Semismooth Newton Method Research Articles

Inexact GMRES Policy Iteration for Large-Scale Markov Decision Processes

Policy iteration enjoys a local quadratic rate of contraction, but its iterations are computationally expensive for Markov decision processes (MDPs) with a large number of states. In light of the connection between policy iteration and the semismooth Newton method and taking inspiration from the inexact variants of the latter, we propose inexact policy iteration, a new class of methods for large-scale finite MDPs with local contraction guarantees. We then design an instance based on the deployment of the generalized minimal residual method (GMRES) for the approximate policy evaluation step, which we call inexact GMRES policy iteration. Finally, we demonstrate the superior practical performance of inexact GMRES policy iteration on an MDP with 10000 states, where it achieves a × 5.8 and × 2.2 speedup with respect to policy iteration and optimistic policy iteration, respectively.

Multi-Agent Dynamic Scheduling With a Posteriori Path Tracking and Collision Avoidance Using Model Predictive Control

This research work investigates a coordinated multi-agent path planning and tracking method. The solution of a pre-processed dynamic scheduling problem performs target assignment and provides optimal starting times and paths for each agent. Afterwards, a linear model predictive controller ensures robust and fast path tracking while preventing agents from collisions. This task is formulated as a discretized quadratic programming (QP) problem and is solved using an in-house developed semi-smooth Newton method. Numerical experiments have demonstrated the efficiency of the approach.

A minimum-type nonlinear complementarity simulator with constrained pressure residual (CPR) methods for wormhole propagation in carbonate acidization

Due to the high computational complexity of wormhole propagation in carbonate acidization, nonphysical oscillations of the numerical solutions often appear, which seriously ruins the physical accuracy or even breaks down the whole process of reservoir flows at large-scale simulation. In this paper, we introduce and study a family of recently developed nonlinear complementarity formulations for handling the nonphysical oscillations. Typically, this bound-preserving process is modeled by the corresponding nonlinear partial differential equations (PDEs) along with the inequality restrictions as a non-smooth system of equations under the application of a minimum-type complementary function. Because of the non-smoothness of the nonlinear complementarity system, the nonlinear algebraic system after the fully implicit discretization is solved by using a semismooth Newton method combined with a generalized Jacobian matrix. To accelerate the convergence and enhance the robustness of the corresponding linear iterations, we employ an improved class of constrained pressure residual (CPR) preconditioners with different combinations of physics-based and domain decomposition methods. Experiments on two- and three-dimensional wormhole propagation problems are presented to demonstrate the applicability of the aforementioned algorithms. Large-scale reservoir simulation with more than one billion degrees of freedom is provided to show the algorithmic scalability by using tens of thousands of processors.

An efficient semismooth Newton method for adaptive sparse signal recovery problems

We know that compressive sensing can establish stable sparse recovery results from highly undersampled data under a restricted isometry property condition. In reality, however, numerous problems are coherent, and vast majority conventional methods might work not so well. Recently, it was shown that using the difference between - and -norm as a regularization always has superior performance. In this paper, we consider an adaptive - model where the -norm with measures the data fidelity and the -term measures the sparsity. This proposed model has the ability to deal with different types of noises and extract the sparse property even under high coherent condition. We use a proximal majorization-minimization technique to handle the non-convex regularization term and then employ a semismooth Newton method to solve the corresponding convex relaxation subproblem. We prove that the sequence generated by the semismooth Newton method admits fast local convergence rate to the subproblem under some technical assumptions. Finally, we do some numerical experiments to demonstrate the superiority of the proposed model and the progressiveness of the proposed algorithm.

A semismooth Newton based augmented Lagrangian method for nonsmooth optimization on matrix manifolds

This paper is devoted to studying an augmented Lagrangian method for solving a class of manifold optimization problems, which have nonsmooth objective functions and nonlinear constraints. Under the constant positive linear dependence condition on manifolds, we show that the proposed method converges to a stationary point of the nonsmooth manifold optimization problem. Moreover, we propose a globalized semismooth Newton method to solve the augmented Lagrangian subproblem on manifolds efficiently. The local superlinear convergence of the manifold semismooth Newton method is also established under some suitable conditions. We also prove that the semismoothness on submanifolds can be inherited from that in the ambient manifold. Finally, numerical experiments on compressed modes and (constrained) sparse principal component analysis illustrate the advantages of the proposed method.

