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The alternating direction method of multipliers (ADMM) has been proved to be effective for solving two-block convex minimization model subject to linear constraints. However, the convergence of multiple-block convex minimization model with linear constraints may not be guaranteed without additional assumptions. Recently, some parallel multi-block ADMM algorithms which solve the subproblems in a parallel way have been proposed. This paper is a further study on this method with the purpose of improving the parallel multi-block ADMM algorithm by introducing more parameters. We propose two multi-parameter parallel ADMM algorithms with proximal point terms attached to all subproblems. Comparing with some popular parallel ADMM-based algorithms, the parameter conditions of the new algorithms are relaxed. Experiments on both real and synthetic problems are conducted to justify the effectiveness of the proposed algorithms compared to several efficient ADMM-based algorithms for multi-block problems.

In this paper, the convergence to minimizers of a convex function of a modified proximal point algorithm involving a single-valued nonexpansive mapping and a multivalued nonexpansive mapping in CAT(0) spaces is studied and a numerical example is given to support our main results.

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.

"In this paper, we study the existence of solutions of equilibrium problems in the setting of Hadamard manifolds under the pseudomonotonicity and geodesic upper sign continuity of the equilibrium bifunction and under different kinds of coercivity conditions. We also study the existence of solutions of the equilibrium problems under properly quasimonotonicity of the equilibrium bifunction. We propose a two-step proximal point algorithm for solving equilibrium problems in the setting of Hadamard manifolds. The convergence of the proposed algorithm is studied under the strong pseudomonotonicity and Lipschitz-type condition. The results of this paper either extend or generalize several known results in the literature."

In this paper, we propose the modified proximal point algorithm with the process for three nearly Lipschitzian asymptotically nonexpansive mappings and multivalued mappings in CAT(0) space under certain conditions. We prove some convergence theorems for the algorithm which was introduced by Shamshad Hussain et al. [18]. A numerical example is given to illustrate the efficiency of proximal point algorithm for supporting our result.

In this article, we introduce a category of mixed hemivariational inequality problems on Hadamard manifolds. Using Fan-KKM lemma we initiate the existence of solutions of mixed hemivariational inequality problems (MHVIP) when the underlying vector field is monotone. We also establish the proximal point algorithm for solving mixed hemivariational inequality problems on these nonlinear spaces. Our outcomes and schemes may stimulate further exploration in this fascinating and interesting area of analysis.

We apply proof mining methods to analyse a result of Boikanyo and Morosanu on the strong convergence of a Halpern-type proximal point algorithm. As a consequence, we obtain quantitative versions of this result, providing uniform effective rates of asymptotic regularity and metastability.

The common peroneal nerve (CPN) can be injured during fibular-based posterolateral reconstructions due to its close relationship to the neck of the fibula. Therefore, the purpose of this study was to observe the course of the CPN and its branches around the fibular head and neck and quantify the position in relation to relevant bony landmarks and observe the relation between tunnel drilling for posterolateral corner reconstruction and both the tunnel entry and exit at the proximal fibula and the CPN and its branches was observed. In 101 (mean age = 70.6 ± 16 years) embalmed cadaver knees, the relationship between bony landmarks (tibial tuberosity, styloid process of fibula (APR)) and the CPN and its branches were established and 8 (M1–M8) distances from these landmarks measured; mean, SD and 95% CI were recorded. In 21 of these knees, a fibula tunnel was drilled as in PLC reconstruction and the association of the CPN and its branches to the tunnel entry and exit were judged by two independent observers. Fisher’s exact test of independence was used to determine significant differences between genders. Tunnel intersection was analysed in a binary yes/no fashion and was described in frequencies and percentages. The mean distance from the APR to where the CPN reaches the fibula neck (M1) was 31.4 ± 8.9 mm (CI:29.8–33.0); from the apex of the styloid process (APR) to where the CPN passes posterior to the broadest point of the fibular head (M3) was 21.7 ± 12.6 mm (CI:19.4–24.0); from the apex of the APR to the most proximal point of the CPN/CPN first branch in the midline of the fibular head (M2) was 37.0 ± 6.7 mm (CI: 35.4–37.7). Out of the 21 randomly selected knees for drilling, the first branch of the CPN was damaged at the tunnel entry point in 7 (33%), and in 5 knees (24%), the CPN was damaged at the tunnel exit. In one knee, at both the tunnel entry and exit, the first branch of the CPN and the CPN were intersected, respectively. The results of this study strongly suggest that the CPN is at risk when drilling the fibula tunnel performing fibula-based posterolateral corner reconstructions. The total injury rate was 57% with a 33% incidence of injury to the first branch of the nerve at the tunnel entry and 24% to the CPN at the tunnel exit. Due to the high incidence of injury, percutaneous placement of guide pins and tunnel drilling is not recommended. The nerve should be visualized and protected by either a traditional open approach or minimally invasive techniques. With a minimally invasive approach, the nerve should be identified at the fibula neck and then followed ante- and retrograde.

