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We study global minimax exactness of merit functions for constrained optimization problems. This concept arises as a natural generalization of the definition of global saddle points in the unified theory of exactness of penalty and augmented Lagrangian functions. We obtain necessary and sufficient conditions for the global minimax exactness of nonlinear augmented Lagrangians in the form of the localization principle, which allow one to reduce the study of the existence of global saddle points (or the existence of solutions of the augmented dual problem) to a local analysis of sufficient optimality conditions. With the use of the localization principle, we obtain simple necessary and sufficient conditions for the existence of global saddle points of He-Wu-Meng's augmented Lagrangian for inequality constrained problems and nonlinear rescaling Lagrangians for nonconvex semidefinite programs. We also introduce and analyze a new nonlinear smooth augmented Lagrangian for constrained minimax problems and provide necessary and sufficient conditions for the existence of a global saddle point of this augmented Lagrangian, which, in particular, expose some limitations of exponential penalty function methods. [ABSTRACT FROM AUTHOR]

We present an algorithm for training Support Vector Machines (SVM) for classification based on fast projected gradient, augmented Lagrangian methods and a working set selection principle. The algorithm is capable of training SVM with tens of thousands data points within seconds on a modern personal computer system. The algorithm can be parallelized and therefore it is suitable for multi processor environment. We describe the algorithm, provide numerical results for solving medium size problems and discuss future directions of speeding up the SVM training. [ABSTRACT FROM AUTHOR]

Purpose: Quantitative Susceptibility Mapping (QSM) is usually performed by minimizing a functional with data fidelity and regularization terms. A weighting parameter controls the balance between these terms. There is a need for techniques to find the proper balance that avoids artifact propagation and loss of details. Finding the point of maximum curvature in the L‐curve is a popular choice, although it is slow, often unreliable when using variational penalties, and has a tendency to yield overregularized results. Methods: We propose 2 alternative approaches to control the balance between the data fidelity and regularization terms: 1) searching for an inflection point in the log‐log domain of the L‐curve, and 2) comparing frequency components of QSM reconstructions. We compare these methods against the conventional L‐curve and U‐curve approaches. Results: Our methods achieve predicted parameters that are better correlated with RMS error, high‐frequency error norm, and structural similarity metric‐based parameter optimizations than those obtained with traditional methods. The inflection point yields less overregularization and lower errors than traditional alternatives. The frequency analysis yields more visually appealing results, although with larger RMS error. Conclusion: Our methods provide a robust parameter optimization framework for variational penalties in QSM reconstruction. The L‐curve–based zero‐curvature search produced almost optimal results for typical QSM acquisition settings. The frequency analysis method may use a 1.5 to 2.0 correction factor to apply it as a stand‐alone method for a wider range of signal‐to‐noise‐ratio settings. This approach may also benefit from fast search algorithms such as the binary search to speed up the process. [ABSTRACT FROM AUTHOR]

Digital volume correlation (DVC), the volumetric extension of the popular digital image correlation (DIC) technique, is a powerful experimental tool for measuring 3D volumetric full-field displacements and strains. Most current DVC algorithms can be categorized into either local or finite-element-based global methods. As with most experimental approaches, there are drawbacks with each of these methods. In the local method the subvolume deformations are estimated independently and the computed displacement field may not necessarily be kinematically compatible. Thus, the deformation gradients can be noisy, especially when using small volumetric subsets. Although the global method often enforces kinematic compatibility, it generally incurs substantially greater computational costs than its local counterpart, which is especially significant for large volumetric data sets. To address these shortcomings, we present a new hybrid DVC algorithm, called augmented Lagrangian digital volume correlation (ALDVC), which combines the advantages of both the local (fast computation time) and global (compatible displacement field) methods. This new algorithm builds on our recent work on the augmented Lagrangian digital image correlation (2D-ALDIC) technique and solves the general motion optimization problem by using the alternating direction method of multipliers (ADMM). We demonstrate that our ALDVC algorithm has high accuracy and precision while maintaining low computational cost, and is a significant improvement compared to current local and global DVC methods. ALDVC is a computationally efficient algorithm to measure 3D volumetric displacements and strains. An open-source Matlab implementation is freely available. [ABSTRACT FROM AUTHOR]

The augmented Lagrangian method (ALM) is a classical optimization tool that solves a given “difficult” (constrained) problem via finding solutions of a sequence of “easier” (often unconstrained) subproblems with respect to the original (primal) variable, wherein constraints satisfaction is controlled via the so-called dual variables. ALM is highly flexible with respect to how primal subproblems can be solved, giving rise to a plethora of different primal–dual methods. The powerful ALM mechanism has recently proved to be very successful in various large-scale and distributed applications. In addition, several significant advances have appeared, primarily on precise complexity results with respect to computational and communication costs in the presence of inexact updates and design and analysis of novel optimal methods for distributed consensus optimization. We provide a tutorial-style introduction to ALM and its variants for solving convex optimization problems in large-scale and distributed settings. We describe control-theoretic tools for the algorithms’ analysis and design, survey recent results, and provide novel insights into the context of two emerging applications: federated learning and distributed energy trading. [ABSTRACT FROM AUTHOR]

