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In this paper, for solving the vertical linear complementarity problem (VLCP) effectively, a two-step new modulus-based matrix splitting (TNMMS) iteration method is introduced. Its convergence properties are discussed in depth. The efficiency of the proposed method is confirmed by some numerical experiments from the 2-D boundary problem. Numerical results show that the proposed method is superior to some existing methods, such as the two modulus-based matrix splitting (TMMS) iteration method (Numer. Algor., 2011, 57: 83–99) and the new modulus-based matrix splitting (NMMS) iteration method (Optim., 2023, 72: 2499–2516). [ABSTRACT FROM AUTHOR]

The paper deals with a variant of the interior-point method for the minimization of strictly quadratic objective function subject to simple bounds and separable quadratic inequality constraints. Such minimizations arise from the finite element approximation of contact problems of linear elasticity with friction in three space dimensions. The main goal of the paper is the convergence analysis of the algorithm and its implementation. The optimal preconditioners for solving ill-conditioned inner linear systems are proposed. Numerical experiments illustrate the computational efficiency for large-scale problems. [ABSTRACT FROM AUTHOR]

This paper first proposes a trust region algorithm to obtain a stationary point of unconstrained multiobjective optimisation problem. Under suitable assumptions, the global convergence of the new algorithm is established. We then extend the trust region method to solve the non-smooth multiobjective optimisation problem. [ABSTRACT FROM PUBLISHER]

In this paper, a mixed QP-free method for constrained optimization problem is proposed. This new method consists of two different sub-methods, sub-method I and sub-method II. These two sub-methods are all QP-free. A switch is designed to decide which sub-method should be adopted in each iteration. Under some mild assumptions, we prove that the method is globally and superlinearly convergent. [ABSTRACT FROM AUTHOR]

In this paper we study numerical stability of path-based algorithms for the traffic assignment problem. These algorithms are based on decomposition of the original problem into smaller sub-problems which are optimized sequentially. Previously, path-based algorithms were numerically tested only in the setting of moderate requirements to the level of solution precision. In this study we analyse convergence of these methods when the convergence measure approaches machine epsilon of IEEE double precision format. In particular, we demonstrate that the straightforward implementation of one of the algorithms of this group (projected gradient) suffers from loss of precision and is not able to converge to highly precise solution. We propose a way to solve this problem and test the proposed adjusted version of the algorithm on various benchmark instances. [ABSTRACT FROM AUTHOR]

In this paper we propose an augmented Lagrangian trust region method for equality constrained optimization. Different from standard augmented Lagrangian methods which minimize the augmented Lagrangian function for fixed Lagrange multiplier and penalty parameter at each iteration, the proposed method tries to minimize its second-order approximation function. We propose a new strategy for adjusting the penalty parameter. With adaptive update of Lagrange multipliers, we prove the global convergence of the proposed method. Numerical results on test problems from the CUTEr collection are also reported. [ABSTRACT FROM PUBLISHER]

Following the classical exponential penalty function method of mathematical programming, the exponential penalty function method of multiobjective programming problems (MOPP) is constructed and its convergence is proved. In addition, the approach is applied to solving a finite min-max MOPP. [ABSTRACT FROM AUTHOR]

In this paper, we propose a derivative-free quasi-Newton condition, which results in a new class of quasi-Newton updating formulas for unconstrained optimization. Each updating formula in this class is a rank-two updating formula and preserves the positive definiteness of the second derivative matrix of the quadratic model. Its first two terms are the same as the first two terms of the BFGS updating formula. We establish global convergence of quasi-Newton methods based upon the updating formulas in this class, and superlinear convergence of a special quasi-Newton method among them. Then we propose a special quasi-Newton updating formula, which repetitively uses the new quasi-Newton condition. This updating formula is derivative-free. Numerical results are reported. [ABSTRACT FROM AUTHOR]

The effect of dimensionality on the widely used Nelder–Mead simplex method for unconstrained optimization is investigated. It is shown that by using the quadratic function f (x)=x T x, the Nelder–Mead simplex method deteriorates as the dimension increases. [ABSTRACT FROM AUTHOR]