We consider the problem of maximizing the $\ell_1$ norm of a linear map over the sphere, which arises in various machine learning applications such as orthogonal dictionary learning (ODL) and robust subspace recovery (RSR). The problem is numerically challenging due to its nonsmooth objective and nonconvex constraint, and its algorithmic aspects have not been well explored. In this paper, we show how the manifold structure of the sphere can be exploited to design fast algorithms for tackling this problem. Specifically, our contribution is threefold. First, we present a manifold proximal point algorithm (ManPPA) for the problem and show that it converges at a sublinear rate. Furthermore, we show that ManPPA can achieve a quadratic convergence rate when applied to the ODL and RSR problems. Second, we propose a stochastic variant of ManPPA called StManPPA, which is well suited for large-scale computation, and establish its sublinear convergence rate. Both ManPPA and StManPPA have provably faster convergence rates than existing subgradient-type methods. Third, using ManPPA as a building block, we propose a new approach to solving a matrix analog of the problem, in which the sphere is replaced by the Stiefel manifold. The results from our extensive numerical experiments on the ODL and RSR problems demonstrate the efficiency and efficacy of our proposed methods., Comment: Accepted in IEEE Transactions on Signal Processing

In this paper, we complement the framework of bilevel unconstrained minimization (BLUM) [Y. Nesterov, Math. Program., to appear] by a new $p$th-order proximal-point method convergent as $O(k^{-(3p+...

Nervus (n.) ischiadicus is a large nerve arising from the gluteal region and extending toward the posterior locations of the femur. It divides into two terminal branches on the proximal point of fossa poplitea: n. tibialis and n. peroneus communis. N. ischiadicus supplies somatic motor fibers to muscles in the feet, legs and posterior locations of the femur, while also providing sensitive fiber to a large part of leg skin and feet. N. ischiadicus was found to have a different course in the dissection performed on the bilateral gluteal region of the case in this study, the cadaver of a male person aged 55. The course of n. ischiadicus is a single branch in the gluteal region but it showed two branches in the case of this study. This course was found to be a bilateral example of Type b based on the classification of Beaton and Anson. It is important to know the course of this nerve for the surgical interventions to be made on this region considering the interaction between n. ischiadicus and other anatomical structures., Nervus (n.) ischiadicus, gluteal bölgeden çıkan uyluk arka bölgede seyreden kalın bir sinirdir. Genellikle fossa poplitea’nın üst ucunda iki terminal dala ayrılır. Bunlar n. tibialis ve n. peroneus communis’tir. N. ischiadicus ayak, bacak ve uyluk arka bölgede bulunan kaslara somatomotor lifler, bacak derisinin büyük bir bölümüne ve ayağa sensitif lifleri yollar. Bu olguda 55 yaşındaki erkek kadavranın bilateral olarak yapılan gluteal bölge diseksiyonunda n. ischiadicus’un farklı seyrettiği görülmüştür. N. ischiadicus’un seyri gluteal bölgede tek dal olarak beklenirken iki dal halinde olduğu kaydedilmiştir. Beaton ve Anson sınıflandırmasına göre bilateral olarak tip b’ye örnek olduğu görülmüştür. N. ischiadicus’un bu bölgedeki diğer anatomik yapılarla etkileşiminden dolayı buraya yapılan cerrahi girişimler açısından seyrini bilmek önemlidir.