In this paper we propose a splitting scheme which hybridizes the generalized conditional gradient with a proximal step and which we call the CGALP algorithm for minimizing the sum of three proper convex and lower-semicontinuous functions in real Hilbert spaces. The minimization is subject to an affine constraint, that, in particular, allows one to deal with composite problems (a sum of more than three functions) in a separable way by the usual product space technique. While classical conditional gradient methods require Lipschitz continuity of the gradient of the differentiable part of the objective, CGALP needs only differentiability (on an appropriate subset) and hence circumvents the intricate question of Lipschitz continuity of gradients. For the two remaining functions in the objective, we do not require any additional regularity assumption. The second function, possibly nonsmooth, is assumed simple; i.e., the associated proximal mapping is easily computable. For the third function, again nonsmooth, we just assume that its domain is weakly compact and that a linearly perturbed minimization oracle is accessible. In particular, this last function can be chosen to be the indicator of a nonempty bounded closed convex set in order to deal with additional constraints. Finally, the affine constraint is addressed by the augmented Lagrangian approach. Our analysis is carried out for a wide choice of algorithm parameters satisfying so-called open loop rules. As main results, under mild conditions, we show asymptotic feasibility with respect to the affine constraint, weak convergence of the dual multipliers, and convergence of the Lagrangian values to the saddle-point optimal value. We also provide pointwise and ergodic rates of convergence for both the feasibility gap and the Lagrangian values. [ABSTRACT FROM AUTHOR]

This paper presents a consistent topology optimization formulation for mass minimization with local stress constraints by means of the augmented Lagrangian method. To solve problems with a large number of constraints in an effective way, we modify both the penalty and objective function terms of the augmented Lagrangian function. The modification of the penalty term leads to consistent solutions under mesh refinement and that of the objective function term drives the mass minimization towards black and white solutions. In addition, we introduce a piecewise vanishing constraint, which leads to results that outperform those obtained using relaxed stress constraints. Although maintaining the local nature of stress requires a large number of stress constraints, the formulation presented here requires only one adjoint vector, which results in an efficient sensitivity evaluation. Several 2D and 3D topology optimization problems, each with a large number of local stress constraints, are provided. [ABSTRACT FROM AUTHOR]

We introduce a filter mechanism to enforce convergence for augmented Lagrangian methods for nonlinear programming. In contrast to traditional augmented Lagrangian methods, our approach does not require the use of forcing sequences that drive the first-order error to zero. Instead, we employ a filter to drive the optimality measures to zero. Our algorithm is flexible in the sense that it allows for equality-constrained quadratic programming steps to accelerate local convergence. We also include a feasibility restoration phase that allows fast detection of infeasible problems. We provide a convergence proof that shows that our algorithm converges to first-order stationary points. We provide preliminary numerical results that demonstrate the effectiveness of our proposed method. [ABSTRACT FROM AUTHOR]

In the context of numerical constrained optimization, we investigate stochastic algorithms, in particular evolution strategies, handling constraints via augmented Lagrangian approaches. In those approaches, the original constrained problem is turned into an unconstrained one and the function optimized is an augmented Lagrangian whose parameters are adapted during the optimization. The use of an augmented Lagrangian however breaks a central invariance property of evolution strategies, namely invariance to strictly increasing transformations of the objective function. We formalize nevertheless that an evolution strategy with augmented Lagrangian constraint handling should preserve invariance to strictly increasing affine transformations of the objective function and the scaling of the constraints—a subclass of strictly increasing transformations. We show that this invariance property is important for the linear convergence of these algorithms and show how both properties are connected. [ABSTRACT FROM AUTHOR]

We propose a new self-adaptive and double-loop smoothing algorithm to solve composite, nonsmooth, and constrained convex optimization problems. Our algorithm is based on Nesterov's smoothing technique via general Bregman distance functions. It self-adaptively selects the number of iterations in the inner loop to achieve a desired complexity bound without requiring to set the accuracy a priori as in variants of augmented Lagrangian methods (ALM). We prove O 1 k -convergence rate on the last iterate of the outer sequence for both unconstrained and constrained settings in contrast to ergodic rates which are common in ALM as well as alternating direction method-of-multipliers literature. Compared to existing inexact ALM or quadratic penalty methods, our analysis does not rely on the worst-case bounds of the subproblem solved by the inner loop. Therefore, our algorithm can be viewed as a restarting technique applied to the ASGARD method in Tran-Dinh et al. (SIAM J Optim 28(1):96–134, 2018) but with rigorous theoretical guarantees or as an inexact ALM with explicit inner loop termination rules and adaptive parameters. Our algorithm only requires to initialize the parameters once, and automatically updates them during the iteration process without tuning. We illustrate the superiority of our methods via several examples as compared to the state-of-the-art. [ABSTRACT FROM AUTHOR]

