Dual Active Set Method Research Articles

A modified combined active-set Newton method for solving phase-field fracture into the monolithic limit

In this work, we examine a numerical phase-field fracture framework in which the crack irreversibility constraint is treated with a primal–dual active set method and a linearization is used in the degradation function to enhance the numerical stability. The first goal is to carefully derive from a complementarity system our primal–dual active set formulation, which has been used in the literature in numerous studies, but for phase-field fracture without its detailed mathematical derivation yet. Based on the latter, we formulate a modified combined active-set Newton approach that significantly reduces the computational cost in comparison to comparable prior algorithms for quasi-monolithic settings. For many practical problems, Newton converges fast, but active set needs many iterations, for which three different efficiency improvements are suggested in this paper. Afterwards, we design an iteration on the linearization in order to iterate the problem to the monolithic limit. Our new algorithms are implemented in the programming framework pfm-cracks from Heister and Wick (2020). In the numerical examples, we conduct performance studies and investigate efficiency enhancements. The main emphasis is on the cost complexity by keeping the accuracy of numerical solutions and goal functionals. Our algorithmic suggestions are substantiated with the help of several benchmarks in two and three spatial dimensions. Therein, predictor–corrector adaptivity and parallel performance studies are explored as well.

A dual active set method for <inline-formula><tex-math id="M1">$ \ell $</tex-math></inline-formula>1-regularized problem

We consider an $ \ell $1-regularized problem, which has many applications such as compressed sensing, MRI, and seismic images. Although there have been some methods for the problem, they cannot obtain a highly accurate solution. To overcome this drawback, we propose a dual active set method, which is an active set method for the Fenchel dual problem of the $ \ell $1-regularized problem. Our method finds a highly accurate solution quickly because of the following features of the dual problem: (i) the dimension of the decision variables is smaller; (ii) since the optimal solution of the primal problem is sparse, the active constraints of the optimal solution of the dual problem are few and can be identified quickly. Then, we show that the method outputs the solution with finite iterations when it solves certain subproblems exactly. Moreover, we give its efficient and concrete implementation for Basis Pursuit Denoising (BPDN), which is a typical $ \ell $1-regularized problem. In numerical experiments for BPDN, our method provides more accurate solutions than all baselines in comparable computing time.

A quasi-monolithic phase-field description for orthotropic anisotropic fracture with adaptive mesh refinement and primal–dual active set method

In this work, thermodynamically consistent phase-field fracture frameworks for transversely isotropic and orthotropic settings are proposed. We formulate an anisotropic crack phase-field via a penalization approach for each family of fibers. The resulting model is augmented with thermodynamical arguments and then carefully analyzed from a mechanical perspective. The fracture dissipation inequality to prevent crack healing is imposed via a primal–dual active set strategy. Predictor–corrector mesh adaptivity allows to work with small length-scale parameters at a reasonable computational cost. Due to the importance of laminated structures for industrial applications, fracture responses for transversely isotropic and orthotropic materials are performed. Therein, several studies are conducted that include comparisons of anisotropic formulations with Griffith’s critical elastic energy release rate and with specific critical fracture energy formulations, as well as non-split and split approaches.

A semi-smooth Newton and Primal–Dual Active Set method for Non-Smooth Contact Dynamics

Multi-rigid-body dynamic contact systems, in other words Non Smooth Contact Dynamics (NSCD) problems, generate some inherent difficulties to multivocal laws, which results in non-linearities and non-smoothness associated to frictional contact models. Recently, Primal–Dual Active Set strategies (PDAS) have emerged as a promising method for solving contact problems. These methods are based on the following principle: the frictional contact conditions are restated as non-linear complementary functions for which the solution is provided by the iterative semi-smooth Newton method. Based on these prerequisites, this contribution aims to provide a generalization of the NSCD-PDAS for dynamic frictional contact problems. Several numerical experiments are reported for algorithm validation purposes and also to assess the efficiency and performances of PDAS methods with respect to the Newton/Augmented Lagrangian and the Bi-Potential methods.

