A non-monotone inexact regularized smoothing Newton method for solving nonlinear complementarity problems

In the paper [S.P. Rui and C.X. Xu, A smoothing inexact Newton method for nonlinear complementarity problems, J. Comput. Appl. Math. 233 (2010), pp. 2332–2338], the authors proposed an inexact smoothing Newton method for nonlinear complementarity problems (NCP) with the assumption that F is a uniform P function. In this paper, we present a non-monotone inexact regularized smoothing Newton method for solving the NCP which is based on Fischer–Burmeister smoothing function. We show that the proposed algorithm is globally convergent and has a locally superlinear convergence rate under the weaker condition that F is a P ₀ function and the solution of NCP is non-empty and bounded. Numerical results are also reported for the test problems, which show the effectiveness of the proposed algorithm.     

The second-order cone eigenvalue complementarity problem

The eigenvalue complementarity problem (EiCP) differs from the traditional eigenvalue problem in that the primal and dual variables belong to a closed and convex cone K and its dual, respectively, and satisfy a complementarity condition. In this paper we investigate the solution of the second-order cone EiCP (SOCEiCP) where K is the Lorentz cone. We first show that the SOCEiCP reduces to a special Variational Inequality Problem on a compact set defined by K and a normalization constraint. This guarantees that SOCEiCP has at least one solution, and a new enumerative algorithm is introduced for finding a solution to this problem. The method is based on finding a global minimum of an appropriate nonlinear programming (NLP) formulation of the SOCEiCP using a special branching scheme along with a local nonlinear optimizer that computes stationary points on subsets of the feasible region of NLP associated with the nodes generated by the algorithm. A semi-smooth Newton's method is combined with this enumerative algorithm to enhance its numerical performance. Our computational experience illustrates the efficacy of the proposed techniques in practice.     

A smooth Newton method with 3-1 piecewise NCP function for generalized nonlinear complementarity problem

In this paper, based on the 3-1 piecewise nonlinear complementarity problem (NCP) function, we proposed a smoothing Newton-type method for the generalized nonlinear complementarity problem (GNCP) with a modified non-monotone line search. The algorithm for a GNCP is more difficult than that for an NCP, because two functions must be considered in the problem. We reformulate the (GNCP) to a smoothing system of equations by two independent variables, and then develop a smoothing Newton-type method for solving it. Under reasonable conditions, we obtain the global convergent properties. Also, the numerical experiments are reported in this paper.     

A globally convergent trust region multidimensional filter SQP algorithm for nonlinear programming

We propose a trust region multidimensional filter SQP algorithm. The multidimensional filter technique proposed by Gould et al. [SIAM J. Optim., 15 (2005), pp. 17–38] is extended to solve constrained optimization problems. The constraints are partitioned into p parts. The entry of our filter consists of these different parts. Not only the criteria for accepting a trial step would be relaxed, but also the individual behaviour of each part of the constraints is considered. The filter's entries and the acceptance criteria are different from other filter-related algorithms in the literature. It should be noted that the undesirable link between the objective function and the constraint violation function in the filter acceptance criteria disappears. Our algorithm is also combined with the non-monotone technique for accepting a trial step, which leads to a more flexible acceptance criteria. Under mild conditions, global convergence is proved. Numerical results show the robustness and efficiency of our algorithm.     

A new SQP algorithm based on semidefinite programming

The paper gives a new filter-SQP algorithm of the semidefinite programming through combining the sequence nonlinear algorithm of solving the semidefinite programming, low rank conversion, and filter method, which, under certain conditions, proves that the algorithm is global convergence.     

Efficient DC programming approaches for the asymmetric eigenvalue complementarity problem

In this paper, we propose nonlinear programming (NLP) formulations and difference of convex functions (DC) programming approaches for the asymmetric eigenvalue complementarity problem (EiCP). The EiCP has a solution if and only if these NLP formulations have zero global optimal value. We reformulate the NLP formulations as DC programs which can be efficiently solved by a DC algorithm. Some preliminary numerical results illustrate the good performance of the proposed methods.     

非线性互补问题的粒子群算法

针对非线性互补问题求解的困难,利用粒子群算法并结合极大熵函数法给出了该类问题的一种新的有效算法。该算法首先利用极大熵函数将非线性互补问题转化为一个无约束最优化问题,将该函数作为粒子群算法的适应值函数;然后应用粒子群算法来优化该问题。数值结果表明,该算法收敛快、数值稳定性较好,是求解非线性互补问题的一种有效算法。     

A monotone Newton multisplitting method for the nonlinear complementarity problem with a nonlinear source term

In this paper, we develop a Newton multisplitting method for the nonlinear complementarity problem with a nonlinear source term in which the multisplitting method is used as secondary iterations to approximate the solutions for the resulting linearized subproblems. We prove the monotone convergence theorem for the proposed method under proper conditions.     

A new non-monotone SQP algorithm for the minimax problem

In this paper, a new sequential quadratic programming (SQP) algorithm is proposed to solve the minimax problem which uses the idea of nonmonotonicity. The problem is transformed into an equivalent inequality constrained nonlinear optimization problem. In order to prevent the scaling problem, we do some modifications to the minimization problem. By the non-monotone SQP method, the new algorithm is globally convergent without using a penalty function. Furthermore, it is shown that the proposed method does not suffer from the Maratos effect, so the locally superlinear convergence is achieved. Numerical results suggest that our algorithm for solving the minmax problem is efficient and robust.     