A Multiple-Shooting Differential Dynamic Programming Algorithm is applied to a variety of constrained nonlinear optimal control problems, including classic benchmark problems, as well as a robotic arm problem and sensitive spacecraft trajectory optimization problems. The multiple-shooting extension presented in Part 1 is validated, as well as the Powell, Hestenes, and Rockafellar approach used in the treatment of equality and inequality path and terminal constraints. The results for example applications demonstrate the applicability of the algorithm to a variety of trajectory optimization problems. The advantages of the multiple-shooting approach over the single-shooting algorithm is evident in problems with high sensitivity. In particular, convergence of the multiple-shooting algorithm is demonstrated for complex spacecraft trajectory problems that are intractable using the single-shooting formulation. • The multiple-shooting differential dynamic programming algorithm is validated. • The multiple-shooting approach is effective for general optimal control problems. • The sensitivities of each subproblem are reduced with multiple-shooting. • The new algorithm can be used to solve complex and sensitive problems robustly. • The Augmented Lagrangian approach to inequality constraints is effective. [ABSTRACT FROM AUTHOR]

At each iteration of the augmented Lagrangian algorithm, a nonlinear subproblem is being solved. The number of inner iterations (of some/any method) needed to obtain a solution of the subproblem, or even a suitable approximate stationary point, is in principle unknown. In this paper we show that to compute an approximate stationary point sufficient to guarantee local superlinear convergence of the augmented Lagrangian iterations, it is enough to solve two quadratic programming problems (or two linear systems in the equality-constrained case). In other words, two inner Newtonian iterations are sufficient. To the best of our knowledge, such results are not available even under the strongest assumptions (of second-order sufficiency, strict complementarity, and the linear independence constraint qualification). Our analysis is performed under second-order sufficiency only, which is the weakest assumption for obtaining local convergence and rate of convergence of outer iterations of the augmented Lagrangian algorithm. The structure of the quadratic problems in question is related to the stabilized sequential quadratic programming and to second-order corrections. [ABSTRACT FROM AUTHOR]

Interior point methods have attracted most of the attention in the recent decades for solving large scale convex quadratic programming problems. In this paper we take a different route as we present an augmented Lagrangian method for convex quadratic programming based on recent developments for nonlinear programming. In our approach, box constraints are penalized while equality constraints are kept within the subproblems. The motivation for this approach is that Newton's method can be efficient for minimizing a piecewise quadratic function. Moreover, since augmented Lagrangian methods do not rely on proximity to the central path, some of the inherent difficulties in interior point methods can be avoided. In addition, a good starting point can be easily exploited, which can be relevant for solving subproblems arising from sequential quadratic programming, in sensitivity analysis and in branch and bound techniques. We prove well-definedness and finite convergence of the method proposed. Numerical experiments on separable strictly convex quadratic problems formulated from the Netlib collection show that our method can be competitive with interior point methods, in particular when a good initial point is available and a second-order Lagrange multiplier update is used. [ABSTRACT FROM AUTHOR]

We consider the problem of minimizing a smooth nonconvex function over a structured convex feasible set, that is, defined by two sets of constraints that are easy to treat when considered separately. In order to exploit the structure of the problem, we define an equivalent formulation by duplicating the variables and we consider the augmented Lagrangian of this latter formulation. Following the idea of the Alternating Direction Method of Multipliers (ADMM), we propose an algorithm where a two-blocks decomposition method is embedded within an augmented Lagrangian framework. The peculiarities of the proposed algorithm are the following: (1) the computation of the exact solution of a possibly nonconvex subproblem is not required; (2) the penalty parameter is iteratively updated once an approximated stationary point of the augmented Lagrangian is determined. Global convergence results are stated under mild assumptions and without requiring convexity of the objective function. Although the primary aim of the paper is theoretical, we perform numerical experiments on a nonconvex problem arising in machine learning, and the obtained results show the practical advantages of the proposed approach with respect to classical ADMM. [ABSTRACT FROM AUTHOR]

An interesting, yet challenging problem in topology optimization consists of finding the lightest structure that is able to withstand a given set of applied loads without experiencing local material failure. Most studies consider material failure via the von Mises criterion, which is designed for ductile materials. To extend the range of applications to structures made of a variety of different materials, we introduce a unified yield function that is able to represent several classical failure criteria including von Mises, Drucker-Prager, Tresca, Mohr-Coulomb, Bresler-Pister and Willam- Warnke, and use it to solve topology optimization problems with local stress constraints. The unified yield function not only represents the classical criteria, but also provides a smooth representation of the Tresca and the Mohr-Coulomb criteria-an attribute that is desired when using gradient-based optimization algorithms. The present framework has been built so that it can be extended to failure criteria other than the ones addressed in this investigation. We present numerical examples to illustrate how the unified yield function can be used to obtain different designs, under prescribed loading or designdependent loading (e.g. self-weight), depending on the chosen failure criterion. [ABSTRACT FROM AUTHOR]