A Unified Primal Dual Active Set Algorithm for Nonconvex Sparse Recovery

In this paper, we consider the problem of recovering a sparse signal based on penalized least squares formulations. We develop a novel algorithm of primal-dual active set type for a class of nonconvex sparsity-promoting penalties, including ℓ0, bridge, smoothly clipped absolute deviation, capped ℓ1 and minimax concavity penalty. First, we establish the existence of a global minimizer for the related optimization problems. Then we derive a novel necessary optimality condition for the global minimizer using the associated thresholding operator. The solutions to the optimality system are coordinatewise minimizers, and under minor conditions, they are also local minimizers. Upon introducing the dual variable, the active set can be determined using the primal and dual variables together. Further, this relation lends itself to an iterative algorithm of active set type which at each step involves first updating the primal variable only on the active set and then updating the dual variable explicitly. When combined with a continuation strategy on the regularization parameter, the primal dual active set method is shown to converge globally to the underlying regression target under certain regularity conditions. Extensive numerical experiments with both simulated and real data demonstrate its superior performance in terms of computational efficiency and recovery accuracy compared with the existing sparse recovery methods.

Inexact primal–dual active set method for solving elastodynamic frictional contact problems

In this paper, several active set methods based on classical problems arising in Contact Mechanics are analyzed, namely unilateral/bilateral contact associated with Tresca’s/Coulomb’s law of friction in small and large deformation. The purpose of this work is to extend an Inexact Primal–Dual Active Set (IPDAS) method already used in Hueber et al. (2008) to the formalism of dynamics and hyper-elasticity. This method permits to solve the unilateral problem with Coulomb’s law of friction by taking into account an alternative for the latter based on the approximation of the Coulomb’s law by a succession of states of Tresca friction in which the friction threshold is fixed at each fixed point iteration. The mechanical formulation in the hyper-elasticity framework is first presented, next, we establish weak formulations of the different cases of frictional contact problems and we give the finite element approximation of the problems. Then, we detail the numerical treatment within the framework of the primal–dual active set strategy for different frictional contact conditions. We finally provide some numerical experiments to bring into light the efficiency of the IPDAS method and to carry out a comparison with the augmented Lagrangian method by considering representative contact problems in both small and large deformation cases.

Matrix-free multigrid solvers for phase-field fracture problems

In this work, we present a framework for the matrix-free solution to a monolithic quasi-static phase-field fracture model with geometric multigrid methods. Using a standard matrix-based approach within the Finite Element Method requires lots of memory, which eventually becomes a serious bottleneck. A matrix-free approach overcomes this problem and greatly reduces the amount of required memory, allowing to solve larger problems on available hardware. One key challenge is concerned with the crack irreversibility for which a primal–dual active set method is employed. Here, the active set values of fine meshes must be available on coarser levels of the multigrid algorithm. The developed multigrid method provides a preconditioner for a generalized minimal residual (GMRES) solver. This method is used for solving the linear equations inside Newton’s method for treating the overall nonlinear-monolithic discrete displacement/phase-field formulation. Several numerical examples demonstrate the performance and robustness of our solution technology. Mesh refinement studies, variations in the phase-field regularization parameter, iterations numbers of the linear and nonlinear solvers, and some parallel performances are conducted to substantiate the efficiency of the proposed solver for single fractures, multiple pressurized fractures, and a L-shaped panel test in three dimensions.