Many engineering and scientific applications, e.g. resource allocation, control of hybrid systems, scheduling, etc., require the solution of mixed-integer non-linear problems (MINLPs). Problems of such class combine the high computational burden arising from considering discrete variables with the complexity of non-linear functions. As a consequence, the development of algorithms able to efficiently solve medium-large MINLPs is still an interesting field of research. In the last decades, several approaches to tackle MINLPs have been developed. Some of such approaches, usually defined as exact methods, aim at finding a globally optimal solution for a given MINLP at expense of a long execution time, while others, generally defined as heuristics, aim at discovering suboptimal feasible solutions in the shortest time possible. Among the various proposed paradigms, outer approximation (OA) and feasibility pump (FP), respectively as exact method and as heuristic, deserve a special mention for the number of relevant publications and successful implementations related to them. In this paper we present a new exact method for convex mixed-integer non-linear programming called proximal outer approximation (POA). POA blends the fundamental ideas behind FP into the general OA scheme that attepts to yield faster and more robust convergence with respect to OA while retaining the good performances in terms of fast generation of feasible solutions of FP.

In this paper, we study the convergence of a proximal point method for solving quasi-equilibrium problems (QEP) in Hilbert spaces. We extent the method proposed by Moudafi [Proximal point algorithm...

In this paper, we combine the S-iteration process introduced by Agarwal et al. (J. Nonlinear Convex Anal., 8(1), 61–79 2007) with the proximal point algorithm introduced by Rockafellar (SIAM J. Control Optim., 14, 877–898 1976) to propose a new modified proximal point algorithm based on the S-type iteration process for approximating a common element of the set of solutions of convex minimization problems and the set of fixed points of nearly asymptotically quasi-nonexpansive mappings in the framework of CAT(0) spaces and prove the △-convergence of the proposed algorithm for solving common minimization problem and common fixed point problem. Our result generalizes, extends and unifies the corresponding results of Dhompongsa and Panyanak (Comput. Math. Appl., 56, 2572–2579 2008), Khan and Abbas (Comput. Math. Appl., 61, 109–116 2011), Abbas et al. (Math. Comput. Modelling, 55, 1418–1427 2012) and many more.

Large sectors of the recent optimization literature focused in the last decade on the development of optimal stochastic first-order schemes for constrained convex models under progressively relaxed...

In this paper, we introduce and study strong convergence of a proximal point algorithm for approximating common solution of a finite family of monotone inclusion problems and fixed point problem of a generalized demimetric mapping in complete CAT(0) spaces. An illustrative example is given to validate theoretical result obtained herein. Our results improve and generalize some well-known results in the literature.

The proximal point algorithm (PPA) has been widely used in convex optimization. Many algorithms fall into the framework of PPA. To guarantee the convergence of PPA, however, existing results conven...

This paper deals with the study of weak sharp solutions for nonsmooth variational inequalities and finite convergence property of the proximal point method. We present several characterizations for weak sharpness of the solutions set of nonsmooth variational inequalities without using the gap functions. We show that under weak sharpness of the solutions set, the sequence generated by proximal point methods terminates after a finite number of iterations. We also give an upper bound for the number of iterations for which the sequence generated by the exact proximal point methods terminates.

It is well-known that the stability of a first-order autonomous system can be determined by testing the symmetric positive definite solutions of associated Lyapunov matrix equations. However, the research on the constrained solutions of the Lyapunov matrix equations is quite few. In this paper, we present three iterative algorithms for symmetric positive semidefinite solutions of the Lyapunov matrix equations. The first and second iterative algorithms are based on the relaxed proximal point algorithm (RPPA) and the Peaceman–Rachford splitting method (PRSM), respectively, and their global convergence can be ensured by corresponding results in the literature. The third iterative algorithm is based on the famous alternating direction method of multipliers (ADMM), and its convergence is subsequently discussed in detail. Finally, numerical simulation results illustrate the effectiveness of the proposed iterative algorithms.

In this paper, we consider a proximal point algorithm for finding zeros of maximal monotone operators in complete CAT(0) spaces. First, a necessary and sufficient condition is presented for the zero set of the operator to be nonempty. Afterwards, we prove that, under suitable conditions, the proposed algorithm converges strongly to the metric projection of some point onto the zero set of the involving maximal monotone operator.