Multiple-shooting benefits a wide variety of optimal control algorithms by alleviating large sensitivities present in highly nonlinear problems, improving robustness to initial guesses, and increasing the potential for parallel implementation. In this paper series, motivated by the challenges of optimizing highly sensitive spacecraft trajectories, the multiple-shooting approach is embedded for the first time in the formulation of a Differential Dynamic Programming algorithm. Contrary to traditional nonlinear programming methods which necessitate minimal modification of the formulation in order to incorporate multiple-shooting principles, DDP requires novel non-trivial derivations, in order to include the initial conditions of the multiple-shooting subintervals and track their sensitivities. The initial conditions are updated in a necessary additional algorithmic step. A null-space trust-region method is used for the solution of the quadratic subproblems, and equality and inequality path and terminal constraints are handled through a general augmented Lagrangian approach. The propagation of the trajectory and the optimization of its controls are decoupled through the use of State-Transition Matrices. Part 1 of the paper series provides the necessary theoretical developments and implementation details for the algorithm. Part 2 contains validation and application cases. • The multiple-shooting framework is applied to Differential Dynamic Programming. • The necessary quadratic expansions feedback control laws are developed. • An Augmented Lagrangian approach is applied for equality and inequality constraints. [ABSTRACT FROM AUTHOR]

Optimal power flow (OPF) is a fundamental problem in power system operating and planning. The need for distributed solutions is rapidly growing in distribution systems with radial networks, discrete equipments and many distributed generation units. Typically the model is difficult to solve due to the highly non-convexity and even non-continuity of the arising OPF problem. In this paper, we address these challenges by developing a penalty dual decomposition (PDD) based algorithm to obtain distributed solutions. The proposed PDD based algorithm mainly consists of two loops: in the outer loop we update the dual variables and the penalty parameter according to the constraint violation, while in the inner loop we divide the primal variables into several blocks and employ the block successive upper-bound minimization (BSUM) method to iteratively optimize each block variables with fixed penalty parameter and dual variables. Each subproblem in the proposed PDD based algorithm can be solved either in closed-form or by the bisection method and every limit point generated by the proposed algorithm is guaranteed to be a stationary point of the OPF problem. Simulation results, and comparison with centralized PDD and concave-convex procedure (CCCP) based algorithms are presented to validate the effectiveness of the proposed algorithm. [ABSTRACT FROM AUTHOR]

We propose an infeasible interior-point algorithm for constrained linear least-squares problems based on the primal-dual regularization of convex programs of Friedlander and Orban. Regularization allows us to dispense with the assumption that the active gradients are linearly independent. At each iteration, a linear system with a symmetric quasi-definite (SQD) matrix is solved. This matrix is shown to be uniformly bounded and nonsingular. While the linear system may be solved using sparse LDL T factorization, we observe that other approaches may be used. In particular, we build on the connection between SQD linear systems and unconstrained linear least-squares problems to solve the linear system with sparse QR factorization. We establish conditions under which a solution of the original, constrained least-squares problem is recovered. We report computational experience with the sparse QR factorization and illustrate the potential advantages of our approach. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new class of imaging denoising models by using the L p -norm of mean curvature of image graphs as regularizers with p ∈ (1 , 2 ] . The motivation of introducing such models is to add stronger regularizations than that of the original mean curvature based image denoising model (Zhu and Chan in SIAM J Imaging Sci 5(1):1–32, 2012) in order to remove noise more efficiently. To minimize these variational models, we develop a novel augmented Lagrangian method, and one thus just needs to solve two linear elliptic equations to find saddle points of the associated augmented Lagrangian functionals. Specifically, we linearize the nonlinear term in one of the two subproblems and minimize a proximal-like functional that can be easily treated. We prove that the minimizer of the substitute functional does reduce the value of the original subproblem under certain conditions. Numerical results are presented to illustrate the features of the proposed models and also the efficiency of the designed algorithm. [ABSTRACT FROM AUTHOR]

In this work, we present an Augmented Lagrangian algorithm for nonlinear semidefinite problems (NLSDPs), which is a natural extension of its consolidated counterpart in nonlinear programming. This method works with two levels of constraints; one that is penalized and other that is kept within the subproblems. This is done to allow exploiting the subproblem structure while solving it. The global convergence theory is based on recent results regarding approximate Karush–Kuhn–Tucker optimality conditions for NLSDPs, which are stronger than the usually employed Fritz John optimality conditions. Additionally, we approach the problem of covering a given object with a fixed number of balls with a minimum radius, where we employ some convex algebraic geometry tools, such as Stengle's Positivstellensatz and its variations, which allows for a much more general model. Preliminary numerical experiments are presented. [ABSTRACT FROM AUTHOR]