Analysis of two active set type methods to solve unilateral contact problems

In this work two active set type methods are considered in order to solve a mathematical problem which describes the frictionless contact between a deformable body and a perfectly rigid obstacle, the so-called Signorini Problem. These methods are the primal dual active set method and the projection iterative method. Our aim, here, is to analyze these two active set type methods and to carry out a comparison with the well-known augmented Lagrangian method by considering two representative contact problems in the case of large and small deformation. After presenting the mechanical formulation in the hyperelasticity framework, we establish weak formulations of the problem and the existence result of the weak solution is recalled. Then, we give the finite element approximation of the problem and a description of the numerical methods is presented. The main result of this work is to provide a convergence result for the projection iterative method. Finally, we present numerical simulations which illustrate the behavior of the solution and allow the comparison of the numerical methods.

Pressure and fluid-driven fracture propagation in porous media using an adaptive finite element phase field model

This work presents phase field fracture modeling in heterogeneous porous media. We develop robust and efficient numerical algorithms for pressure-driven and fluid-driven settings in which the focus relies on mesh adaptivity in order to save computational cost for large-scale 3D applications. In the fluid-driven framework, we solve for three unknowns pressure, displacements and phase field that are treated with a fixed-stress iteration in which the pressure and the displacement–phase-field system are decoupled. The latter subsystem is solved with a combined Newton approach employing a primal–dual active set method in order to account for crack irreversibility. Numerical examples for pressurized fractures and fluid filled fracture propagation in heterogeneous porous media demonstrate our developments. In particular, mesh refinement allows us to perform systematic studies with respect to the spatial discretization parameter.

Dual active set method for support vector machines under multi-constraint activation

Active set methods are competitive alternatives of SVM optimizers to working set or decomposition techniques for moderate scale problems. While there are many works on applying the primal active set methods to SVM, few study the dual active set methods. Comparing with the primal active set method, the dual active set method is more efficient and numerically stable due to using an unconstrained minimum as a feasible start point and having choices of which constraints to add to the active set. However, since the conventional dual active set method can only add one violated constraint in one step, this paper proposes a novel dual active set method, which allows to add multiple constraints and accelerates the convergence. Moreover, the new method does not require to compute the matrix inversion when applied to SVM, and essentially reduces the training time. Experiment results on several benchmark data sets validate that the proposed method is indeed more efficient than the primal and the conventional dual active set methods.

A primal dual active set with continuation algorithm for the [formula omitted]-regularized optimization problem

We develop a primal dual active set with continuation algorithm for solving the ℓ0-regularized least-squares problem that frequently arises in compressed sensing. The algorithm couples the primal dual active set method with a continuation strategy on the regularization parameter. At each inner iteration, it first identifies the active set from both primal and dual variables, and then updates the primal variable by solving a (typically small) least-squares problem defined on the active set, from which the dual variable can be updated explicitly. Under certain conditions on the sensing matrix, i.e., mutual incoherence property or restricted isometry property, and the noise level, a finite step global convergence of the overall algorithm is established. Extensive numerical examples are presented to illustrate the efficiency and accuracy of the algorithm and its convergence behavior.

Optimal control for an elliptic system with convex polygonal control constraints

In recent years, optimal control problems governed by partial differential equations with pointwise constraints on the controls have received a considerable amount of attention. The challenge consists in finding efficient numerical methods in spite of the nonsmoothness introduced by pointwise inequality constraints. Most investigations so far have considered the case of scalar-valued control with simple constraints (e.g. unilateral or bilateral constraints). For vector-valued controls, which appear, for example, in the context of the Navier–Stokes equations, the constraints that were investigated were of a box type. For scalar-valued controls with unilateral or bilateral constraints, the numerical realization of optimal control problems can advantageously be performed with the semismooth Newton method or, equivalently, by the primal–dual active set method. We refer to Hintermuller et al. (2003), Ito & Kunisch (2004), De Los Reyes & Kunisch (2005), Hintermuller & Hinze (2006), Ito & Kunisch (2008) and Ulbrich (2011) and the references cited therein. An alternative approach is based on interior point methods. They were analysed, for example, in Schiela & Weiser (2008) and Weiser et al. (2008). The semismooth Newton method is different from penalty—or barrier—function methods which rely on penalty—or barrier— parameters, respectively. It was proven to be locally superlinearly convergent, and for unilaterally constrained problems to be globally convergent; cf. Hintermuller et al. (2003). For vector-valued controls, there are only a few papers that deal with the case of general convex control constraints. In Wachsmuth (2006a,b) and Wachsmuth (2007), the first-order necessary and the second-order sufficient optimality conditions for general convex constraints are obtained. In Wachsmuth (2006b), a numerical procedure is proposed which, however, does not concretely exploit the structure of

