A globally linearly convergent method for pointwise quadratically supportable convex–concave saddle point problems

We study the Proximal Alternating Predictor–Corrector (PAPC) algorithm introduced recently by Drori, Sabach and Teboulle [8] to solve nonsmooth structured convex–concave saddle point problems consisting of the sum of a smooth convex function, a finite collection of nonsmooth convex functions and bilinear terms. We introduce the notion of pointwise quadratic supportability, which is a relaxation of a standard strong convexity assumption and allows us to show that the primal sequence is R-linearly convergent to an optimal solution and the primal-dual sequence is globally Q-linearly convergent. We illustrate the proposed method on total variation denoising problems and on locally adaptive estimation in signal/image deconvolution and denoising with multiresolution statistical constraints.     

A convergent FEM-DG method for the compressible Navier–Stokes equations

This paper presents a new numerical method for the compressible Navier–Stokes equations governing the flow of an ideal isentropic gas. To approximate the continuity equation, the method utilizes a discontinuous Galerkin discretization on piecewise constants and a basic upwind flux. For the momentum equation, the method is a new combined discontinuous Galerkin and finite element method approximating the velocity in the Crouzeix–Raviart finite element space. While the diffusion operator is discretized in a standard fashion, the convection and time-derivative are discretized using discontinuous Galerkin on the element average velocity and a Lax–Friedrich type flux. Our main result is convergence of the method to a global weak solution as discretization parameters go to zero. The convergence analysis constitutes a numerical version of the existence analysis of Lions and Feireisl.     

A Gauss—Newton method for convex composite optimization

An extension of the Gauss—Newton method for nonlinear equations to convex composite optimization is described and analyzed. Local quadratic convergence is established for the minimization ofh F under two conditions, namelyh has a set of weak sharp minima,C, and there is a regular point of the inclusionF(x) C. This result extends a similar convergence result due to Womersley (this journal, 1985) which employs the assumption of a strongly unique solution of the composite functionh F. A backtracking line-search is proposed as a globalization strategy. For this algorithm, a global convergence result is established, with a quadratic rate under the regularity assumption.This material is based on research supported by National Science Foundation Grants CCR-9157632 and DMS-9102059, the Air Force Office of Scientific Research Grant F49620-94-1-0036, and the United States—Israel Binational Science Foundation Grant BSF-90-00455.     

A globally convergent primal—dual interior point algorithm for convex programming

In this paper, we study the global convergence of a large class of primal—dual interior point algorithms for solving the linearly constrained convex programming problem. The algorithms in this class decrease the value of a primal—dual potential function and hence belong to the class of so-called potential reduction algorithms. An inexact line search based on Armijo stepsize rule is used to compute the stepsize. The directions used by the algorithms are the same as the ones used in primal—dual path following and potential reduction algorithms and a very mild condition on the choice of the centering parameter is assumed. The algorithms always keep primal and dual feasibility and, in contrast to the polynomial potential reduction algorithms, they do not need to drive the value of the potential function towards — in order to converge. One of the techniques used in the convergence analysis of these algorithms has its root in nonlinear unconstrained optimization theory.Research supported in part by NSF Grant DDM-9109404.     

A branch-and-bound method for the minimum k−enclosing ball problem

The minimum k-enclosing ball problem seeks the ball with smallest radius that contains at least k of m given points. This problem is NP-hard. We present a branch-and-bound algorithm on the tree of the subsets of k points to solve this problem. Our method is able to solve the problem exactly in a short amount of time for small and medium sized datasets.     

A Newton’s method for the continuous quadratic knapsack problem

We introduce a new efficient method to solve the continuous quadratic knapsack problem. This is a highly structured quadratic program that appears in different contexts. The method converges after $O(n)$ iterations with overall arithmetic complexity $O(n^2)$ . Numerical experiments show that in practice the method converges in a small number of iterations with overall linear complexity, and is faster than the state-of-the-art algorithms based on median finding, variable fixing, and secant techniques.     

A Jacobi–Davidson type method for the product eigenvalue problem

We propose a Jacobi–Davidson type method to compute selected eigenpairs of the product eigenvalue problem _Am?_A1x=λx,$A_m?A_{1}x=\lambda x,$ where the matrices may be large and sparse. To avoid difficulties caused by a high condition number of the product matrix, we split up the action of the product matrix and work with several search spaces. We generalize the Jacobi–Davidson correction equation and the harmonic and refined extraction for the product eigenvalue problem. Numerical experiments indicate that the method can be used to compute eigenvalues of product matrices with extremely high condition numbers.     