This paper presents a supervised data-driven algorithm for impulse noise removal via iterative scheme-inspired network (IIN). IIN is defined over a data flow graph, which is derived from the iterative procedures in Alternating Direction Method of Multipliers (ADMM) algorithm for optimizing the L1-guided variational model. In the training phase, the L1-minimization is reformulated into an augmented Lagrangian scheme through adding a new auxiliary variable. In the testing phase, it has computational overhead similar to ADMM but uses optimized parameters learned from the training data for restoration task. Experimental results demonstrate that the newly proposed method can obtain very significantly superior performance than current state-of-the-art variational and dictionary learning-based approaches for salt-and-pepper noise removal. [ABSTRACT FROM AUTHOR]

Sdpnal+ is a MATLAB software package that implements an augmented Lagrangian based method to solve large scale semidefinite programming problems with bound constraints. The implementation was initially based on a majorized semismooth Newton-CG augmented Lagrangian method, here we designed it within an inexact symmetric Gauss-Seidel based semi-proximal ADMM/ALM (alternating direction method of multipliers/augmented Lagrangian method) framework for the purpose of deriving simpler stopping conditions and closing the gap between the practical implementation of the algorithm and the theoretical algorithm. The basic code is written in MATLAB, but some subroutines in C language are incorporated via Mex files. We also design a convenient interface for users to input their SDP models into the solver. Numerous problems arising from combinatorial optimization and binary integer quadratic programming problems have been tested to evaluate the performance of the solver. Extensive numerical experiments conducted in [L.Q. Yang, D.F. Sun, and K.C. Toh, SDPNAL+: A majorized semismooth Newton-CG augmented Lagrangian method for semidefinite programming with nonnegative constraints, Math. Program. Comput. 7 (2015), pp. 331–366] show that the proposed method is quite efficient and robust, in that it is able to solve 98.9% of the 745 test instances of SDP problems arising from various applications to the accuracy of 10 − 6 in the relative KKT residual. [ABSTRACT FROM AUTHOR]

• A pure dual version of the alternating direction method of multiplier is derived. • The linear convergence of the method is proved in an algebraic level. • The convergence depends on the condition number of an auxiliary matrix. • A procedure is proposed to approximate the optimal penalty parameter. We propose an alternating direction method of multiplier (ADMM) for the unilateral (frictionless) contact problem with an optimal parameter selection. We first introduce an auxiliary unknown to seprate the linear elasticity subproblem from the unilateral contact condition. Then an alternating direction is applied to the corresponding augmented Lagrangian. By eliminating the primal and auxiliary unknowns, at the discrete level, we derive a pure dual algorithm, starting point for the convergence analysis and the optimal parameter approximation. Numerical experiments are proposed to illustrate the efficiency of the proposed (optimal) penalty parameter selection method. [ABSTRACT FROM AUTHOR]

Numerically robust algorithmic formulations suitable for rate-independent crystal plasticity are presented. They cover classic local models as well as gradient-enhanced theories in which the gradients of the plastic slips are incorporated by means of the micromorphic approach. The elaborated algorithmic formulations rely on the underlying variational structure of (associative) crystal plasticity. To be more precise and in line with so-called variational constitutive updates or incremental energy minimization principles, an incrementally defined energy derived from the underlying time-continuous constitutive model represents the starting point of the novel numerically robust algorithmic formulations. This incrementally defined potential allows to compute all variables jointly as minimizers of this energy. While such discrete variational constitutive updates are not new in general, they are considered here in order to employ powerful techniques from non-linear constrained optimization theory in order to compute robustly the aforementioned minimizers. The analyzed prototype models are based on (1) nonlinear complementarity problem (NCP) functions as well as on (2) the augmented Lagrangian formulation. Numerical experiments show the numerical robustness of the resulting algorithmic formulations. Furthermore, it is shown that the novel algorithmic ideas can also be integrated into classic, non-variational, return-mapping schemes. • Novel algorithmic formulations for rate-independent crystal plasticity theory based on incremental energy minimization. • Algorithms can also be applied to conventional, non-variational plasticity models. • Algorithms can be applied to local as well as to gradient-enhanced plasticity models. • Algorithm guarantees a regular tangent matrix and identification of active slip systems. • Reinterpretation of active-set algorithms as nonlinear complementarity problem (NCP). [ABSTRACT FROM AUTHOR]

