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In this paper, we propose a general framework that provides a unified convergence analysis of the derivative-free projection-based method (DFPM) for solving large-scale constrained nonlinear monotone equations. The new results provide a complete picture on the convergence guarantees of DFPM and cover the existing relevant convergence results as special cases. Preliminary numerical experiment results are also reported to show the numerical performance of six line search schemes used in the existing DFPM. [ABSTRACT FROM AUTHOR]

The classical Dai–Kou scheme (SIAM J Optim 23(1):296–320, 2013), which is popularly referred to as CGOPT, represents one of the most numerically efficient methods for unconstrained optimization. This article exploits nice properties of the scheme together with the projection method to present an adaptive Dai–Kou type method for solving constrained system of nonlinear monotone equations. The new scheme utilizes the backtracking line search strategy by Zhang and Zhou (J Comput Appl Math 196:478–484, 2006) to determine the step-size in the algorithm. In addition, the scheme requires less memory to implement as it avoids computing Jacobian matrix. This attribute makes it an ideal choice for large-scale problems. Other attributes of the scheme include its ability to generate search directions that satisfy the vital condition for global convergence and its applications to signal and image reconstruction problems in compressive sensing. The global convergence of the new scheme is proved using basic assumptions and numerical experiments conducted suggests that the new approach has a clear edge over four iterative schemes in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we present two Dai-Yuan type iterative methods for solving large-scale systems of nonlinear monotone equations. The methods can be considered as extensions of the classical Dai-Yuan conjugate gradient method for unconstrained optimization. By employing two different approaches, the Dai-Yuan method is modified to develop two different search directions, which are combined with the hyperplane projection technique of Solodov and Svaiter. The first search direction was obtained by carrying out eigenvalue study of the search direction matrix of an adaptive DY scheme, while the second is obtained by minimizing the distance between two adaptive versions of the DY method. Global convergence of the methods are established under mild conditions and preliminary numerical results show that the proposed methods are promising and more effective compared to some existing methods in the literature. [ABSTRACT FROM AUTHOR]

Very recently, Gibali et al. (Optimization 66, 417-437 2017) proposed a method, called selective projection method (SPM) in this paper, for solving the variational inequality problem (VIP) defined on C:=⋂i=1mCi≠∅, where m ≥ 1 is an integer and {Ci}i=1m is a finite level set family of convex functions on a real Hilbert space H. For the current iterate xn, SPM updates xn+ 1 by projecting onto a half-space Cnin(⊃Cin) constructed by using the input data, where in ∈{1,2,⋯,m} is selected by a special rule. The prominent advantage of SPM is that it is concise and easy to implement. Gibali et al. proved its convergence in the Euclidean space H=ℝd. In this paper, we firstly prove the strong convergence of SPM in a general Hilbert space. The proof given in this paper is very different from that given by Gibali et al. We also extend SPM to solve VIP defined on the common fixed point set of finite nonexpansive self-mappings of H. Then, we estimate the convergence rate of SPM and its extension in the nonasymptotic sense. Finally, we give some preliminary numerical experiments which illustrate the advantage of SPM. [ABSTRACT FROM AUTHOR]

In this paper, based on a merit function of the split feasibility problem (SFP), we present a Newton projection method for solving it and analyze the convergence properties of the method. The merit function is differentiable and convex. But its gradient is a linear composite function of the projection operator, so it is nonsmooth in general. We prove that the sequence of iterates converges globally to a solution of the SFP as long as the regularization parameter matrix in the algorithm is chosen properly. Especially, under some local assumptions which are necessary for the case where the projection operator is nonsmooth, we prove that the sequence of iterates generated by the algorithm superlinearly converges to a regular solution of the SFP. Finally, some numerical results are presented. [ABSTRACT FROM AUTHOR]

This paper conducts variational analysis of circular programs, which form a new class of optimization problems in nonsymmetric conic programming, important for optimization theory and its applications. First, we derive explicit formulas in terms of the initial problem data to calculate various generalized derivatives/co-derivatives of the projection operator associated with the circular cone. Then we apply generalized differentiation and other tools of variational analysis to establish complete characterizations of full and tilt stability of locally optimal solutions to parameterized circular programs. [ABSTRACT FROM AUTHOR]

The problem of finding a vector with the fewest nonzero elements that satisfies an underdetermined system of linear equations is an NP-complete problem that is typically solved numerically via convex heuristics or nicely behaved nonconvex relaxations. In this paper we consider the elementary method of alternating projections (MAP) for solving the sparsity optimization problem without employing convex heuristics. In a parallel paper we recently introduced the restricted normal cone which generalizes the classical Mordukhovich normal cone and reconciles some fundamental gaps in the theory of sufficient conditions for local linear convergence of the MAP algorithm. We use the restricted normal cone together with the notion of superregularity, which is inherently satisfied for the affine sparse optimization problem, to obtain local linear convergence results with estimates for the radius of convergence of the MAP algorithm applied to sparsity optimization with an affine constraint. [ABSTRACT FROM AUTHOR]

The method of alternating projections (MAP) is a common method for solving feasibility problems. While employed traditionally to subspaces or to convex sets, little was known about the behavior of the MAP in the nonconvex case until 2009, when Lewis, Luke, and Malick derived local linear convergence results provided that a condition involving normal cones holds and at least one of the sets is superregular (a property less restrictive than convexity). However, their results failed to capture very simple classical convex instances such as two lines in a three-dimensional space. In this paper, we extend and develop the Lewis-Luke-Malick framework so that not only any two linear subspaces but also any two closed convex sets whose relative interiors meet are covered. We also allow for sets that are more structured such as unions of convex sets. The key tool required is the restricted normal cone, which is a generalization of the classical Mordukhovich normal cone. Numerous examples are provided to illustrate the theory. [ABSTRACT FROM AUTHOR]

Abstract: It is well known that the general variational inequalities are equivalent to the fixed point problems and the Wiener–Hopf equations. In this paper, we use these alternative equivalent formulations to suggest and analyze some new self-adaptive iterative methods for solving the general variational inequalities. Our results can be viewed as a significant extension of the previously known results for variational inequalities. An example is given to illustrate the efficiency of the proposed method. [Copyright &y& Elsevier]