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Mesh denoising is to recover high quality meshes from noisy inputs scanned from the real world. It is a crucial step in geometry processing, computer vision, computer-aided design, etc. Yet, state-of-the-art denoising methods still fall short of handling meshes containing both sharp features and fine details. Besides, some of the methods usually introduce undesired staircase effects in smoothly curved regions. These issues become more severe when a mesh is corrupted by various kinds of noise, including Gaussian, impulsive, and mixed Gaussian–impulsive noise. In this paper, we present a novel optimization method for robustly denoising the mesh. The proposed method is based on a triple sparsity prior: a double sparse prior on first order and second order variations of the face normal field and a sparse prior on the residual face normal field. Numerically, we develop an efficient algorithm based on variable-splitting and augmented Lagrange method to solve the problem. The proposed method can not only effectively recover various features (including sharp features, fine details, smoothly curved regions, etc), but also be robust against different kinds of noise. We testify effectiveness of the proposed method on synthetic meshes and a broad variety of scanned data produced by the laser scanner, Kinect v1, Kinect v2, and Kinect-fusion. Intensive numerical experiments show that our method outperforms all of the compared select-of-the-art methods qualitatively and quantitatively.

The convergence rate of the augmented Lagrangian method (ALM) for solving the nonlinear semidefinite optimization problem is studied. Under the Jacobian uniqueness conditions, when a multiplier vector (π,Y) and the penalty parameter σ are chosen such that σ is larger than a threshold σ*>0 and the ratio ∥(π,Y)−(π*,Y*)∥/σ is small enough, it is demonstrated that the convergence rate of the augmented Lagrange method is linear with respect to ∥(π,Y)−(π*,Y*)∥ and the ratio constant is proportional to 1/σ, where (π*,Y*) is the multiplier corresponding to a local minimizer. Furthermore, by analyzing the second-order derivative of the perturbation function of the nonlinear semidefinite optimization problem, we characterize the rate constant of local linear convergence of the sequence of Lagrange multiplier vectors produced by the augmented Lagrange method. This characterization shows that the sequence of Lagrange multiplier vectors has a Q-linear convergence rate when the sequence of penalty parameters {σk} has an upper bound and the convergence rate is superlinear when {σk} is increasing to infinity.

There has been much research about regularizing optimal portfolio selections through ℓ1 norm and/or ℓ2-norm squared. The common consensuses are (i) ℓ1 leads to sparse portfolios and there exists a theoretical bound that limits extreme shorting of assets; (ii) ℓ2 (norm-squared) stabilizes the computation by improving the condition number of the problem resulting in strong out-of-sample performance; and (iii) there exist efficient numerical algorithms for those regularized portfolios with closed-form solutions each step. When combined such as in the well-known elastic net regularization, theoretical bounds are difficult to derive so as to limit extreme shorting of assets. In this paper, we propose a minimum variance portfolio with the regularization of ℓ1 and ℓ2 norm combined (namely ℓ1,2-norm). The new regularization enjoys the best of the two regularizations of ℓ1 norm and ℓ2-norm squared. In particular, we derive a theoretical bound that limits short-sells and develop a closed-form formula for the proximal term of the ℓ1,2 norm. A fast proximal augmented Lagrange method is applied to solve the ℓ1,2-norm regularized problem. Extensive numerical experiments confirm that the new model often results in high Sharpe ratio, low turnover and small amount of short sells when compared with several existing models on six datasets.

In this paper we apply an augmented Lagrange method to a class of semilinear elliptic optimal control problems with pointwise state constraints. We show strong convergence of subsequences of the primal variables to a local solution of the original problem as well as weak convergence of the adjoint states and weak-* convergence of the multipliers associated to the state constraint. Moreover, we show existence of stationary points in arbitrary small neighborhoods of local solutions of the original problem. Additionally, various numerical results are presented.

