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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.

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.

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

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.

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 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.

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 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.

The linearized alternating direction method of multipliers (ADMM), with indefinite proximal regularization, has been proved to be efficient for solving separable convex optimization subject to linear constraints. In this paper, we present a generalization of linearized ADMM (G-LADMM) to solve two-block separable convex minimization model, which linearizes all the subproblems by choosing a proper positive-definite or indefinite proximal term and updates the Lagrangian multiplier twice in different ways. Furthermore, the proposed G-LADMM can be expressed as a proximal point algorithm (PPA), and all the subproblems are just to estimate the proximity operator of the function in the objective. We specify the domain of the proximal parameter and stepsizes to guarantee that G-LADMM is globally convergent. It turns out that our convergence domain of the proximal parameter and stepsizes is significantly larger than other convergence domains in the literature. The numerical experiments illustrate the improvements of the proposed G-LADMM to solve LASSO and image decomposition problems.

This note is a reaction to the recent paper by Rouhani and Moradi (J Optim Theory Appl 172:222–235, 2017), where a proximal point algorithm proposed by Boikanyo and Morosanu (Optim Lett 7:415–420, 2013) is discussed. Noticing the inappropriate formulation of that algorithm, we propose a more general algorithm for approximating zeros of a maximal monotone operator on a Hilbert space. Besides the main result on the strong convergence of the sequences generated by this new algorithm, we discuss some particular cases, including the approximation of minimizers of convex functionals and present two examples to illustrate the applicability of the algorithm. The note clarifies and extends both the papers quoted above.

We develop model-based methods for solving stochastic convex optimization problems, introducing the approximate-proximal point, or aProx, family, which includes stochastic subgradient, proximal point, and bundle methods. When the modeling approaches we propose are appropriately accurate, the methods enjoy stronger convergence and robustness guarantees than classical approaches, even though the model-based methods typically add little to no computational overhead over stochastic subgradient methods. For example, we show that improved models converge with probability 1 and enjoy optimal asymptotic normality results under weak assumptions; these methods are also adaptive to a natural class of what we term easy optimization problems, achieving linear convergence under appropriate strong growth conditions on the objective. Our substantial experimental investigation shows the advantages of more accurate modeling over standard subgradient methods across many smooth and non-smooth optimization problems., Comment: To appear in SIAM Journal on Optimization

In this paper, we introduce and study the class of demimetric mappings in CAT(0) spaces.We then propose a modified proximal point algorithm for approximating a common solution of a finite family of minimization problems and fixed point problems in CAT(0) spaces. Furthermore,we establish strong convergence of the proposed algorithm to a common solution of a finite family of minimization problems and fixed point problems for a finite family of demimetric mappings in complete CAT(0) spaces. A numerical example which illustrates the applicability of our proposed algorithm is also given. Our results improve and extend some recent results in the literature.

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.

The proximal gradient method is a highly powerful tool for solving the composite convex optimization problem. In this paper, firstly, we propose inexact inertial acceleration methods based on the viscosity approximation and proximal scaled gradient algorithm to accelerate the convergence of the algorithm. Under reasonable parameters, we prove that our algorithms strongly converge to some solution of the problem, which is the unique solution of a variational inequality problem. Secondly, we propose an inexact alternated inertial proximal point algorithm. Under suitable conditions, the weak convergence theorem is proved. Finally, numerical results illustrate the performances of our algorithms and present a comparison with related algorithms. Our results improve and extend the corresponding results reported by many authors recently.

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.

In this paper, we introduce three new iterative methods for finding a common point of the set of fixed points of a symmetric generalized hybrid mapping and the set of solutions of an equilibrium problem in a real Hilbert space. Each method can be considered as an combination of Ishikawa's process with the proximal point algorithm, the extragradient algorithm with or without linesearch. Under certain conditions on parameters, the iteration sequences generated by the proposed methods are proved to be weakly convergent to a solution of the problem. These results extend the previous results given in the literature. A numerical example is also provided to illustrate the proposed algorithms.

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

This paper develops a novel Continuous-time Accelerated Proximal Point Algorithm (CAPPA) for $\ell_1$-minimization problems with provable fixed-time convergence guarantees. The problem of $\ell_1$-minimization appears in several contexts, such as sparse recovery (SR) in Compressed Sensing (CS) theory, and sparse linear and logistic regressions in machine learning to name a few. Most existing algorithms for solving $\ell_1$-minimization problems are discrete-time, inefficient and require exhaustive computer-guided iterations. CAPPA alleviates this problem on two fronts: (a) it encompasses a continuous-time algorithm that can be implemented using analog circuits; (b) it betters LCA and finite-time LCA (recently developed continuous-time dynamical systems for solving SR problems) by exhibiting provable fixed-time convergence to optimal solution. Consequently, CAPPA is better suited for fast and efficient handling of SR problems. Simulation studies are presented that corroborate computational advantages of CAPPA., Comment: 6 pages, 5 figures

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.

We modify a hybrid method and a proximal point algorithm to iteratively find a zero point of the sum of two monotone operators and fixed point of nonspreading multivalued mappings in a Hilbert space by using the technique of forward-backward splitting method. The strong convergence theorem is established and the illustrative numerical example is presented on this work. The results of this paper extend and improve some well-known results in the literature.

We obtain existence and convergence theorems on two variants of the proximal point algorithm for proper lower semicontinuous convex functions in complete geodesic spaces with curvature bounded above., 22 pages; some minor corrections were made

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.

We introduce various versions of cyclic pseudomonotonicity and study the relations between them. Some examples about the relation between them and monotonicity are also presented. By imposing some assumptions on the cyclic pseudomonotone bifunctions, we study the convergence analysis of the proximal point algorithm which has been studied by Iusem and Sosa [ 5 ] for pseudomonotone bifunctions, with better assumptions.

In this paper, we extend the proximal point algorithm proposed by Chang et al.[8] for total asymptotically nonexpansive mapping in CAT(0) spaces. We also demonstrate the ?-convergence and strong convergence to a common element of the set of minimizers of a convex function and the set of fixed points of the Ces?ro type mean of total asymptotically nonexpansive mappings in CAT(0) spaces.

In this paper, we first reformulate the max-min dispersion problem as a saddle-point problem. Specifically, we introduce an auxiliary problem whose optimum value gives an upper bound on that of the original problem. Then we propose the saddle-point problem to be solved by an adaptive custom proximal point algorithm. Numerical results show that the proposed algorithm is efficient.

The Krasnosel’skiĭ–Mann (KM) iteration is a widely used method to solve fixed point problems. This paper investigates the convergence rate for the KM iteration. We first establish a new convergence rate for the KM iteration which improves the known big-$O$ rate to little-$o$ without any other restrictions. The proof relies on the connection between the KM iteration and a useful technique on the convergence rate of summable sequences. Then we apply the result to give new results on convergence rates for the proximal point algorithm and the Douglas–Rachford method.

The main purpose of this paper is to revisit the proximal point algorithms with over-relaxed A -maximal m -relaxed monotone mappings for solving variational inclusions in Hilbert spaces without Lipschitz continuity requirement to overcome the drawbacks of the paper (Verma, 2009) [5] . We affirmatively answer the open question mentioned in the paper (Huang and Noor, 2012) [6] .