Presolving for Mixed-Integer Semidefinite Optimization

This paper provides a discussion and evaluation of presolving methods for mixed-integer semidefinite programs. We generalize methods from the mixed-integer linear case and introduce new methods that depend on the semidefinite condition. The methods considered include adding linear constraints, deriving bounds relying on 2 × 2 minors of the semidefinite constraints, tightening of variable bounds based on solving a semidefinite program with one variable, and scaling of the matrices in the semidefinite constraints. Tightening the bounds of variables can also be used in a node presolving step. Along the way, we discuss how to solve semidefinite programs with one variable using a semismooth Newton method and the convergence of iteratively applying bound tightening. We then provide an extensive computational comparison of the different presolving methods, demonstrating their effectiveness with an improvement in running time of about 22% on average. The impact depends on the instance type and varies across the methods. Funding: This work was supported by the EXPRESS II project within the German Research Foundation (Deutsche Forschungsgemeinschaft, DFG) priority program CoSIP (DFG-SPP 1798). It was also partly supported by the DFG within Project A4 in the SFB 805.

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.

A global convergent semi-smooth Newton method for semi-linear elliptic optimal control problem

Global convergent semi-smooth Newton (GCSSN) method for L 2 -norm control constrained elliptic optimal control problem with L 1 -control cost is discussed. The first order necessary optimality conditions are analyzed and reduced system is proposed. After finite difference discretization, we propose the global convergent semi-smooth Newton method for discrete reduced system with the nonmonotone line search. The local superlinear convergence rate and convergence are proved theoretically. The demonstrated theoretical properties are verified with numerical results. To illustrate the effectiveness and efficiency, we compare the proposed method with semi-smooth Newton method in the simulation part. • Global convergent semi-smooth Newton method for semi-linear elliptic optimal control problem is proposed. • We obtain the first order optimality necessary condition for optimal control problem. • The convergence properties of algorithm are proven theoretically and verified by simulations.

An Investigation on Semismooth Newton based Augmented Lagrangian Method for Image Restoration

The augmented Lagrangian method (also called as method of multipliers) is an important and powerful optimization method for lots of smooth or nonsmooth variational problems in modern signal processing, imaging and optimal control. However, one usually needs to solve a coupled and nonlinear system of equations, which is very challenging. In this paper, we propose several semismooth Newton methods to solve arising nonlinear subproblems for image restoration in finite dimensional spaces, which leads to several highly efficient and competitive algorithms for imaging processing. With the analysis of the metric subregularities of the corresponding functions, we give both the global convergence and local linear convergence rate for the proposed augmented Lagrangian methods with semismooth Newton solvers.

Scalable semismooth Newton methods with multilevel domain decomposition for subsurface flow and reactive transport in porous media

Large-scale modeling and predictive simulations of the subsurface flow and reactive transport system in porous media is significantly challenging, due to the high nonlinearity of the governing equations and the strong heterogeneity of material coefficients. The design of novel mathematical models and state-of-the-art methods for the flow simulation through porous media typically needs to satisfy the so-called bound-preserving property, i.e., the computed solution should stay inside a physically meaningful range. This paper presents a robust, scalable numerical framework based on the variational inequality formulation and the semismooth Newton method in a fully implicit manner, to model and simulate highly nonlinear flows without violating the boundedness requirement of the solution. Rigorous theoretical analysis for the variational inequality formulation of the problem is provided for facilitating the design of algorithms. Specifically, our approach further enhances the numerical formulation by utilizing a family of multilevel monolithic overlapping Schwarz methods for efficiently preconditioning, and the parallel implementation of the bound-preserving solver is based on the fast and robust domain decomposition technique. Numerical experiments are presented to demonstrate the efficiency and parallel scalability of the solution strategy for both standard benchmarks as well as realistic flow problems involving strong heterogeneity and high nonlinearity. We also show that the proposed framework is more robust and efficient than the commonly used inexact Newton algorithm in terms of the bound-preserving property.