In this paper we will present a review of recent advances in the application of the augmented Lagrange multiplier method as a general approach for generating multiplier-free stabilised methods. The augmented Lagrangian method consists of a standard Lagrange multiplier method augmented by a penalty term, penalising the constraint equations, and is well known as the basis for iterative algorithms for constrained optimisation problems. Its use as a stabilisation methods in computational mechanics has, however, only recently been appreciated. We first show how the method generates Galerkin/Least Squares type schemes for equality constraints and then how it can be extended to develop new stabilised methods for inequality constraints. Application to several different problems in computational mechanics is given.

We present a general framework for the analysis and modelling of frictional contact involving composite materials. The study has focused on composite materials formed by a matrix of rubber and synthetic or metallic fibres, which is the case of standard tires. We detail the numerical treatment of incompressibility at large deformations that rubber can experience, as well as the stiffening effect that properly oriented fibres will induce within the rubber. To solve the frictional contact between solids, a Dual Augmented Lagrangian Multiplier Method is used together with the Mortar method. This ensures a variationally consistent estimation of the contact forces. A modified Serial-Parallel Rule of Mixtures is employed to model the behaviour of composite materials. This is a simple and novel methodology that allows the blending of constitutive behaviours as diverse as rubber (very low stiffness and incompressible behaviour) and steel (high stiffness and compressible behaviour) taking into account the orientation of the fibres within the material. The locking due to the incompressibility constraint in the rubber material has been overcome by using Total Lagrangian mixed displacement-pressure elements. A collection of numerical examples is provided to show the accuracy and consistency of the methodology presented when solving frictional contact, incompressibility and composite materials under finite strains.

In the present work we apply an augmented Lagrange method to solve pointwise state constrained elliptic optimal control problems. We prove strong convergence of the primal variables as well as weak convergence of the adjoint states and weak-* convergence of the multipliers associated to the state constraint. In addition, we show that the sequence of generated penalty parameters is bounded only in exceptional situations, which is different from classical results in finite-dimensional optimization. In addition, numerical results are presented.

A novel data representation method of convolutional neural network (CNN) is proposed in this paper to represent data of different modalities. We learn a CNN model for the data of each modality to map the data of different modalities to a common space and regularize the new representations in the common space by a cross-model relevance matrix. We further impose that the class label of data points can also be predicted from the CNN representations in the common space. The learning problem is modeled as a minimization problem, which is solved by an augmented Lagrange method with updating rules of Alternating direction method of multipliers. The experiments over benchmark of sequence data of multiple modalities show its advantage.

This paper presents a novel chaotic augmented Lagrange method for solving constrained optimization problems. The algorithm employs chaotic maps to reduce the search space and to get the best parameters for handling the problem constraints. Then, the first carrier wave method can be applied to obtain a solution as an initial point of simplex method to find optimal solution. To verify the efficiency of the proposed algorithm, an empirical study is conducted in three groups: mathematical, challenging, and structural optimization problems. The experimental results show that the proposed method can solve different kinds of constrained optimization problems with great precision.

Many practical engineering design problems need constrained optimization. The literature reports several meta-heuristic algorithms have been applied to solve constrained optimization problems. In many cases, the algorithms fail due to violation of constraints. Recently in 2014, a new meta-heuristic algorithm known as symbiotic organism search (SOS) is reported by Cheng and Prayogo. It is inspired by the natural phenomenon of interaction between organisms in an ecosystem which help them to survive and grow. In this paper, the SOS algorithm is combined with augmented Lagrange multiplier (ALM) method to solve the constrained optimization problems. The ALM is accurate and effective as the constraints in this case do not have the power to restrict the search space or search direction. The orthogonal array strategies have gained popularity among the meta-heuristic researchers due to its potentiality to enhance the exploitation process of the algorithms. Simultaneously, researchers are also looking at designing parallel version of the meta-heuristics to reduce the computational burden. In order to enhance the performance, an Orthogonal Parallel SOS (OPSOS) is developed. The OPSOS along with ALM method is a suitable combination which is used here to solve twelve benchmark nonlinear constrained problems and four engineering design problems. Simulation study reveals that the proposed approach has almost similar accuracy with lower run time than ALM with Orthogonal SOS. Comparative analysis also establish superior performance over ALM with orthogonal colliding bodies optimization, modified artificial bee colony, augmented Lagrangian-based particle swarm optimization and Penalty function-based genetic algorithm.