arXiv:1403.0150

In this paper, we provide a simple convergence analysis of proximal gradient algorithm with Bregman distance, which provides a tighter bound than existing result. In particular, for the problem of minimizing a class of convex objective functions, we show that proximal gradient algorithm with Bregman distance can be viewed as proximal point algorithm that incorporates another Bregman distance. Consequently, the convergence result of the proximal gradient algorithm with Bregman distance follows directly from that of the proximal point algorithm with Bregman distance, and this leads to a simpler convergence analysis with a tighter convergence bound than existing ones. We further propose and analyze the backtracking line search variant of the proximal gradient algorithm with Bregman distance. Simulation results show that the line search method significantly improves the convergence performance of the algorithm.