Abstract Background Biomarker detection presents itself as a major means of translating biological data into clinical applications. Due to the recent advances in high throughput sequencing technologies, an increased number of metagenomics studies have suggested the dysbiosis in microbial communities as potential biomarker for certain diseases. The reproducibility of the results drawn from metagenomic data is crucial for clinical applications and to prevent incorrect biological conclusions. The variability in the sample size and the subjects participating in the experiments induce diversity, which may drastically change the outcome of biomarker detection algorithms. Therefore, a robust biomarker detection algorithm that ensures the consistency of the results irrespective of the natural diversity present in the samples is needed. Results Toward this end, this paper proposes a novel Regularized Low Rank-Sparse Decomposition (RegLRSD) algorithm. RegLRSD models the bacterial abundance data as a superposition between a sparse matrix and a low-rank matrix, which account for the differentially and non-differentially abundant microbes, respectively. Hence, the biomarker detection problem is cast as a matrix decomposition problem. In order to yield more consistent and solid biological conclusions, RegLRSD incorporates the prior knowledge that the irrelevant microbes do not exhibit significant variation between samples belonging to different phenotypes. Moreover, an efficient algorithm to extract the sparse matrix is proposed. Comprehensive comparisons of RegLRSD with the state-of-the-art algorithms on three realistic datasets are presented. The obtained results demonstrate that RegLRSD consistently outperforms the other algorithms in terms of reproducibility performance and provides a marker list with high classification accuracy. Conclusions The proposed RegLRSD algorithm for biomarker detection provides high reproducibility and classification accuracy performance regardless of the dataset complexity and the number of selected biomarkers. This renders RegLRSD as a reliable and powerful tool for identifying potential metagenomic biomarkers.

We consider an overdetermined problem of Serrin-type with respect to an operator in divergence form with piecewise constant coefficients. We give sufficient condition for unique solvability near radially symmetric configurations by means of a perturbation argument relying on shape derivatives and the implicit function theorem. This problem is also treated numerically, by means of a steepest descent algorithm based on a Kohn–Vogelius functional. [ABSTRACT FROM AUTHOR]

Mixed integer quadratic programming (MIQP) is the problem of minimizing a quadratic function over mixed integer points in a rational polyhedron. This paper focuses on the augmented Lagrangian dual (ALD) for MIQP. ALD augments the usual Lagrangian dual with a weighted nonlinear penalty on the dualized constraints. We first prove that ALD will reach a zero duality gap asymptotically as the weight on the penalty goes to infinity under some mild conditions on the penalty function. We next show that a finite penalty weight is enough for a zero gap when we use any norm as the penalty function. Finally, we prove a polynomial bound on the weight on the penalty term to obtain a zero gap. [ABSTRACT FROM AUTHOR]

Natural image priors play an important role in image denoising, and various prior-based methods have been widely proposed for noise removal. However, these methods tend to smooth the fine image textures while suppressing noise, degrading the image visual quality. To address this problem, in this paper, we propose an adaptive texture-preserving denoising method. In contrast to most existing prior-based denoising methods, two types of priors [gradient histogram matching priors and nonlocal self-similarity (NSS) priors] are proposed, and their combination is used for image denoising. We introduce a hyper-Laplacian distribution of the gradient histogram matching prior, which enforces the gradient histogram of the denoised image to be as close as possible to the estimated reference histogram from the original image. Meanwhile, the proposed model obtained by introducing the NSS priors effectively preserves fine image details and generates sharp image edges. To improve the accuracy of the method, a content-adaptive parameter selection scheme based on edge detection filters is proposed. Moreover, the optimization problem with two types of priors and the content-adaptive parameter added into the objective function becomes a challenging non-convex optimization problem. To effectively solve this problem, we have developed a new numerical solution based on augmented Lagrangian multipliers and alternating minimization scheme. The experimental results demonstrate that the proposed method effectively preserves the texture features of the denoised images and outperforms several variational methods and other state-of-the-art methods in terms of various evaluation indices and visual quality, especially at medium and high noise levels. [ABSTRACT FROM AUTHOR]

A static frictional contact problem between an elasto-plastic body and a rigid foundation is considered. The material's behavior is described by the nonlinear elastic constitutive Hencky's law. The contact is modeled with the Signorini condition and a version of Coulomb's law in which the coefficient of friction depends on the slip. The existence of a weak solution is proved by using Schauder's fixed-point theorem combined with arguments of abstract variational inequalities. Afterward, a successive iteration technique, based on the Kačanov method, to solve the problem numerically is proposed, and its convergence is established. Then, to improve the conditioning of the iterative problem, an appropriate Augmented Lagrangian formulation is used and that will lead us to Uzawa block relaxation method in every iteration. Finally, numerical experiments of two-dimensional test problems are carried out to illustrate the performance of the proposed algorithm. [ABSTRACT FROM AUTHOR]