Two transformations of gradient-descent iterative methods for solving unconstrained optimization are proposed. The first transformation is called modification and it is defined using a small enlargement of the step size in various gradient-descent methods. The second transformation is termed as hybridization and it is defined as a composition of gradient-descent methods with the Picard–Mann hybrid iterative process. As a result, several accelerated gradient-descent methods for solving unconstrained optimization problems are presented, investigated theoretically and numerically compared. The proposed methods are globally convergent for uniformly convex functions satisfying certain condition under the assumption that the step size is determined by the backtracking line search. In addition, the convergence on strictly convex quadratic functions is discussed. Numerical comparisons show better behaviour of the proposed methods with respect to some existing methods in view of the Dolan and Moré's performance profile with respect to all analysed characteristics: number of iterations, the CPU time, and the number of function evaluations. [ABSTRACT FROM AUTHOR]

Nonlinear gradient dynamic approach for solving the tensor complementarity problem (TCP) are presented. Theoretical analysis shows that each of the defined dynamical system models ensures the convergence. The computer-simulation results further substantiate that the considered dynamical system can solve the TCP. [ABSTRACT FROM AUTHOR]

Two different approaches based on eigenvalues and singular values of the matrix representing the search direction in conjugate gradient algorithms are considered. Using a special approximation of the inverse Hessian of the objective function, which depends by a positive parameter, we get the search direction which satisfies both the sufficient descent condition and the Dai–Liao’s conjugacy condition. In the first approach the parameter in the search direction is determined by clustering the eigenvalues of the matrix defining it. The second approach uses the minimizing the condition number of the matrix representing the search direction. In this case the obtained conjugate gradient algorithm is exactly the three-term conjugate gradient algorithm proposed by Zhang, Zhou and Li. The global convergence of the algorithms is proved for uniformly convex functions. Intensive numerical experiments, using 800 unconstrained optimization test problems, prove that both these approaches have similar numerical performances. We prove that both algorithms are significantly more efficient and more robust than CG-DESCENT algorithm by Hager and Zhang. By solving five applications from the MINPACK-2 test problem collection, withvariables, we show that the suggested conjugate gradient algorithms are top performer versus CG-DESCENT. [ABSTRACT FROM AUTHOR]

This article deals with minimizing quadratic functions with a special form of quadratic constraints that arise in 3D contact problems of linear elasticity with isotropic friction [Haslinger, J., Kucera, R. and Dostál, Z., 2004, An algorithm for the numerical realization of 3D contact problems with Coulomb friction. Journal of Computational and Applied Mathematics, 164/165, 387-408.]. The proposed algorithm combines the active set method with the conjugate gradient method. Its general scheme is similar to the algorithms of Polyak's type that solve the quadratic programming problems with simple bounds. As the algorithm does not terminate in a finite number of steps, the convergence is proved. The implementation uses an adaptive precision control of the conjugate gradient loops. Numerical experiments demonstrate the computational efficiency of the method. [ABSTRACT FROM AUTHOR]

The approximate solution of the problem of controlling an initial value problem for a linear system of autonomous ordinary differential equations is considered. The corresponding homogeneous solution to the differential equation is assumed to be non-expansive and the inhomogeneity is a linear function of the control variable that is constant along a priori given sub-intervals. The optimal control minimises a convex functional that depends, possibly in a non-linear way, on the solution of the differential equation. Infinite time horizons are allowed. In view of the piecewise constant control, the corresponding Lagrangian can be split into the sum of Lagrangians acting on sub-intervals. The two algorithms suggested are based on an iterative process that takes advantage of this splitting as well as of the explicit solution to the differential constraints. Convergence results are provided under suitable assumptions on the problem’s data. Finally, numerical tests for a model of global warming demonstrate the performance of the algorithms. [ABSTRACT FROM AUTHOR]

We consider the stochastic nonlinear complementarity problem (SNCP). We first formulate the problem as a stochastic mathematical program with equilibrium constraints and then, in order to develop efficient algorithms, we give some reformulations of the problem. Furthermore, based on the reformulations, we propose a smoothed penalty method for solving SNCP. A rigorous convergence analysis is also given. [ABSTRACT FROM AUTHOR]