The present paper first provides sufficient conditions and characterizations for linearly conditioned bifunction associated with an equilibrium problem. It then introduces the notion of weak sharp solution for equilibrium problems which is analogous to the linear conditioning notion. This new notion generalizes and unifies the notion of weak sharp minima for optimization problems as well as the notion of weak sharp solutions for variational inequality problems. Some sufficient conditions and characterizations of weak sharpness are also presented. Finally, we study the finite convergence property of sequences generated by some algorithms for solving equilibrium problems under linear conditioning and weak shapness assumptions. An upper bound of the number of iterations by which the sequence generated by proximal point algorithm converges to a solution of equilibrium problems is also given.

The purpose of this paper is to use a modified proximal point algorithm for solving variational inclusion problem in real Hilbert spaces. It is proven that the sequence generated by the proposed iterative algorithm converges strongly to the common solution of the convex minimization and variational inclusion problems.

The tight sublinear convergence rate of the proximal point algorithm for maximal monotone inclusion problems is established based on the squared fixed point residual. By using the performance estim...

The main aim of this paper is to consider the proximal point method for solving multiobjective optimization problem under the differentiability, locally Lipschitz and quasi-convex conditions of the objective function. The control conditions to guarantee that the accumulation points of any generated sequence, are Pareto critical points are provided.

The classical proximal point algorithm (PPA) requires a metric proximal parameter, which is positive definite and symmetric, because it plays the role of the measurement matrix of a norm in the convergence proof. In this paper, our main goal is to show that the metric proximal parameter can be nonsymmetric if the proximal center is shifted appropriately. The resulting nonsymmetric PPA with moving proximal centers maintains the same implementation difficulty and convergence properties as the original PPA, while the nonsymmetry of the metric proximal parameter allows us to design highly customized algorithms that can effectively take advantage of the structures of the model under consideration. We present both the exact and inexact versions of the nonsymmetric PPA with moving proximal centers, and analyze their convergence including the estimate of their worst-case convergence rates measured by the iteration complexity under mild assumptions and their asymptotically linear convergence rates under stronger assumptions.

Purpose: The purpose of this study was to measure the detailed morphology of the femoral anterior cruciate ligament (ACL) footprint. The correlation and the comparison between the measured area and the area which mathematically calculated as elliptical were also evaluated. Materials and Methods: Thirty nine nonpaired human cadaver knees were used. The ACL was cut in the middle, and the femoral bone was cut at the most proximal point of the femoral notch. The ACL was carefully dissected, and the periphery of the ACL insertion site was outlined on both the whole footprint and the midsubstance insertion. Lateral view of the femoral condyle was photographed with a digital camera, and the images were downloaded to a personal computer. The area, length, and width of the femoral ACL footprint were measured with Image J software (National Institution of Health). Using the length and width of the femoral ACL footprint, the elliptical area was calculated as 0.25 π (length × width). Statistical analysis was performed to reveal the correlation and the comparison of the measured and elliptically calculated area. Results: The sizes of the whole and midsubstance femoral ACL footprints were 127.6 ± 41.7 mm[2] and 61 ± 20.2 mm[2], respectively. The sizes of the elliptically calculated whole and midsubstance femoral ACL footprints were 113.9 ± 4.5 mm[2] and 58.4 ± 3 mm[2], respectively. Significant difference was observed between the measured and the elliptically calculated area. In the midsubstance insertion, significant correlation was observed between the measured and the elliptically calculated area (Pearson's correlation coefficient = 0.603, P= 0.001). However, no correlation was observed in the whole ACL insertion area. Conclusion: The morphology of the femoral ACL insertion resembles an elliptical shape. However, due to the wide variation in morphology, the femoral ACL insertion cannot be considered mathematically elliptical.

In this paper, we propose a general scheme for the double-accelerated stochastic method (DASM) to minimize the ill-conditioned regularized empirical risk minimization (ERM) problem, which includes the sum of two convex functions: one is the sum of the smooth component functions, and the other is the block separable structure. We use an inner–outer iteration procedure to accelerate such convex functions, which considers the accelerated randomized proximal coordinate gradient method as the inner procedure and the accelerated proximal point algorithm as the outer procedure. In order to make a connection between the inner and outer procedures, we enhance the dependence on the condition number and redefine the regularization parameter to improve runtime and convergence rates. We also incorporate the non-smooth regularization term into the inner accelerated procedure. Furthermore, we give a theoretical analysis of the iteration complexity and runtime, which is justified in practical applications. We show that the proposed DASM has better or comparable performance than current state-of-the-art accelerated methods for the ill-conditioned ERM problem.