Preconditioning for Allen–Cahn variational inequalities with non-local constraints

The solution of Allen–Cahn variational inequalities with mass constraints is of interest in many applications. This problem can be solved both in its scalar and vector-valued form as a PDE-constrained optimization problem by means of a primal–dual active set method. At the heart of this method lies the solution of linear systems in saddle point form. In this paper we propose the use of Krylov-subspace solvers and suitable preconditioners for the saddle point systems. Numerical results illustrate the competitiveness of this approach.

A Comparative Study of Dual Active-Set and Primal-Dual Interior-Point Method

The relative performances of two optimization algorithms – dual active-set method (DASM) and primal-dual interior-point method (PDIPM) in the implementation of model predictive control (MPC) in terms of input response, CPU time, and number of iteration have been compared with the help of three case studies including MPC of continuous stirred tank reactor (CSTR) in series. The study shows significant improvement of convergence, CPU time, and cost per iteration using DASM for small-scale problem. Both DASM and PDIPM have been observed to be efficient, as the number of iterations is found to be independent of the problem size. Our study also shows that DASM gives better performance over PDIPM.

Preconditioning for partial differential equation constrained optimization with control constraints

SUMMARYOptimal control problems with partial differential equations play an important role in many applications. The inclusion of bound constraints for the control poses a significant additional challenge for optimization methods. In this paper, we propose preconditioners for the saddle point problems that arise when a primal–dual active set method is used. We also show for this method that the same saddle point system can be derived when the method is considered as a semismooth Newton method. In addition, the projected gradient method can be employed to solve optimization problems with simple bounds, and we discuss the efficient solution of the linear systems in question. In the case when an acceleration technique is employed for the projected gradient method, this again yields a semismooth Newton method that is equivalent to the primal–dual active set method. We also consider the Moreau–Yosida regularization method for control constraints and efficient preconditioners for this technique. Numerical results illustrate the competitiveness of these approaches. Copyright © 2011 John Wiley & Sons, Ltd.

Primal–dual active set method for control constrained optimal control of the Stokes equations

In this paper, we study the distributed and boundary optimal control of the Stokes equations in the presence of pointwise control constraints. The analysis of the problems involve existence results, as well as necessary conditions and optimality systems. A primal–dual active set strategy is studied for the numerical solution of the problems. For the distributed case, global and local convergence results for the infinite dimensional method are presented. For the boundary control case, due to the lack of regularity of the pointwise constraint multiplier, the method is applied to a discretized version of the original problem. Finally, some numerical experiments, which illustrate the performance of the method for both cases, are commented.

Optimal Control of Elliptic Variational Inequalities

Optimality systems for optimal control problems governed by elliptic variational inequalities are derived. Existence of appropriately defined Lagrange multipliers is proved. A primal—dual active set method is proposed to solve the optimality systems numerically. Examples with and without lack of strict complementarity are included.

A Dual Method for Certain Positive Semidefinite Quadratic Programming Problems

This paper presents a dual active set method for minimizing a sum of piecewise linear functions and a strictly convex quadratic function, subject to linear constraints. It may be used for direction finding in nondifferentiable optimization algorithms and for solving exact penalty formulations of (possibly inconsistent) strictly convex quadratic programming problems. An efficient implementation is described extending the Goldfarb and Idnani algorithm, which includes Powell's refinements. Numerical results indicate excellent accuracy of the implementation.