A control type method for solving the Cauchy–Stokes problem

In this work, we propose an approximate optimal control formulation of the Cauchy problem for the Stokes system. Here the problem is converted into an optimization one. In order to handle the instability of the solution of this ill-posed problem, a regularization technique is developed. We add a term in the least square function which happens to vanish while the algorithm converges. The efficiency of the proposed method is illustrated by numerical experiments.     

Polyak’s gradient method for split feasibility problem constrained by level sets

In this paper, we are concerned with the split feasibility problem (SFP) whenever the convex sets involved are composed of level sets. By applying Polyak’s gradient method, we get a new and simple algorithm for such a problem. Under standard assumptions, we prove that the whole sequence generated by the algorithm weakly converges to a solution. We also modify the proposed algorithm and state the strong convergence without regularity conditions on the sets involved. Numerical experiments are included to illustrate its applications in signal processing.     

A mixed finite element method for the unilateral contact problem in elasticity

In this paper, we provide a new mixed finite element approximation of the varia-tional inequality resulting from the unilateral contact problem in elasticity. We use the continuous piecewise P2-P1 finite element to approximate the displacement field and the normal stress component on the contact region. Optimal convergence rates are obtained under the reasonable regularity hypotheses. Numerical example verifies our results.     

A smoothing Levenberg–Marquardt method for the extended linear complementarity problem

We consider the extended linear complementarity problem (XLCP), of which the linear and horizontal linear complementarity problems are two special cases. We reformulate the XLCP to a smooth equation by using some smoothing functions and propose a Levenberg–Marquardt method to solve the system of smooth equation. The global convergence and local superlinear convergence rate are established under certain conditions. Numerical tests show the effectiveness of the proposed algorithm.     

A globally and quadratically convergent method for absolute value equations

We investigate the NP-hard absolute value equation (AVE) Ax−|x|=b, where A is an arbitrary n×n real matrix. In this paper, we propose a smoothing Newton method for the AVE. When the singular values of A exceed 1, we show that this proposed method is globally convergent and the convergence rate is quadratic. Preliminary numerical results show that this method is promising.     

Comment on “A new three-term conjugate gradient method for unconstrained problem”

Here, necessary corrections on computing an optimal parameter of the TTDES method of Andrei (Numer. Math. 68(2), 305–321 2015) are stated in brief. Throughout, we use the same notations and equation numbers as in N.Andrei.     

Bifurcation near the origin for the Robin problem with concave–convex nonlinearities

In this Note, we deal with the Robin parametric elliptic equation driven by a nonhomogeneous differential operator and with a reaction that exhibits competing terms (concave–convex nonlinearities). Without employing the Ambrosetti–Rabinowitz condition, we prove a bifurcation theorem for small positive values of the real parameter.     

A primal–dual regularized interior-point method for convex quadratic programs

Interior-point methods in augmented form for linear and convex quadratic programming require the solution of a sequence of symmetric indefinite linear systems which are used to derive search directions. Safeguards are typically required in order to handle free variables or rank-deficient Jacobians. We propose a consistent framework and accompanying theoretical justification for regularizing these linear systems. Our approach can be interpreted as a simultaneous proximal-point regularization of the primal and dual problems. The regularization is termedexact to emphasize that, although the problems are regularized, the algorithm recovers a solution of the original problem, for appropriate values of the regularization parameters.     

Successive projection method for solving the unbalanced Procrustes problem

We present a successive projection method for solving the unbalanced Procrustes problem: given matrix A∈Rn×n and B∈Rn×k, n>k, minimize the residual‖AQ-B‖F with the orthonormal constraint QTQ = Ik on the variant Q∈Rn×k. The presented algorithm consists of solving k least squares problems with quadratic constraints and an expanded balance problem at each sweep. We give a detailed convergence analysis. Numerical experiments reported in this paper show that our new algorithm is superior to other existing methods.     

Bifurcation near infinity for the Neumann problem with concave–convex nonlinearities

In this Note, we study a class of Neumann parametric elliptic equations driven by a nonhomogeneous differential operator and with a reaction that exhibits competing terms (concave–convex nonlinearities). Using the Ambrosetti–Rabinowitz condition and related topological and variational arguments, we prove a bifurcation result for large values of the parameter.     

A new method of solving the coefficient inverse problem

This paper is concerned with the new method for solving the coefficient inverse problem in the reproducing kernel space. It is different from the previous studies. This method gives accurate results and shows that it is valid by the numerical example.