In [M. Benzi and M. A. Olshanskii, SIAM J. Sci. Comput., 28 (2006), pp. 2095-2113] a preconditioner of augmented Lagrangian type was presented for the two-dimensional stationary incompressible Navier-Stokes equations that exhibits convergence almost independent of Reynolds number. The algorithm relies on a highly specialized multigrid method involving a custom prolongation operator and for robustness requires the use of piecewise constant finite elements for the pressure. However, the prolongation operator and velocity element used do not directly extend to three dimensions: the local solves necessary in the prolongation operator do not satisfy the inf-sup condition. In this work we generalize the preconditioner to three dimensions, proposing alternative finite elements for the velocity and prolongation operators for which the preconditioner works robustly. The solver is effective at high Reynolds number: on a three-dimensional lid-driven cavity problem with approximately one billion degrees of freedom, the average number of Krylov iterations per Newton step varies from 4.5 at Re = 10 to 3 at Re = 1000 and 5 at Re = 5000. [ABSTRACT FROM AUTHOR]

We present a new incremental mesh morphing method obtained by proposing and discretizing a solution procedure for the continuous morphing problem. Our method seeks a diffeomorphism that transforms an initial domain to a final domain by only prescribing the boundary displacement. To this end, we propose to minimize the distortion of the morphing mapping constrained to satisfy the imposed boundary displacement. To solve this problem, we consider an augmented Lagrangian method in Hilbert spaces that incorporates the boundary condition in the objective function using the Lagrange multipliers and a penalty parameter. The distortion is devised to penalize the appearance of non-invertible mappings and therefore, we do not need to equip our discrete implementation with untangling capabilities. Moreover, we introduce a weight function to improve the quality of the deformation and thus, the robustness of the non-linear solver. The discretization of the continuous augmented Lagrangian method leads to a mesh morphing method suitable for large displacements and rotations of meshes with non-uniform sizing, and mesh curving of highly stretched high-order meshes. • A new way to impose automatically incremental boundary conditions for mesh morphing. • Enhancing mesh quality with a weight function for the corresponding inner products. • Mesh morphing and mesh curving without the need of untangling capabilities. • The method is derived from a continuous domain morphing problem. • This work improves, in several aspects, our 26th IMR work. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we introduce a total variation l 1 - l 2 regularization scheme with adapting the parameter for image restoration involving blurry and noisy colour images. Numerically, an efficient augmented Lagrangian method associated with alternating minimization method is described to obtain the optimal solution recursively. We provide the convergence analysis for the resulting algorithm. Experimental results show that our proposed model and algorithm have good signal to noise ratio (SNR) and improvement in signal to noise ratio (ISNR) values for a motion blur with different kinds of noises. Highlights • Total variation l 1 - l 2 regularization scheme with adapting the parameter for image restoration involving blurry and noisy colour images. • Efficient augmented Lagrangian method associated with alternating minimization method for restoration. • Proposed model and algorithm have good signal to noise ratio (SNR) and improvement in signal to noise ratio (ISNR) values for a motion blur with different kinds of noises. [ABSTRACT FROM AUTHOR]

Simulating viscoelastic polymers and polymeric fluids requires a robust and accurate capture of elasticity and viscosity. The computation is known to become very challenging under large deformations and high viscosity. Drawing inspirations from return mapping based elastoplasticity treatment for granular materials, we present a finite strain integration scheme for general viscoelastic solids under arbitrarily large deformation and non-equilibrated flow. Our scheme is based on a predictor-corrector exponential mapping scheme on the principal strains from the deformation gradient, which closely resembles the conventional treatment for elastoplasticity and allows straightforward implementation into any existing constitutive models. We develop a new Material Point Method that is fully implicit on both elasticity and inelasticity using augmented Lagrangian optimization with various preconditioning strategies for highly efficient time integration. Our method not only handles viscoelasticity but also supports existing elastoplastic models including Drucker-Prager and von-Mises in a unified manner. We demonstrate the efficacy of our framework on various examples showing intricate and characteristic inelastic dynamics with competitive performance. [ABSTRACT FROM AUTHOR]

We develop three algorithms to solve the subproblems generated by the augmented Lagrangian methods introduced by Iusem-Nasri (2010) for the equilibrium problem. The first algorithm that we propose incorporates the Newton method and the other two are instances of the subgradient projection method. One of our algorithms is also capable of solving nondifferentiable equilibrium problems. Using well-known test problems, all algorithms introduced here are implemented and numerical results are reported to compare their performances. [ABSTRACT FROM AUTHOR]

We study the defocusing nonlinear Schrödinger (NLS) equation written in hydrodynamic form through the Madelung transform. From the mathematical point of view, the hydrodynamic form can be seen as the Euler–Lagrange equations for a Lagrangian submitted to a differential constraint corresponding to the mass conservation law. The dispersive nature of the NLS equation poses some major numerical challenges. The idea is to introduce a two‐parameter family of extended Lagrangians, depending on a greater number of variables, whose Euler–Lagrange equations are hyperbolic and accurately approximate NLS equation in a certain limit. The corresponding hyperbolic equations are studied and solved numerically using Godunov‐type methods. Comparison of exact and asymptotic solutions to the one‐dimensional cubic NLS equation ("gray" solitons and dispersive shocks) and the corresponding numerical solutions to the extended system was performed. A very good accuracy of such a hyperbolic approximation was observed. [ABSTRACT FROM AUTHOR]