A facial reduction approach for the single source localization problem

The single source localization problem (SSLP) appears in several fields such as signal processing and global positioning systems. The optimization problem of SSLP is nonconvex and difficult to find its globally optima solution. It can be reformulated as a rank constrained Euclidean distance matrix (EDM) completion problem with a number of equality constraints. In this paper, we propose a facial reduction approach to solve such an EDM completion problem. For the constraints of fixed distances between sensors, we reduce them to a face of the EDM cone and derive the closed formulation of the face. We prove constraint nondegeneracy for each feasible point of the resulting EDM optimization problem without a rank constraint, which guarantees the quadratic convergence of semismooth Newton's method. To tackle the nonconvex rank constraint, we apply the majorized penalty approach developed by Zhou et al. (IEEE Trans Signal Process 66(3):4331-4346, 2018). Numerical results verify the fast speed of the proposed approach while giving comparable quality of solutions as other methods.

Global implicit solver for multiphase multicomponent flow in porous media with multiple gas components and general reactions

In order to study the efficiency of the various forms of trapping including mineral trapping scenarios for CO2 storage behavior in deep layers of porous media, highly nonlinear coupled diffusion-advection-reaction partial differential equations (PDEs) including kinetic and equilibrium reactions modeling the miscible multiphase multicomponent flow have to be solved. We apply the globally fully implicit PDE reduction method (PRM) developed 2007 by Kräutle and Knabner for one-phase flow, which was extended 2019 to the case of two-phase flow with a pure gas in the study of Brunner and Knabner. We extend the method to the case of an arbitrary number of gases in gaseous phase, because CO2 is not the only gas that threats the climate, and usually is accompanied by other climate killing gases. The application of the PRM leads to an equation system consisting of PDEs, ordinary differential equations, and algebraic equations. The Finite Element discretized / Finite Volume stabilized equations are separated into a local and a global system but nevertheless coupled by the resolution function and evaluated with the aid of a nested Newton solver, so our solver is fully global implicit. For the phase disappearance, we use persistent variables which lead to a semismooth formulation that is solved with a semismooth Newton method. We present scenarios of the injection of a mixture of various gases into deep layers, we investigate phase change effects in the context of various gases, and study the mineral trapping effects of the storage technique. The technical framework also applies to other fields such as nuclear waste storage or oil recovery.

A dual active-set proximal Newton algorithm for sparse approximation of correlation matrices

In this paper, we propose a novel dual active-set algorithm that is based on proximal gradient and semi-smooth Newton iterations for the sparse approximation of correlation matrices in the Frobenius norm. A new dual formulation with upper and lower bounds is derived. To solve the dual, the proximal gradient method is developed to guarantee global convergence. Also, it provides information to estimate active/inactive constraints. Then, the semi-smooth Newton method is applied to accelerate the convergence of the proximal gradient method, which is the key ingredient of our algorithm. It is shown that the proposed algorithm for the dual is globally convergent under certain conditions. Some preliminary numerical results are given to illustrate the effectiveness of our algorithm on synthetic and real data sets.

PRIMAL-DUAL ALGORITHM FOR QUASI-STATIC CONTACT PROBLEM WITH COULOMB'S FRICTION

This paper presents a fast first-order method for solving the quasi-static contact problem with the Coulomb friction. It is known that this problem can be formulated as a second-order cone linear complementarity problem, for which regularized or semi-smooth Newton methods are widely used. As an alternative approach, this paper develops a method based on an accelerated primal-dual algorithm. The proposed method is easy to implement, as most of computation consists of additions and multiplications of vectors and matrices. Numerical experiments demonstrate that this method outperforms a regularized and smoothed Newton method for the second-order cone complementarity problem.

