Finding the projection of a point onto the intersection of convex sets via projections onto half-spaces

We present a modification of Dykstra's algorithm which allows us to avoid projections onto general convex sets. Instead, we calculate projections onto either a half-space or onto the intersection of two half-spaces. Convergence of the algorithm is established and special choices of the half-spaces are proposed.The option to project onto half-spaces instead of general convex sets makes the algorithm more practical. The fact that the half-spaces are quite general enables us to apply the algorithm in a variety of cases and to generalize a number of known projection algorithms.The problem of projecting a point onto the intersection of closed convex sets receives considerable attention in many areas of mathematics and physics as well as in other fields of science and engineering such as image reconstruction from projections.In this work we propose a new class of algorithms which allow projection onto certain super half-spaces, i.e., half-spaces which contain the convex sets. Each one of the algorithms that we present gives the user freedom to choose the specific super half-space from a family of such half-spaces. Since projecting a point onto a half-space is an easy task to perform, the new algorithms may be more useful in practical situations in which the construction of the super half-spaces themselves is not too difficult.     

A storm of feasibility pumps for nonconvex MINLP

One of the foremost difficulties in solving Mixed-Integer Nonlinear Programs, either with exact or heuristic methods, is to find a feasible point. We address this issue with a new feasibility pump algorithm tailored for nonconvex Mixed-Integer Nonlinear Programs. Feasibility pumps are algorithms that iterate between solving a continuous relaxation and a mixed-integer relaxation of the original problems. Such approaches currently exist in the literature for Mixed-Integer Linear Programs and convex Mixed-Integer Nonlinear Programs: both cases exhibit the distinctive property that the continuous relaxation can be solved in polynomial time. In nonconvex Mixed-Integer Nonlinear Programming such a property does not hold, and therefore special care has to be exercised in order to allow feasibility pump algorithms to rely only on local optima of the continuous relaxation. Based on a new, high level view of feasibility pump algorithms as a special case of the well-known successive projection method, we show that many possible different variants of the approach can be developed, depending on how several different (orthogonal) implementation choices are taken. A remarkable twist of feasibility pump algorithms is that, unlike most previous successive projection methods from the literature, projection is ??naturally?? taken in two different norms in the two different subproblems. To cope with this issue while retaining the local convergence properties of standard successive projection methods we propose the introduction of appropriate norm constraints in the subproblems; these actually seem to significantly improve the practical performance of the approach. We present extensive computational results on the MINLPLib, showing the effectiveness and efficiency of our algorithm.     

Projected equation methods for approximate solution of large linear systems

We consider linear systems of equations and solution approximations derived by projection on a low-dimensional subspace. We propose stochastic iterative algorithms, based on simulation, which converge to the approximate solution and are suitable for very large-dimensional problems. The algorithms are extensions of recent approximate dynamic programming methods, known as temporal difference methods, which solve a projected form of Bellman’s equation by using simulation-based approximations to this equation, or by using a projected value iteration method.     

Linear Convergence of Iterative Soft-Thresholding

In this article a unified approach to iterative soft-thresholding algorithms for the solution of linear operator equations in infinite dimensional Hilbert spaces is presented. We formulate the algorithm in the framework of generalized gradient methods and present a new convergence analysis. As main result we show that the algorithm converges with linear rate as soon as the underlying operator satisfies the so-called finite basis injectivity property or the minimizer possesses a so-called strict sparsity pattern. Moreover it is shown that the constants can be calculated explicitly in special cases (i.e. for compact operators). Furthermore, the techniques also can be used to establish linear convergence for related methods such as the iterative thresholding algorithm for joint sparsity and the accelerated gradient projection method.     

Parareal Algorithms Applied to Stochastic Differential Equations with Conserved Quantities

In this paper, we couple the parareal algorithm with projection methods of the trajectory on a specific manifold, defined by the preservation of some conserved quantities of stochastic differential equations. First, projection methods are introduced as the coarse and fine propagators. Second, we apply the projection methods for systems with conserved quantities in the correction step of original parareal algorithm. Finally, three numerical experiments are performed by different kinds of algorithms to show the property of convergence in iteration, and preservation in conserved quantities of model systems.     

Optimization methods and stability of inclusions in Banach spaces

Our paper deals with the interrelation of optimization methods and Lipschitz stability of multifunctions in arbitrary Banach spaces. Roughly speaking, we show that linear convergence of several first order methods and Lipschitz stability mean the same. Particularly, we characterize calmness and the Aubin property by uniformly (with respect to certain starting points) linear convergence of descent methods and approximate projection methods. So we obtain, e.g., solution methods (for solving equations or variational problems) which require calmness only. The relations of these methods to several known basic algorithms are discussed, and errors in the subroutines as well as deformations of the given mappings are permitted. We also recall how such deformations are related to standard algorithms like barrier, penalty or regularization methods in optimization.     