In this work we propose a variational model for multi-modal image registration. It minimizes a new functional based on using reformulated normalized gradients of the images as the fidelity term and higher-order derivatives as the regularizer. We first present a theoretical analysis of the proposed model. Then, to solve the model numerically, we use an augmented Lagrangian method (ALM) to reformulate it to a few more amenable subproblems (each giving rise to an Euler-Lagrange equation that is discretized by finite difference methods) and solve iteratively the main linear systems by the fast Fourier transform; a multilevel technique is employed to speed up the initialisation and avoid likely local minima of the underlying functional. Finally we show the convergence of the ALM solver and give numerical results of the new approach. Comparisons with some existing methods are presented to illustrate its effectiveness and advantages. [ABSTRACT FROM AUTHOR]

For the power law Stokes equations driven by nonlinear slip boundary conditions of friction type, we propose three iterative schemes based on augmented Lagrangian approach and interior point method to solve the finite element approximation associated to the continuous problem. We formulate the variational problem which in this case is a variational inequality and construct the weak solution of the continuous problem. Next, we formulate two alternating direction methods based on augmented Lagrangian formalism in order to separate the velocity from the symmetric part the velocity gradient and tangential part of the velocity. Thirdly, we present some salient points of a path‐following variant of the interior point method associated to the finite element approximation of the problem. Some numerical experiments are performed to confirm the validity of the schemes and allow us to compare them. [ABSTRACT FROM AUTHOR]

Abstract This work presents a level set framework to solve topology optimization problems subject to local stress constraints considering design-dependent pressure loads. Two technical difficulties are related to this problem. The first one is the local nature of stresses. To deal with this issue, stress constraints are included to the problem by means of an Augmented Lagrangian scheme. The second is the adequate association between the moving boundary and the pressure acting on it. This difficulty is easily overcome by the level set method that allows for a clear tracking of the boundary along the optimization process. In the present approach, a reaction–diffusion equation substitutes the classical Hamilton–Jacobi equation to control the level set evolution. This choice has the advantage of allowing the nucleation of holes inside the domain and the elimination of the undesirable level set reinitializations. In addition, the optimization algorithm allows the rupture of loading boundaries, that is, the crossing of the pressured (wet) boundary with the traction free boundary is not avoided. This gives more freedom to the algorithm for topological changes. In order to validate the proposed scheme against stress concentrations, all numerical examples are performed on constrained domains containing singularities. Moreover, optimized designs obtained from stress-constrained and compliance problems are compared. [ABSTRACT FROM AUTHOR]

Digital image correlation (DIC) is a powerful experimental technique for measuring full-field displacement and strain. The basic idea of the method is to compare images of an object decorated with a speckle pattern before and after deformation, and thereby to compute the displacement and strain fields. Local subset DIC and finite element-based global DIC are two widely used image matching methods. However there are some drawbacks to these methods. In local subset DIC, the computed displacement field may not be compatible, and the deformation gradient may be noisy, especially when the subset size is small. Global DIC incorporates displacement compatibility, but can be computationally expensive. In this paper, we propose a new method, the augmented-Lagrangian digital image correlation (ALDIC), that combines the advantages of both the local (fast) and global (compatible) methods. We demonstrate that ALDIC has higher accuracy and behaves more robustly compared to both local subset DIC and global DIC. [ABSTRACT FROM AUTHOR]

Purpose: Background‐field removal is a crucial preprocessing step for quantitative susceptibility mapping (QSM). Remnants from this step often contaminate the estimated local field, which in turn leads to erroneous tissue‐susceptibility reconstructions. The present work aimed to mitigate this undesirable behavior with the development of a new approach that simultaneously decouples background contributions and local susceptibility sources on QSM inversion. Methods: Input phase data for QSM can be seen as a composite scalar field of local effects and residual background components. We developed a new weak‐harmonic regularizer to constrain the latter and to separate the 2 components. The resulting optimization problem was solved with the alternating directions of multipliers method framework to achieve fast convergence. In addition, for convenience, a new alternating directions of multipliers method–based preconditioned nonlinear projection onto dipole fields solver was developed to enable initializations with wrapped‐phase distributions. Weak‐harmonic QSM, with and without nonlinear projection onto dipole fields preconditioning, was compared with the original (alternating directions of multipliers method–based) total variation QSM algorithm in phantom and in vivo experiments. Results: Weak‐harmonic QSM returned improved reconstructions regardless of the method used for background‐field removal, although the proposed nonlinear projection onto dipole fields method often obtained better results. Streaking and shadowing artifacts were substantially suppressed, and residual background components were effectively removed. Conclusion: Weak‐harmonic QSM with field preconditioning is a robust dipole inversion technique and has the potential to be extended as a single‐step formulation for initialization with uncombined multi‐echo data. [ABSTRACT FROM AUTHOR]