Primal-dual Active Set Method Research Articles

Primal-dual active set strategy for large scale optimization of cardiac defibrillation

A feasibility study of optimal control techniques for cardiac defibrillation on anatomical three spatial dimensional rabbit ventricle geometry in the presence of bilateral control constraints is presented. The work addresses the numerical treatment of multi-scale and multi-domain simulations of the bidomain equations and is based on the primal-dual active set method to solve the optimality system for this large scale optimization problem. Numerical results are presented for a successful defibrillation study. Robustness of the optimization algorithm w.r.t to variations in the model parameters is demonstrated. A feasibility study for multiple small boundary control support is included as well. Finally, the numerical convergence of the optimization algorithm and the parallel efficiency is demonstrated.

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.

The Nonnegative Matrix Factorization: Regularization and Complexity

Data continues to grow, and it has become ever important to find effective big data analysis techniques. Computational tools, such as singular value decomposition, have been employed in the interpretation of big data. Another tool has recently gained popularity and comparative success: the nonnegative matrix factorization (NMF). The NMF method is a feature selection, dimension reduction algorithm which takes a data matrix and finds a simpler representation. In this paper, we discuss the choice of regularization parameters for a regularized NMF problem, and we develop a primal-dual active set method which enhances the representation. In addition, we examine rank determination for the case of the NMF problem. We develop a method to choose the proper complexity based on the concept of NMF-singular values.

A primal-dual active set method and predictor-corrector mesh adaptivity for computing fracture propagation using a phase-field approach

Abstract In this paper, we consider phase-field based fracture propagation in elastic media. The main purpose is the development of a robust and efficient numerical scheme. To enforce crack irreversibility as a constraint, we use a primal-dual active set strategy, which can be identified as a semi-smooth Newton method. The active set iteration is merged with the Newton iteration for solving the fully-coupled nonlinear partial differential equation discretized using finite elements, resulting in a single, rapidly converging nonlinear scheme. It is well known that phase-field models require fine meshes to accurately capture the propagation dynamics of the crack. Because traditional estimators based on adaptive mesh refinement schemes are not appropriate, we develop a predictor-corrector scheme for local mesh adaptivity to reduce the computational cost. This method is both robust and efficient and allows us to treat temporal and spatial refinements and to study the influence of model regularization parameters. Finally, our proposed approach is substantiated with different numerical tests for crack propagation in elastic media and pressurized fracture propagation in homogeneous and heterogeneous media.

Primal-Dual Active-Set Methods for Large-Scale Optimization

In this paper, we introduce two primal-dual active-set methods for solving large-scale constrained optimization problems. The first method minimizes a sequence of primal-dual augmented Lagrangian functions subject to bounds on the primal variables and artificial bounds on the dual variables. The basic structure is similar to the well-known optimization package Lancelot (Conn, et al. in SIAM J Numer Anal 28:545---572, 1991), which uses the traditional primal augmented Lagrangian function. Like Lancelot, our algorithm may use gradient projection-based methods enhanced by subspace acceleration techniques to solve each subproblem and therefore may be implemented matrix-free. The artificial bounds on the dual variables are a unique feature of our method and serve as a form of dual regularization. Our second algorithm is a two-phase method. The first phase computes iterates using our primal-dual augmented Lagrangian algorithm, which benefits from using cheap gradient projections and matrix-free linear CG calculations. The final iterate produced during this phase is then used as input for phase two, which is a stabilized sequential quadratic programming method (Gill and Robinson in SIAM J Opt 1---45, 2013). Obtaining superlinear local convergence under weak assumptions is an important benefit of the transition to a stabilized sequential quadratic programming algorithm. Interestingly, the bound-constrained subproblem used in phase one is equivalent to the stabilized subproblem used in phase two under certain assumptions. This fact makes our choice of algorithms a natural one.

A Feasible Active Set Method for Strictly Convex Quadratic Problems with Simple Bounds

A primal-dual active set method for quadratic problems with bound constraints is presented which extends the infeasible active set approach of Kunisch and Rendl [SIAM J. Optim., 14 (2003), pp. 35--52]. Based on a guess of the active set, a primal-dual pair ($x$,$\alpha$) is computed that satisfies stationarity and the complementary condition. If $x$ is not feasible, the variables connected to the infeasibilities are added to the active set, and a new primal-dual pair ($x$,$\alpha$) is computed. This process is iterated until a primal feasible solution is generated. Then a new active set is determined based on the feasibility information of the dual variable $\alpha$. Strict convexity of the quadratic problem is sufficient for the algorithm to stop after a finite number of steps with an optimal solution. Computational experience indicates that this approach also performs well in practice.

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.

Nonlocal Allen-Cahn systems: analysis and a primal-dual active set method

We show existence and uniqueness of a solution for the non-local vector-valued Allen-Cahn variational inequality in a formulation involving Lagrange multipliers for local and non-local constraints. Furthermore, we propose and analyze a primal-dual active set method for local and non-local vector-valued Allen-Cahn variational inequalities. The local convergence behaviour of the primal-dual active set algorithm is studied by interpreting the approach as a semi-smooth Newton method and numerical simulations are presented demonstrating its efficiency.

