An inexact algorithm with proximal distances for variational inequalities

In this paper we introduce an inexact proximal point algorithm using proximal distances for solving variational inequality problems when the mapping is pseudomonotone or quasimonotone. Under some natural assumptions we prove that the sequence generated by the algorithm is convergent for the pseudomonotone case and assuming an extra condition on the solution set we prove the convergence for the quasimonotone case. This approach unifies the results obtained by Auslender et al. [Math Oper. Res. 24 (1999) 644–688] and Brito et al. [J. Optim. Theory Appl. 154 (2012) 217–234] and extends the convergence properties for the class of φ-divergence distances and Bregman distances.

The Viscosity Approximation Forward-Backward Splitting Method for Zeros of the Sum of Monotone Operators

We investigate the convergence analysis of the following general inexact algorithm for approximating a zero of the sum of a cocoercive operatorAand maximal monotone operatorsBwithD(B)⊂H:xn+1=αnf(xn)+γnxn+δn(I+rnB)-1(I-rnA)xn+en, forn=1,2,…,for givenx1in a real Hilbert spaceH, where(αn),(γn), and(δn)are sequences in(0,1)withαn+γn+δn=1for alln≥1,(en)denotes the error sequence, andf:H→His a contraction. The algorithm is known to converge under the following assumptions onδnanden: (i)(δn)is bounded below away from 0 and above away from 1 and (ii)(en)is summable in norm. In this paper, we show that these conditions can further be relaxed to, respectively, the following: (i)(δn)is bounded below away from 0 and above away from 3/2 and (ii)(en)is square summable in norm; and we still obtain strong convergence results.

A Smoothing Inexact Newton Method for Nonlinear Complementarity Problems

A smoothing inexact Newton method is presented for solving nonlinear complementarity problems. Different from the existing exact methods, the associated subproblems are not necessary to be exactly solved to obtain the search directions. Under suitable assumptions, global convergence and superlinear convergence are established for the developed inexact algorithm, which are extensions of the exact case. On the one hand, results of numerical experiments indicate that our algorithm is effective for the benchmark test problems available in the literature. On the other hand, suitable choice of inexact parameters can improve the numerical performance of the developed algorithm.