Primal-dual Active Set Method Research Articles

Parallel primal-dual active-set algorithm with nonlinear and linear preconditioners

The primal-dual active-set (PDAS) algorithm is a well-established and efficient method for addressing complementarity problems. However, the majority of existing approaches primarily concentrate on solving this non-smooth system with linear cases, and the straightforward extension of the primal-dual active-set method for solving nonlinear large-scale engineering problems does not work as well as expected, due to the unbalanced nonlinearities that bring about the difficulty of the slow convergence or stagnation. In the paper, we present the primal-dual active-set method with backtracking on the parallel computing framework for solving the nonlinear complementarity problem (NCP) arising from the discretization of partial differential equations. Some adaptive nonlinear preconditioning strategies based on nonlinear elimination are presented to handle the high nonlinearity of the nonsmooth system, and a family of linear preconditioners based on domain decomposition is developed to enhance the efficiency and scalability of this Newton-type method. Moreover, rigorous proof to establish both the monotone and superlinear convergence of the primal-dual active-set algorithm is also provided for the theoretical analysis. A series of numerical experiments for a family of multiphase reservoir problems, i.e., the CO2 injection model, are carried out to demonstrate the robustness and efficiency of the proposed parallel algorithm.

Primal-dual active set method for evaluating American put options on zero-coupon bonds

Primal-dual active set method for evaluating American put options on zero-coupon bonds

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.

Primal-Dual Active-Set Method for the Valuation of American Exchange Options

Primal-Dual Active-Set Method for the Valuation of American Exchange Options

Unified primal-dual active set method for dynamic frictional contact problems

In this paper, we propose a semi-smooth Newton method and a primal-dual active set strategy to solve dynamical contact problems with friction. The conditions of contact with Coulomb’s friction can be formulated in the form of a fixed point problem related to a quasi-optimization one thanks to the semi-smooth Newton method. This method is based on the use of the primal-dual active set (PDAS) strategy. The main idea here is to find the correct subset mathcal{A} of nodes that are in contact (active) opposed to those which are not in contact (inactive). For each case, the nonlinear boundary condition is replaced by a suitable linear one. Numerical experiments on both hyper-elastic problems and rigid granular materials are presented to show the efficiency of the proposed method.

Primal-dual active-set method for solving the unilateral pricing problem of American better-of options on two assets

<abstract><p>In this paper, an efficient numerical algorithm is proposed for the valuation of unilateral American better-of options with two underlying assets. The pricing model can be described as a backward parabolic variational inequality with variable coefficients on a two-dimensional unbounded domain. It can be transformed into a one-dimensional bounded free boundary problem by some conventional transformations and the far-field truncation technique. With appropriate boundary conditions on the free boundary, a bounded linear complementary problem corresponding to the option pricing is established. Furthermore, the full discretization scheme is obtained by applying the backward Euler method and the finite element method in temporal and spatial directions, respectively. Based on the symmetric positive definite property of the discretized matrix, the value of the option and the free boundary are obtained simultaneously by the primal-dual active-set method. The error estimation is established by the variational theory. Numerical experiments are carried out to verify the efficiency of our method at the end.</p></abstract>

Adaptive neural network surrogate model for solving the implied volatility of time-dependent American option via Bayesian inference

<abstract><p>In this paper, we propose an adaptive neural network surrogate method to solve the implied volatility of American put options, respectively. For the forward problem, we give the linear complementarity problem of the American put option, which can be transformed into several standard American put option problems by variable substitution and discretization in the temporal direction. Thus, the price of the option can be solved by primal-dual active-set method using numerical transformation and finite element discretization in spatial direction. For the inverse problem, we give the framework of the general Bayesian inverse problem, and adopt the direct Metropolis-Hastings sampling method and adaptive neural network surrogate method, respectively. We perform some simulations of volatility in the forward model with one- and four-dimension to compare the point estimates and posterior density distributions of two sampling methods. The superiority of adaptive surrogate method in solving the implied volatility of time-dependent American options are verified.</p></abstract>

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.

Parallel Matrix-Free Higher-Order Finite Element Solvers for Phase-Field Fracture Problems

Phase-field fracture models lead to variational problems that can be written as a coupled variational equality and inequality system. Numerically, such problems can be treated with Galerkin finite elements and primal-dual active set methods. Specifically, low-order and high-order finite elements may be employed, where, for the latter, only few studies exist to date. The most time-consuming part in the discrete version of the primal-dual active set (semi-smooth Newton) algorithm consists in the solutions of changing linear systems arising at each semi-smooth Newton step. We propose a new parallel matrix-free monolithic multigrid preconditioner for these systems. We provide two numerical tests, and discuss the performance of the parallel solver proposed in the paper. Furthermore, we compare our new preconditioner with a block-AMG preconditioner available in the literature.

A fuzzy system based active set algorithm for the numerical solution of the optimal control problem governed by partial differential equation

A computational indirect method based on a fuzzy system technique and active set strategy is presented. This new method has been used for the numerical solution of an optimal control problem governed by elliptic partial differential equation. To solve this problem, we start with obtaining the first-order necessary optimality conditions, which contain a mixed variational inequality in function space. Then a fuzzy basis functions technique combined with a primal-dual active set method to derive a new efficient algorithm for solving the given problem. Finally, some numerical examples are given to illustrate the accuracy and efficiency of the proposed technique.

