Nonlinear Saddle Point Problems Research Articles

Generalized variational framework with minimax optimization for parametric blind deconvolution

Blind deconvolution (BD), which aims to separate unknown convolved signals, is a fundamental problem in signal processing. Due to the ill-posedness and underdetermination of the convolution system, it is a challenging nonlinear inverse problem. This paper is devoted to the algorithmic studies of parametric BD, which is typically applied to recover images from ad hoc optical modalities. We propose a generalized variational framework for parametric BD with various priors and potential functions. By using the conjugate theory in convex analysis, the framework can be cast into a nonlinear saddle point problem. We employ the recent advances in minimax optimization to solve the parametric BD by the nonlinear primal-dual hybrid gradient method, with all subproblems admitting closed-form solutions. Numerical simulations on synthetic and real datasets demonstrate the compelling performance of the minimax optimization approach for solving parametric BD.

Efficient and effective algebraic splitting‐based solvers for nonlinear saddle point problems

The incremental Picard Yosida (IPY) method has recently been developed as an iteration for nonlinear saddle point problems that is as effective as Picard but more efficient. By combining ideas from algebraic splitting of linear saddle point solvers with incremental Picard‐type iterations and grad‐div stabilization, IPY improves on the standard Picard method by allowing for easier linear solves at each iteration—but without creating more total nonlinear iterations compared to Picard. This paper extends the IPY methodology by studying it together with Anderson acceleration (AA). We prove that IPY for Navier–Stokes and regularized Bingham fits the recently developed analysis framework for AA, which implies that AA improves the linear convergence rate of IPY by scaling the rate with the gain of the AA optimization problem. Numerical tests illustrate a significant improvement in convergence behavior of IPY methods from AA, for both Navier–Stokes and regularized Bingham.

A Peaceman–Rachford Splitting Method with Monotone Plus Skew-Symmetric Splitting for Nonlinear Saddle Point Problems

This paper is devoted to solving the linearly constrained convex optimization problems by Peaceman–Rachford splitting method with monotone plus skew-symmetric splitting on KKT operators. This approach generalizes the Hermitian and skew-Hermitian splitting method, an unconditionally convergent algorithm for non-Hermitian positive definite linear systems, to the nonlinear scenario. The convergence of the proposed algorithm is guaranteed under some mild assumptions, e.g., the strict convexity on objective functions and the consistency on constraints, even though the Lions–Mercier property is not fulfilled. In addition, we explore an inexact version of the proposed algorithm, which allows solving the subproblems approximately with some inexactness criteria. Numerical simulations on an image restoration problem demonstrate the compelling performance of the proposed algorithm.

Convergence Analysis of the Inexact Uzawa Algorithm for Nonlinear Saddle Point Problems

Convergence Analysis of the Inexact Uzawa Algorithm for Nonlinear Saddle Point Problems

An Inexact Uzawa Algorithmic Framework for Nonlinear Saddle Point Problems with Applications to Elliptic Optimal Control Problem

We consider a class of nonlinear saddle point problems with various applications in PDEs and optimal control problems and propose an algorithmic framework based on some inexact Uzawa methods in the...

Nonlinear multigrid solvers exploiting AMGe coarse spaces with approximation properties

Abstract This paper introduces a nonlinear multigrid solver for mixed finite element discretizations based on the Full Approximation Scheme (FAS) and element-based Algebraic Multigrid (AMGe). The AMGe coarse spaces with approximation properties used in this work enable us to overcome the difficulties in evaluating the nonlinear coarse operators and the degradation in convergence rates that characterized previous attempts to extend FAS to algebraic multilevel hierarchies on general unstructured grids. Specifically, the AMGe technique employed in this paper allows to derive stable and accurate coarse discretizations on general unstructured grids for a large class of nonlinear partial differential equations, including saddle point problems. The approximation properties of the coarse spaces ensure that our FAS approach for general unstructured meshes leads to optimal mesh-independent convergence rates similar to those achieved by geometric FAS on a nested hierarchy of refined meshes. In the numerical results, Newton’s method and Picard iterations with state-of-the-art inner linear solvers are compared to our FAS algorithm for the solution of a nonlinear saddle point problem arising from porous media flow applications. Our approach outperforms – both in terms of number of iterations and computational time – traditional methods in all the experiments.

On the Numerical Solution of the Stationary Power-Law Stokes Equations: A Penalty Finite Element Approach

In this work, we study the penalty finite element approximation of the stationary power law Stokes problem. We prove uniform convergence of the finite element solution with respect to the penalized parameter under classical assumptions on the weak solution. We formulate and analyze the convergence of a nonlinear saddle point problem by adopting a particular algorithm based on vanishing viscosity approach and long time behavior of an initial value problem. Finally, the predictions observed theoretically are validated by means of numerical experiments.

The nonlinear inexact Uzawa hybrid algorithms based on one-step Newton method for solving nonlinear saddle-point problems

This paper mainly consider the construction of efficient Uzawa-type algorithms for solving the nonlinear saddle-point problem (NSPP). Two nonlinear inexact Uzawa hybrid algorithms based on one-step Newton method are proposed for solving the NSPPs. By using the so-called energy-norm, the convergence of the proposed algorithms is verified under some practical conditions. In addition, some numerical results are reported to illustrate the behavior of the considered algorithm.

