Sanja Rapajić

Globally convergent Jacobian smoothing inexact Newton methods for NCP

Abstract A new smoothing algorithm for the solution of nonlinear complementarity problems (NCP) is introduced in this paper. It is based on semismooth equation reformulation of NCP by Fischer–Burmeister function and its related smooth approximation. In each iteration the corresponding linear system is solved only approximately. Since inexact directions are not necessarily descent, a nonmonotone technique is used for globalization procedure. Numerical results are also presented.

How to improve MAOR method convergence area for linear complementarity problems

The linear complementarity problem can be solved by modified AOR method given in [Appl. Math. Comput. 140 (2003) 53]. In the same paper the convergence was proved for the H-matrix case, using the estimation of spectral radius of corresponding matrix. In this paper we present the other possibility for obtaining convergence result. We use the estimation of maximum norm, and surprisingly, obtain convergence area which can be better. First, we consider SDD (strictly diagonally dominant) matrix case, and after that H-matrix case.

FR type methods for systems of large-scale nonlinear monotone equations

A large class of iterative methods for solving nonlinear monotone systems is developed in recent years. In this paper we propose some new FR type directions in the frame of algorithm which is a combination of conjugate gradient approach and hyperplane projection technique. Derivative-free, function-value-based line search combined with projection procedure is used for globalization strategy. Numerical performances of methods with different search directions are compared.

Conductivity and dielectric behaviour of indium substituted zinc ferrites prepared by coprecipitation method

This paper presents the results concerning dielectric behavior and conductivity of the nanosized Zn1−xInxFe2O4 powders (x = 0, 0.15, 0.2, and 0.3), obtained by coprecipitation method. The frequency dependence of dielectric permittivity and conductivity of the samples is determined in the frequency range of 1–105 Hz, at temperatures from 300–350 K, while the temperature dependence of conductivity was recorded at 100 Hz, 10 kHz and 100 kHz. The ac conductivity was found to follow universal dielectric response, which is typical for charge transport by hopping or tunneling processes. Analyzing the variation of the parameter n (as a measure of the degree of correlation between conductivity and frequency), with the temperatures we discuss the possible conduction mechanism in investigated samples. Qualitatively, non-overlapping small polarons (NSPT) are usually associated with increase in n with increasing temperature, while correlated barrier hopping (CBH) shows a decrease in n with increasing T.

Finite-difference method for singular nonlinear systems

This paper presents a method for solving nonlinear system with singular Jacobian at the solution. The convergence rate in the case of singularity deteriorates and one way to accelerate convergence is to form bordered system. A local algorithm, with finite-difference approximations, for forming and solving such system is proposed in this paper. To overcome the need that initial approximation has to be very close to the solution, we also propose a method which is a combination of descent method with finite-differences and local algorithm. Some numerical results obtained on relevant examples are presented.

A nonmonotone Jacobian smoothing inexact Newton method for NCP

In this paper we propose Jacobian smoothing inexact Newton method for nonlinear complementarity problems (NCP) with derivative-free nonmonotone line search. This nonmonotone line search technique ensures globalization and is a combination of Grippo-Lampariello-Lucidi (GLL) and Li-Fukushima (LF) strategies, with the aim to take into account their advantages. The method is based on very well known Fischer-Burmeister reformulation of NCP and its smoothing Kanzow’s approximation. The mixed Newton equation, which combines the semismooth function with the Jacobian of its smooth operator, is solved approximately in every iteration, so the method belongs to the class of Jacobian smoothing inexact Newton methods. The inexact search direction is not in general a descent direction and this is the reason why nonmonotone scheme is used for globalization. Global convergence and local superlinear convergence of method are proved. Numerical performances are also analyzed and point out that high level of nonmonotonicity of this line search rule enables robust and efficient method.

Barzilai–Borwein method with variable sample size for stochastic linear complementarity problems

A smoothing method for solving stochastic linear complementarity problems is proposed. The expected residual minimization reformulation of the problem is considered, and it is approximated by the sample average approximation (SAA). The proposed method is based on sequential solving of a sequence of smoothing problems where each of the smoothing problems is defined with its own sample average approximation. A nonmonotone line search with a variant of the Barzilai–Borwein (BB) gradient direction is used for solving each of the smoothing problems. The BB search direction is efficient and low cost, particularly suitable for nonmonotone line search procedure. The variable sample size scheme allows the sample size to vary across the iterations and the method tends to use smaller sample size far away from the solution. The key point of this strategy is a good balance between the variable sample size strategy, the smoothing sequence and nonmonotonicity. Eventually, the maximal sample size is used and the SAA problem is solved. Presented numerical results indicate that the proposed strategy reduces the overall computational cost.

Integration of pseudo-polynomials based on g-integrals

The natural generalization of classical polynomials are the pseudo-polynomials and the pseudo-polynomials with interval coefficients too. The main topic of this paper is the g-integration of special functions, i.e. the g-integration of pseudo-polynomials.

Iterative method with modification of the right-hand side vector for nonlinear complementarity problems

Nonlinear complementarity problems (NCPs) are often solved by iterative methods based on a generalization of the classical Newton method and its modifications for smooth equations. We consider a method with modification of the right-hand side vector for a reformulation of the semi-smooth equation arising from NCPs and prove local convergence of the proposed method. Some numerical results are presented.