On variable-step relaxed projection algorithm for variational inequalities

Projection algorithms are practically useful for solving variational inequalities (VI). However some among them require the knowledge related to VI in advance, such as Lipschitz constant. Usually it is impossible in practice. This paper studies the variable-step basic projection algorithm and its relaxed version under weakly co-coercive condition. The algorithms discussed need not know constant/function associated with the co-coercivity or weak co-coercivity and the step-size is varied from one iteration to the next. Under certain conditions the convergence of the variable-step basic projection algorithm is established. For the practical consideration, we also give the relaxed version of this algorithm, in which the projection onto a closed convex set is replaced by another projection at each iteration and latter is easy to calculate. The convergence of relaxed scheme is also obtained under certain assumptions. Finally we apply these two algorithms to the Split Feasibility Problem (SFP).     

凸可行问题的平行近似次梯度投影算法

对凸可行问题提出了包括上松弛的平行近似次梯度投影算法和加速平行近似次梯度投影算法．与序列近似次梯度投影算法相比, 平行近似次梯度投影算法(每次迭代同时运用多个凸集的近似次梯度超平面上的投影)能够保证迭代序列收敛到离各个凸集最近的点． 上松弛的迭代技术和含有外推因子的加速技术的应用, 减少了数据存储量, 提高了收 敛速度. 最后在较弱的条件下证明了算法的收敛性, 数值实验结果验证了算法的有效性和优越性.     

Shrinking gradient descent algorithms for total variation regularized image denoising

Total variation regularization introduced by Rudin, Osher, and Fatemi (ROF) is widely used in image denoising problems for its capability to preserve repetitive textures and details of images. Many efforts have been devoted to obtain efficient gradient descent schemes for dual minimization of ROF model, such as Chambolle’s algorithm or gradient projection (GP) algorithm. In this paper, we propose a general gradient descent algorithm with a shrinking factor. Both Chambolle’s and GP algorithm can be regarded as the special cases of the proposed methods with special parameters. Global convergence analysis of the new algorithms with various step lengths and shrinking factors are present. Numerical results demonstrate their competitiveness in computational efficiency and reconstruction quality with some existing classic algorithms on a set of gray scale images.     

The relaxed inexact projection methods for the split feasibility problem

In this paper we present several relaxed inexact projection methods for the split feasibility problem (SFP). Each iteration of the first proposed algorithm consists of a projection onto a halfspace containing the given closed convex set. The algorithm can be implemented easily and its global convergence to the solution can be established under suitable conditions. Moreover,we present some modifications of the relaxed inexact projection method with constant stepsize by adopting Armijo-like search. We furthermore present a variable-step relaxed inexact projection method which does not require the computation of the matrix inverses and the largest eigenvalue of the matrix A^TA, and the objective function can decrease sufficiently at each iteration. We show convergence of these modified algorithms under mild conditions. Finally, we perform some numerical experiments, which show the behavior of the algorithms proposed.     

Interior projection-like methods for monotone variational inequalities

We propose new interior projection type methods for solving monotone variational inequalities. The methods can be viewed as a natural extension of the extragradient and hyperplane projection algorithms, and are based on using non Euclidean projection-like maps. We prove global convergence results and establish rate of convergence estimates. The projection-like maps are given by analytical formulas for standard constraints such as box, simplex, and conic type constraints, and generate interior trajectories. We then demonstrate that within an appropriate primal-dual variational inequality framework, the proposed algorithms can be applied to general convex constraints resulting in methods which at each iteration entail only explicit formulas and do not require the solution of any convex optimization problem. As a consequence, the algorithms are easy to implement, with low computational cost, and naturally lead to decomposition schemes for problems with a separable structure. This is illustrated through examples for convex programming, convex-concave saddle point problems and semidefinite programming.The work of this author was partially supported by the United States–Israel Binational Science Foundation, BSF Grant No. 2002-2010.     

Two algorithms for solving systems of inclusion problems

The goal of this paper is to present two algorithms for solving systems of inclusion problems, with all components of the systems being a sum of two maximal monotone operators. The algorithms are variants of the forward-backward splitting method and one being a hybrid with the alternating projection method. They consist of approximating the solution sets involved in the problem by separating half-spaces which is a well-studied strategy. The schemes contain two parts, the first one is an explicit Armijo-type search in the spirit of the extragradient-like methods for variational inequalities. The second part is the projection step, this being the main difference between the algorithms. While the first algorithm computes the projection onto the intersection of the separating half-spaces, the second chooses one component of the system and projects onto the separating half-space of this case. In the iterative process, the forward-backward operator is computed once per inclusion problem, representing a relevant computational saving if compared with similar algorithms in the literature. The convergence analysis of the proposed methods is given assuming monotonicity of all operators, without Lipschitz continuity assumption. We also present some numerical experiments.     

Projected subgradient methods with non-Euclidean distances for non-differentiable convex minimization and variational inequalities

We study subgradient projection type methods for solving non-differentiable convex minimization problems and monotone variational inequalities. The methods can be viewed as a natural extension of subgradient projection type algorithms, and are based on using non-Euclidean projection-like maps, which generate interior trajectories. The resulting algorithms are easy to implement and rely on a single projection per iteration. We prove several convergence results and establish rate of convergence estimates under various and mild assumptions on the problem’s data and the corresponding step-sizes. We dedicate this paper to Boris Polyak on the occasion of his 70th birthday.     