In this paper, we propose a new customized proximal point algorithm for linearly constrained convex optimization problem, and further extend the proposed method to separable convex optimization problem. Unlike the existing customized proximal point algorithms, the proposed algorithms do not involve relaxation step, but still ensure the convergence. We obtain the particular iteration schemes and the unified variational inequality perspective. The global convergence and $${\mathcal {O}}(1/k)$$-convergence rate of the proposed methods are investigated under some mild assumptions. Numerical experiments show that compared to some state-of-the-art methods, the proposed methods are effective.

We prove some important properties of the p-resolvent mapping recently introduced by B. J. Choi and U. C. Ji, The proximal point algorithm in uniformly convex metric spaces, Commun. Korean Math. Soc. 31 2016, 4, 845–855, in p-uniformly convex metric space. Furthermore, we introduce a modified Mann-type PPA involving nonexpansive mapping and prove that the sequence generated by the algorithm converges to a common solution of a finite family of minimization problems, which is also a fixed point of a nonexpansive mapping in the framework of a complete p-uniformly convex metric space.

In this paper, we introduce a new modified proximal point algorithm based on M-iteration to approximate a common element of the set of solutions of convex minimization problems and the set of fixed points of nearly asymptotically quasi-nonexpansive mapping in CAT(0) space. We also prove the $$\Delta $$ -convergence of the proposed algorithm for solving common minimization problem and fixed point problem. We also provide an application and numerical results based on our proposed algorithm.

A class of proximal gradient-type algorithm for bilevel nonlinear nondifferentiable programming problems with smooth substructure is developed in this paper. The original problem is approximately reformulated by explicit slow control technique to a parameterized family function which makes full use of the information of smoothness. At each iteration, we only need to calculate one proximal point analytically or with low computational cost. We prove that the accumulation iterations generated by the algorithms are solutions of the original problem. Moreover, some results of complexity of the algorithms are presented in convergence analysis. Numerical experiments are implemented to verify the efficiency of the proximal gradient algorithms for solving this kind of bilevel programming problems.

When solving a quadratic program (QP), one can improve the numerical stability of any QP solver by performing proximal-point outer iterations, resulting in solving a sequence of better conditioned QPs. In this letter we present a method which, for a given multi-parametric quadratic program (mpQP) and any polyhedral set of parameters, determines which sequences of QPs will have to be solved when using outer proximal-point iterations. By knowing this sequence, bounds on the worst-case complexity of the method can be obtained, which is of importance in, for example, real-time model predictive control (MPC) applications. Moreover, we combine the proposed method with previous work on complexity certification for active-set methods to obtain a more detailed certification of the proximal-point methods complexity, namely the total number of inner iterations. Funding Agencies|Swedish Research Council (VR)Swedish Research Council [2017-04710]

In this paper, we propose a modified proximal point algorithm based on the Thakur iteration process to approximate the common element of the set of solutions of convex minimization problems and the fixed points of two nearly asymptotically quasi-nonexpansive mappings in the framework of $\operatorname{CAT}(0)$ CAT ( 0 ) spaces. We also prove the Δ-convergence of the proposed algorithm. We also provide an application and numerical result based on our proposed algorithm as well as the computational result by comparing our modified iteration with previously known Sahu’s modified iteration.

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.

This study uses the collection of fifteenth- to twentieth-century travel literature from the BSA library to consider issues of (elite) mobility in the Aegean Sea. These texts contain a wealth of information on the routes chosen by travellers between various islands and mainland ports, and on the navigability of the Aegean basin. By using a combination of computational Proximal Point Analysis and Social Network Analysis, these routes are visualised, and discussion focuses on how navigation varied both between centuries and according to the traveller's place of origin. It is suggested that travellers were dependent on other sorts of networks, and that routes travelled were shaped greatly by the economics and politics of the day. It is also proposed that methodologies used in this paper offer great potential for engaging broad-scale sets of archive data.