Primal-dual active set method for pricing American better-of option on two assets

An American better-of option is a rainbow option with two underlying assets. It can be described by a parabolic variational inequality (VI) on a two-dimensional unbounded domain, which can also be characterized by a two-dimensional free boundary problem. Based on the numeraire transformation and the known information on the free boundary, we derive a one-dimensional linear complementarity problem (LCP) related to options on a bounded domain. Moreover, the full discretization scheme of LCP is constructed by finite difference and finite element methods in temporal and spatial directions, respectively. The primal-dual active-set (PDAS) method is adopted for solving the resulting large-scale discretized system. In each PDAS iteration, a unique index set of primal-dual variables will be computed by solving a linear system. A systematical convergent analysis is presented on our proposed method for pricing American better-of options. One of the desirable features of our method is that we can get the option value and two free boundaries simultaneously. Numerical simulations are performed to verify the efficiency of our proposed method.

The Primal-Dual Active Set Method for a Class of Nonlinear Problems withT-Monotone Operators

The family of primal-dual active set methods is drawing more attention in scientific and engineering applications due to its effectiveness and robustness for variational inequality problems. In this work, we introduce and study a primal-dual active set method for the solution of the variational inequality problems withT-monotone operators. We show that the sequence generated by the proposed method globally and monotonously converges to the unique solution of the variational inequality problem. Moreover, the convergence rate of the proposed scheme is analyzed under the framework of the algebraic setting; i.e., the established convergence results show that the iteration number of the methods is bounded by the number of the unknowns. Finally, numerical results show that the efficiency can be achieved by the primal-dual active set method.

A FE-ADMM algorithm for Lavrentiev-regularized state-constrained elliptic control problem

In this paper, Elliptic control problems with pointwise box constraints on the state is considered, where the corresponding Lagrange multipliers in general only represent regular Borel measure functions. To tackle this difficulty, the Lavrentiev regularization is employed to deal with the state constraints. To numerically discretize the resulted problem, full piecewise linear finite element discretization is employed. Estimation of the error produced by regularization and discretization is done. The error order of full discretization is not inferior to that of variational discretization because of the Lavrentiev-regularization. Taking the discretization error into account, algorithms of high precision do not make much sense. Utilizing efficient first-order algorithms to solve discretized problems to moderate accuracy is sufficient. Then a heterogeneous alternating direction method of multipliers (hADMM) is proposed. Different from the classical ADMM, our hADMM adopts two different weighted norms in two subproblems respectively. Additionally, to get more accurate solution, a two-phase strategy is presented, in which the primal-dual active set (PDAS) method is used as a postprocessor of the hADMM. Numerical results not only verify error estimates but also show the efficiency of the hADMM and the two-phase strategy.

An FE-Inexact Heterogeneous ADMM for Elliptic Optimal Control Problems with L1-Control Cost

Elliptic PDE-constrained optimal control problems with L1-control cost (L1-EOCP) are considered. To solve L1-EOCP, the primal-dual active set (PDAS) method, which is a special semismooth Newton (SSN) method, used to be a priority. However, in general solving Newton equations is expensive. Motivated by the success of alternating direction method of multipliers (ADMM), we consider extending the ADMM to L1-EOCP. To discretize L1-EOCP, the piecewise linear finite element (FE) is considered. However, different from the finite dimensional l1-norm, the discretized L1-norm does not have a decoupled form. To overcome this difficulty, an effective approach is utilizing nodal quadrature formulas to approximately discretize the L1-norm and L2-norm. It is proved that these approximation steps will not change the order of error estimates. To solve the discretized problem, an inexact heterogeneous ADMM (ihADMM) is proposed. Different from the classical ADMM, the ihADMM adopts two different weighted inner products to define the augmented Lagrangian function in two subproblems, respectively. Benefiting from such different weighted techniques, two subproblems of ihADMM can be efficiently implemented. Furthermore, theoretical results on the global convergence as well as the iteration complexity results o(1/k) for ihADMM are given. In order to obtain more accurate solution, a two-phase strategy is also presented, in which the primal-dual active set (PDAS) method is used as a postprocessor of the ihADMM. Numerical results not only confirm error estimates, but also show that the ihADMM and the two-phase strategy are highly efficient.

Sensitivity Analysis for Nonlinear Programming in CasADi

We present an extension of the CasADi numerical optimization framework that allows arbitrary order NLP sensitivities to be calculated automatically and efficiently. The approach, which can be used together with any NLP solver available in CasADi, is based on a sparse QR factorization and an implementation of a primal-dual active set method. The whole toolchain is freely available as open-source software and allows generation of thread-safe, self-contained C code with small memory footprint. We illustrate the toolchain using three examples; a sparse QP, an optimal control problem and a parameter estimation problem.

A primal-dual active set method for solving multi-rigid-body dynamic contact problems

In this work, an active set type method is considered in order to solve a mathematical problem that describes the frictionless dynamic contact of a multi-body rigid system, the so-called nonsmooth contact dynamics (NSCD) problem. Our aim, here, is to present the local treatment of contact conditions by an active set type method dedicated to NSCD and to carry out a comparison with the various well-known methods based on the bipotential theory and the augmented Lagrangian theory. After presenting the mechanical formulation of the NSCD and the resolution of the global problem concerning the equations of motion, we focus on the local level devoted to the resolution of the contact law. Then we detail the numerical treatment of the contact conditions within the framework of the primal-dual active set strategy. Finally, numerical experiments are presented to establish the efficiency of the proposed method by considering the comparison with the other numerical methods.