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.

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.

Solving the Cahn-Hilliard variational inequality with a semi-smooth Newton method

The Cahn-Hilliard variational inequality is a non-standard parabolic variational inequality of fourth order for which straightforward numerical approaches cannot be applied. We propose a primal-dual active set method which can be interpreted as a semi-smooth Newton method as solution technique for the discretized Cahn-Hilliard variational inequality. A (semi-)implicit Euler discretization is used in time and a piecewise linear finite element discretization of splitting type is used in space leading to a discrete variational inequality of saddle point type in each time step. In each iteration of the primal-dual active set method a linearized system resulting from the discretization of two coupled elliptic equations which are defined on different sets has to be solved. We show local convergence of the primal-dual active set method and demonstrate its efficiency with several numerical simulations.

A Primal-Dual Active-Set Method for Non-Negativity Constrained Total Variation Deblurring Problems

This paper studies image deblurring problems using a total variation-based model, with a non-negativity constraint. The addition of the non-negativity constraint improves the quality of the solutions, but makes the solution process a difficult one. The contribution of our work is a fast and robust numerical algorithm to solve the non-negatively constrained problem. To overcome the nondifferentiability of the total variation norm, we formulate the constrained deblurring problem as a primal-dual program which is a variant of the formulation proposed by Chan, Golub, and Mulet for unconstrained problems. Here, dual refers to a combination of the Lagrangian and Fenchel duals. To solve the constrained primal-dual program, we use a semi-smooth Newton's method. We exploit the relationship between the semi-smooth Newton's method and the primal-dual active set method to achieve considerable simplification of the computations. The main advantages of our proposed scheme are: no parameters need significant adjustment, a standard inverse preconditioner works very well, quadratic rate of local convergence (theoretical and numerical), numerical evidence of global convergence, and high accuracy of solving the optimality system. The scheme shows robustness of performance over a wide range of parameters. A comprehensive set of numerical comparisons are provided against other methods to solve the same problem which show the speed and accuracy advantages of our scheme.

Mesh independence and fast local convergence of a primal-dual active-set method for mixed control-state constrained elliptic control problems

A class of mixed control-state constrained optimal control problems for elliptic partial differential equations arising, for example, in Lavrentiev-type regularized state constrained optimal control is considered. Its numerical solution is obtained via a primal-dual activeset method, which is equivalent to a class of semi-smooth Newton methods. The locally superlinear convergence of the active-set method in function space is established, and its mesh independence is proved. The paper contains a report on numerical test runs including a comparison with a short-step path-following interior-point method and a coarse-to-fine mesh sweep, that is, a nested iteration technique, for accelerating the overall solution process. Finally, convergence and regularity properties of the regularized problems with respect to a vanishing Lavrentiev parameter are considered. 2000 Mathematics subject classification: primary 65K05; secondary 90C33.

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.

A Primal-Dual Active Set Method for Bilaterally Control Constrained Optimal Control of the Navier–Stokes Equations

This paper is devoted to the analysis and numerical solution of distributed optimal control of the Navier–Stokes equations in presence of bilateral pointwise control constraints. The analysis of the problem involves the proof of existence of an optimal solution, as well as the presentation of necessary and sufficient conditions for optimality. For the numerical solution of the problem we apply a primal-dual active set strategy and show global and local convergence properties of the method. Finally, some numerical experiments, which illustrate the performance of the method, are discussed.

The primal-dual active set method for a crack problem with non-penetration

The Lame problem in a 2D domain with a crack under a non-penetration condition is considered as a variational inequality. A primal-dual active set method is proposed as an efficient numerical solution technique and compared to a previously employed iterative method for a penalized formulation. Sufficient conditions for monotonic convergence of a discretized version of the proposed algorithm are given and numerical experiments are presented.

The Primal-Dual Active Set Method for Nonlinear Optimal Control Problems with Bilateral Constraints

The primal-dual active set method has proved to be an efficient numerical tool in the context of diverse applications. So far it has been investigated mainly for linear problems. This paper is devoted to the study of global convergence of the primal-dual active set method for nonlinear problems with bilateral constraints. Utilizing the close relationship between the primal-dual active set method and semismooth Newton methods, local superlinear convergence of the method is investigated as well.

Total Bounded Variation Regularization as a Bilaterally Constrained Optimization Problem

It is demonstrated that the predual for problems with total bounded variation regularization terms can be expressed as a bilaterally constrained optimization problem. Existence of a Lagrange multiplier and an optimality system are established. This allows us to utilize efficient optimization methods developed for problems with box constraints in the context of bounded variation formulations. Here, in particular, the primal-dual active set method, considered as a semismooth Newton method, is analyzed, and superlinear convergence is proved. As a by-product we obtain that the Lagrange multiplier associated with the box constraints acts as an edge detector. Numerical results for image denoising and zooming/resizing show the efficiency of the new approach.