Computational acceleration of projection algorithms for the linear best approximation problem

This is an experimental computational account of projection algorithms for the linear best approximation problem. We focus on the sequential and simultaneous versions of Dykstra’s algorithm and the Halpern-Lions-Wittmann-Bauschke algorithm for the best approximation problem from a point to the intersection of closed convex sets in the Euclidean space. These algorithms employ different iterative approaches to reach the same goal but no mathematical connection has yet been found between their algorithmic schemes. We compare these algorithms on linear best approximation test problems that we generate so that the solution will be known a priori and enable us to assess the relative computational merits of these algorithms. For the simultaneous versions we present a new component-averaging variant that substantially accelerates their initial behavior for sparse systems.     

Convex feasibility modeling and projection methods for sparse signal recovery

A computationally-efficient method for recovering sparse signals from a series of noisy observations, known as the problem of compressed sensing (CS), is presented. The theory of CS usually leads to a constrained convex minimization problem. In this work, an alternative outlook is proposed. Instead of solving the CS problem as an optimization problem, it is suggested to transform the optimization problem into a convex feasibility problem (CFP), and solve it using feasibility-seeking sequential and simultaneous subgradient projection methods, which are iterative, fast, robust and convergent schemes for solving CFPs. As opposed to some of the commonly-used CS algorithms, such as Bayesian CS and Gradient Projections for sparse reconstruction, which become inefficient as the problem dimension and sparseness degree increase, the proposed methods exhibit robustness with respect to these parameters. Moreover, it is shown that the CFP-based projection methods are superior to some of the state-of-the-art methods in recovering the signal’s support. Numerical experiments show that the CFP-based projection methods are viable for solving large-scale CS problems with compressible signals.     

Inertial projection and contraction algorithms for variational inequalities

In this article, we introduce an inertial projection and contraction algorithm by combining inertial type algorithms with the projection and contraction algorithm for solving a variational inequality in a Hilbert space H. In addition, we propose a modified version of our algorithm to find a common element of the set of solutions of a variational inequality and the set of fixed points of a nonexpansive mapping in H. We establish weak convergence theorems for both proposed algorithms. Finally, we give the numerical experiments to show the efficiency and advantage of the inertial projection and contraction algorithm.     

Further applications of a splitting algorithm to decomposition in variational inequalities and convex programming

A classical method for solving the variational inequality problem is the projection algorithm. We show that existing convergence results for this algorithm follow from one given by Gabay for a splitting algorithm for finding a zero of the sum of two maximal monotone operators. Moreover, we extend the projection algorithm to solveany monotone affine variational inequality problem. When applied to linear complementarity problems, we obtain a matrix splitting algorithm that is simple and, for linear/quadratic programs, massively parallelizable. Unlike existing matrix splitting algorithms, this algorithm converges under no additional assumption on the problem. When applied to generalized linear/quadratic programs, we obtain a decomposition method that, unlike existing decomposition methods, can simultaneously dualize the linear constraints and diagonalize the cost function. This method gives rise to highly parallelizable algorithms for solving a problem of deterministic control in discrete time and for computing the orthogonal projection onto the intersection of convex sets.This research is partially supported by the U.S. Army Research Office, contract DAAL03-86-K-0171 (Center for Intelligent Control Systems), and by the National Science Foundation under grant NSF-ECS-8519058.Thanks are due to Professor J.-S. Pang for his helpful comments.     

一族超线性收敛的投影拟牛顿算法

本文将梯度投影与拟牛顿法相结合，给出了求解一般线性约束非线性规划问题含两组参数的算法族．在一定的条件下证明了算法族的全局收敛性与它的子族的超线性收敛速度，并给出了投影Ｄ．Ｆ．Ｐ方法、投影ＢＦＧＳ方法等一些特例．     

Some properties of the recursive projection and interpolation algorithms

The recursive projection algorithm (rpa), derived from the vectorSylvester identity, has a connection with the recursive interpolationalgorithm (ria). These algorithms have many applications andconnections with some other methods used in various areas ofnumerical analysis. The aim of this paper is to study some propertiesof these algorithms. We use properties of projectors, a matriximplementation and the Schur complement extended to the vectorcase, to give a result about their finite termination.     

Charged ball method for solving some computational geometry problems

The concept of replacement of the initial stationary optimization problem with some nonstationary mechanical system tending with time to the position of equilibrium, which coincides with the solution of the initial problem, makes it possible to construct effective numerical algorithms. First, differential equations of the movement should be derived. Then we pass to the difference scheme and define the iteration algorithm. There is a wide class of optimization methods constructed in such a way. One of the most known representatives of this class is the heavy ball method. As a rule, such type of algorithms includes parameters that highly affect the convergence rate. In this paper, the charged ball method, belonging to this class, is proposed and investigated. It is a new effective optimization method that allows solving some computational geometry problems. A problem of orthogonal projection of a point onto a convex closed set with a smooth boundary and the problem of finding the minimum distance between two such sets are considered in detail. The convergence theorems are proved, and the estimates for the convergence rate are found. Numerical examples illustrating the work of the proposed algorithms are given.