Analysis of the inexact Uzawa algorithms for nonlinear saddle point problems

Inexact Uzawa algorithms for solving nonlinear saddle-point problems are proposed. A simple sufficient condition for the convergence of the inexact Uzawa algorithms is obtained. Numerical experiments show that the inexact Uzawa algorithms are convergent. doi:10.1017/S1446181110000829

ANALYSIS OF THE INEXACT UZAWA ALGORITHMS FOR NONLINEAR SADDLE-POINT PROBLEMS

AbstractInexact Uzawa algorithms for solving nonlinear saddle-point problems are proposed. A simple sufficient condition for the convergence of the inexact Uzawa algorithms is obtained. Numerical experiments show that the inexact Uzawa algorithms are convergent.

A Saddle Point Approach to the Computation of Harmonic Maps

In this paper we consider numerical approximations of a constraint minimization problem, where the object function is a quadratic Dirichlet functional for vector fields and the interior constraint is given by a convex function. The solutions of this problem are usually referred to as harmonic maps. The solution is characterized by a nonlinear saddle point problem, and the corresponding linearized problem is well-posed near strict local minima. The main contribution of the present paper is to establish a corresponding result for a proper finite element discretization in the case of two space dimensions. Iterative schemes of Newton type for the discrete nonlinear saddle point problems are investigated, and mesh independent preconditioners for the iterative methods are proposed.

Nonlinear Inexact Uzawa Algorithms for Linear and Nonlinear Saddle-point Problems

This paper proposes some nonlinear Uzawa methods for solving linear and nonlinear saddle-point problems. A nonlinear inexact Uzawa algorithm is first introduced for linear saddle-point problems. Two different PCG techniques are allowed in the inner and outer iterations of the algorithm. This algorithm is then extended for a class of nonlinear saddle-point problems arising from some convex optimization problems with linear constraints. For this extension, some PCG method used in the inner iteration needs to be carefully constructed so that it converges in a certain energy norm instead of the usual $l^2$-norm. It is shown that the new algorithm converges under some practical conditions and there is no need for any a priori estimates on the minimal and maximal eigenvalues of the two local preconditioned systems involved. The two new methods perform more efficiently than the existing methods in the cases where no good preconditioners are available for the Schur complements.

An Interior-Point Method for a Class of Saddle-Point Problems

We present a polynomial-time interior-point algorithm for a class of nonlinear saddle-point problems that involve semidefiniteness constraints on matrix variables. These problems originate from robust optimization formulations of convex quadratic programming problems with uncertain input parameters. As an application of our approach, we discuss a robust formulation of the Markowitz portfolio selection model.

A Dual-Dual Formulation for the Coupling of Mixed-FEM and BEM in Hyperelasticity

We combine a dual-mixed finite element method with the boundary integral equation method, to study the solvability and Galerkin approximations of a linear-nonlinear transmission problem arising in plane elastostatics. Our approach is based on the introduction of the strain tensor as an auxiliary unknown in the finite element domain, and it leads to what we call a dual-dual mixed formulation since the corresponding operator equation becomes of a twofold saddle-point structure. We derive existence and uniqueness of solution for the continuous and discrete formulations and provide the associated error analysis. In particular, we extend the finite element space for plane elasticity given by the PEERS element and define an explicit finite element/boundary element subspace satisfying all the required compatibility conditions. Most of our analysis makes use of an extension of the classical Babuska--Brezzi theory to a class of nonlinear saddle-point problems.

Global and Superlinear Convergence of Inexact Uzawa Methods for Saddle Point Problems with Nondifferentiable Mappings

This paper investigates inexact Uzawa methods for nonlinear saddle point problems. We prove that the inexact Uzawa method converges globally and superlinearly even if the derivative of the nonlinear mapping does not exist. We show that the Newton-type decomposition method for saddle point problems is a special case of a Newton--Uzawa method. We discuss applications of inexact Uzawa methods to separable convex programming problems and coupling of finite elements/boundary elements for nonlinear interface problems.

A Predictor-Corrector Algorithm for a Class of Nonlinear Saddle Point Problems

An interior path-following algorithm is proposed for solving the nonlinear saddle point problem $$ {\rm minimax}\ c^Tx+\ph(x)+b^Ty-\psi(y)-y^TAx $$ \vspace*{-18pt} $$ {\rm subject\ to\ }(x,y)\in \X\ti \Y\su R^n\ti R^m, $$ \noindent where $\ph(x)$ and $\ps(y)$ are smooth convex functions and $\X$ and $\Y$ are boxes (hyperrectangles). This problem is closely related to the models in stochastic programming and optimal control studied by Rockafellar and Wets (Math. Programming Studies, 28 (1986), pp. 63--93; SIAM J. Control Optim., 28 (1990), pp. 810--822). Existence and error-bound results on a central path are derived. Starting from an initial solution near the central path with duality gap $O(\mu)$, the algorithm finds an $\ep$-optimal solution of the problem in $O(\sqrt{m+n}\,|\log\mu/\ep|)$ iterations if both $\ph(x)$ and $\ps(y)$ satisfy a scaled Lipschitz condition.