Naihua Xiu

A note on the CQ algorithm for the split feasibility problem

Let C and Q be nonempty closed convex sets in N and M, respectively, and A an M ? N real matrix. The split feasibility problem (SFP) is to find x C with Ax Q, if such x exist. Byrne (2002 Inverse Problems 18 441?53) proposed a CQ algorithm with the following iterative scheme: where ? (0, 2/L), L denotes the largest eigenvalue of the matrix ATA, and PC and PQ denote the orthogonal projections onto C and Q, respectively. In his algorithm, Byrne assumed that the projections PC and PQ are easily calculated. However, in some cases it is impossible or needs too much work to exactly compute the orthogonal projection. Recently, Yang (2004 Inverse Problems 20 1261?6) presented a relaxed CQ algorithm, in which he replaced PC and PQ by and , that is, the orthogonal projections onto two halfspaces Ck and Qk, respectively. Clearly, the latter is easy to implement. One common advantage of the CQ algorithm and the relaxed CQ algorithm is that computation of the matrix inverses is not necessary. However, they use a fixed stepsize related to the largest eigenvalue of the matrix ATA, which sometimes affects convergence of the algorithms. In this paper, we present modifications of the CQ algorithm and the relaxed CQ algorithm by adopting Armijo-like searches. The modified algorithms need not compute the matrix inverses and the largest eigenvalue of the matrix ATA, and make a sufficient decrease of the objective function at each iteration. We also show convergence of the modified algorithms under mild conditions.

Some recent advances in projection-type methods for variational inequalities

Projection-type methods are a class of simple methods for solving variational inequalities, especially for complementarity problems. In this paper we review and summarize recent developments in this class of methods, and focus mainly on some new trends in projection-type methods.

A Regularized Smoothing Newton Method for Symmetric Cone Complementarity Problems

This paper extends the regularized smoothing Newton method in vector complementarity problems to symmetric cone complementarity problems (SCCP), which includes the nonlinear complementarity problem, the second-order cone complementarity problem, and the semidefinite complementarity problem as special cases. In particular, we study strong semismoothness and Jacobian nonsingularity of the total natural residual function for SCCP. We also derive the uniform approximation property and the Jacobian consistency of the Chen-Mangasarian smoothing function of the natural residual. Based on these properties, global and quadratical convergence of the proposed algorithm is established.

Unified framework of extragradient-type methods for pseudomonotone variational inequalities

In this paper, we propose a unified framework of extragradient-type methods for solving pseudomonotone variational inequalities, which allows one to take different stepsize rules and requires the computation of only two projections at each iteration. It is shown that the modified extragradient method of Ref. 1 falls within this framework with a short stepsize and so does the method of Ref. 2 with a long stepsize. It is further demonstrated that the algorithmic framework is globally convergent under mild assumptions and is sublinearly convergent if in addition a projection-type error bound holds locally. Preliminary numerical experiments are reported.

The sparsest solutions to Z-tensor complementarity problems

Finding the sparsest solutions to a tensor complementarity problem is generally NP-hard due to the nonconvexity and noncontinuity of the involved

Some projection-like methods for the generalized Nash equilibria

\ell _0

Improved Approximation Algorithms for the Facility Location Problems with Linear/Submodular Penalties

ℓ0 norm. In this paper, a special type of tensor complementarity problems with Z-tensors has been considered. Under some mild conditions, we show that to pursuit the sparsest solutions is equivalent to solving polynomial programming with a linear objective function. The involved conditions guarantee the desired exact relaxation and also allow to achieve a global optimal solution to the relaxed nonconvex polynomial programming problem. Particularly, in comparison to existing exact relaxation conditions, such as RIP-type ones, our proposed conditions are easy to verify.

Regularizing active set method for nonnegatively constrained ill-posed multichannel image restoration problem

The implicit Lagrangian was first proposed by Mangasarian and Solodov as a smooth merit function for the nonnegative orthant complementarity problem. It has attracted much attention in the past ten years because of its utility in reformulating complementarity problems as unconstrained minimization problems. In this paper, exploiting the Jordan-algebraic structure, we extend it to the vector-valued implicit Lagrangian for symmetric cone complementary problem (SCCP), and show that it is a continuously differentiable complementarity function for SCCP and whose Jacobian is strongly semismooth. As an application, we develop the real-valued implicit Lagrangian and the corresponding smooth merit function for SCCP, and give a necessary and sufficient condition for the stationary point of the merit function to be a solution of SCCP. Finally, we show that this merit function can provide a global error bound for SCCP with the uniform Cartesian P-property.