On a primal-dual Newton proximal method for convex quadratic programs

This paper introduces QPDO, a primal-dual method for convex quadratic programs which builds upon and weaves together the proximal point algorithm and a damped semismooth Newton method. The outer proximal regularization yields a numerically stable method, and we interpret the proximal operator as the unconstrained minimization of the primal-dual proximal augmented Lagrangian function. This allows the inner Newton scheme to exploit sparse symmetric linear solvers and multi-rank factorization updates. Moreover, the linear systems are always solvable independently from the problem data and exact linesearch can be performed. The proposed method can handle degenerate problems, provides a mechanism for infeasibility detection, and can exploit warm starting, while requiring only convexity. We present details of our open-source C implementation and report on numerical results against state-of-the-art solvers. QPDO proves to be a simple, robust, and efficient numerical method for convex quadratic programming.

Variable fixing method by weighted average for the continuous quadratic knapsack problem

<p style='text-indent:20px;'>We analyze the method of solving the separable convex continuous quadratic knapsack problem by weighted average from the viewpoint of variable fixing. It is shown that this method, considered as a variant of the variable fixing algorithms, and Kiwiel's variable fixing method generate the same iterates. We further improve the algorithm based on the analysis regarding the semismooth Newton method. Computational results are given and comparisons are made among the state-of-the-art algorithms. Experiments show that our algorithm has significantly good performance; it behaves very much like an <inline-formula><tex-math id="M1">\begin{document}$ O(n) $\end{document}</tex-math></inline-formula> algorithm with a very small constant.</p>

Dynamic Programming Through the Lens of Semismooth Newton-Type Methods

Policy iteration and value iteration are at the core of many (approximate) dynamic programming methods. For Markov Decision Processes with finite state and action spaces, we show that they are instances of semismooth Newton-type methods for solving the Bellman equation. In particular, we prove that policy iteration is equivalent to the exact semismooth Newton method and enjoys a local quadratic convergence rate. This finding is corroborated by extensive numerical evidence in the fields of control and operations research, which confirms that policy iteration generally requires relatively few iterations to achieve convergence even in presence of a large number of admissible policies. We then show that value iteration is an instance of the fixed-point iteration method and develop a novel locally accelerated version of value iteration with global convergence guarantees and negligible extra computational costs.

An efficient augmented Lagrangian method with semismooth Newton solver for total generalized variation

<p style='text-indent:20px;'>Total generalization variation (TGV) is a very powerful and important regularization for various inverse problems and computer vision tasks. In this paper, we propose a semismooth Newton based augmented Lagrangian method for solving this problem. The augmented Lagrangian method (also called as method of multipliers) is widely used for lots of smooth or nonsmooth variational problems. However, its efficiency heavily depends on solving the corresponding coupled and nonlinear system together and simultaneously. With efficient primal-dual semismooth Newton methods for the challenging and highly coupled nonlinear subproblems involving total generalized variation, we develop a highly efficient and competitive augmented Lagrangian method compared with some fast first-order method. With the analysis of the metric subregularities of the corresponding functions, we give both the global convergence and local linear convergence rate for the proposed augmented Lagrangian methods.</p>

Optimal Borel measure controls for the two-dimensional stationary Boussinesq system

We analyze an optimal control problem for the stationary two-dimensional Boussinesq system with controls taken in the space of regular Borel measures. Such measure-valued controls are known to produce sparse solutions. First-order and second-order necessary and sufficient optimality conditions are established. Following an optimize-then-discretize strategy, the corresponding finite element approximation will be solved by a semi-smooth Newton method initialized by a continuation strategy. The controls are discretized by finite linear combinations of Dirac measures concentrated at the nodes associated with the degrees of freedom for the mini-finite element.

Numerical Modeling of the Leak through Semipermeable Walls for 2D/3D Stokes Flow: Experimental Scalability of Dual Algorithms

The paper deals with the Stokes flow subject to the threshold leak boundary conditions in two and three space dimensions. The velocity–pressure formulation leads to the inequality type problem that is approximated by the P1-bubble/P1 mixed finite elements. The resulting algebraic system is nonsmooth. It is solved by the path-following variant of the interior point method, and by the active-set implementation of the semi-smooth Newton method. Inner linear systems are solved by the preconditioned conjugate gradient method. Numerical experiments illustrate scalability of the algorithms. The novelty of this work consists in applying dual strategies for solving the problem.