The monotone convergence of a class of parallel nonlinear relaxation methods for nonlinear complementarity problems

We set up a class of parallel nonlinear multisplitting AOR methods by directly multisplitting the nonlinear mapping involved in the nonlinear complementarity problems. The different choices of the relaxation parameters can yield all the known and a lot of new relaxation methods, as well as a lot of new relaxed parallel nonlinear multisplitting methods for solving the nonlinear complementarity problems. The two-sided approximation properties and the influences on the convergence rates from the relaxation parameters about our new methods are shown, and sufficient conditions guaranteeing the methods to converge globally are discussed. Finally, a lot of numerical results show that our new methods are feasible and efficient.     

A feasible directions algorithm for nonlinear complementarity problems and applications in mechanics

Complementarity problems are involved in mathematical models of several applications in engineering, economy and different branches of physics. We mention contact problems and dynamics of multiple bodies systems in solid mechanics. In this paper we present a new feasible direction algorithm for nonlinear complementarity problems. This one begins at an interior point, strictly satisfying the inequality conditions, and generates a sequence of interior points that converges to a solution of the problem. At each iteration, a feasible direction is obtained and a line search performed, looking for a new interior point with a lower value of an appropriate potential function. We prove global convergence of the present algorithm and present a theoretical study about the asymptotic convergence. Results obtained with several numerical test problems, and also application in mechanics, are described and compared with other well known techniques. All the examples were solved very efficiently with the present algorithm, employing always the same set of parameters.     

A neural network based on the generalized Fischer-Burmeister function for nonlinear complementarity problems

In this paper, we consider a neural network model for solving the nonlinear complementarity problem (NCP). The neural network is derived from an equivalent unconstrained minimization reformulation of the NCP, which is based on the generalized Fischer-Burmeister function ?_p(a,b)=‖(a,b)‖_p-(a+b). We establish the existence and the convergence of the trajectory of the neural network, and study its Lyapunov stability, asymptotic stability as well as exponential stability. It was found that a larger p leads to a better convergence rate of the trajectory. Numerical simulations verify the obtained theoretical results.     

Hierarchical gradient methods for nonlinear LSQ problems

The idea of hierarchical gradient methods for optimization is considered. It is shown that the proposed approach provides powerful means to cope with some global convergence problems characteristic of the classical gradient methods. Concerning global convergence problems, four topics are addressed: The detour effect, the problem of multiscale models, the problem of highly ill-conditioned objective functions, and the problem of local-minima traps related to ambiguous regions of attractions. The great potential of hierarchical gradient algorithms is revealed through a hierarchical Gauss-Newton algorithm for unconstrained nonlinear least-squares problems. The algorithm, while maintaining a superlinear convergence rate like the common conjugate gradient or quasi-Newton methods, requires the evaluation of partial derivatives with respect to only one variable on each iteration. This property enables economized consumption of CPU time in case the computer codes for the derivatives are intensive CPU consumers, e.g., when the gradient evaluations of ODE or PDE models are produced by numerical differentiation. The hierarchical Gauss-Newton algorithm is extended to handle interval constraints on the variables and its effectiveness demonstrated by computational results.     

A fixed-point approach for nonlinear discrete boundary value problems

Existence results are established for second-order discrete boundary value problems.     

非线性互补问题的光滑化拟牛顿算法

为求解非线性互补问题,给出了一种新的基于光滑对称扰动Fischer-Burmeister函数的光滑化拟牛顿算法。该算法利用了无导数线搜索。数值实验表明,算法是有效的。     

Accelerating iterative algorithms for symmetric linear complementarity problems

We propose a method for accelerating iterative algorithms for solving symmetric linear complementarity problems. The method consists in performing a one-dimensional optimization in the direction generated by a splitting method even for non-descent directions. We give strong convergence proofs and present numerical experiments that justify using this acceleration.     

Minimax问题的一个超线性收敛的SQP算法

本文提出一个求解Minimax问题改进的SQP算法.对已有的算法降低计算工作量,减弱假设条件,并得到更好的全局收敛和超线性收敛性结果.最后数值实验表明算法是有效的.     

自由时间最优控制问题的一种控制向量参数化方法

针对自由时间最优控制问题,提出一种控制向量参数化（CVP）方法.通过引入时间尺度因子,将自由时间最优控制问题转化为固定时间问题,并将终端时刻作为优化参数.基于CVP方法,最优控制问题被转化为一个非线性规划（NLP）问题.建立目标和约束函数的Hamiltonian函数,通过求解伴随方程获得目标和约束函数的梯度,采用序列二次规划（SQP）方法获得问题的数值解.对于控制有切换结构的优化问题,给出了一种网格精细化策略,以提高控制质量.补料分批反应器最优控制问题的仿真实验验证了所提出方法的有效性.     

Stability results for polyhedral complementarity problems

In this work, we study the multivalued complementarity problem for polyhedral multifunctions under homogeneity assumptions. We employ an approach that consists in approximating the equivalent variational inequality formulation of the problem and studying the asymptotic behavior of sequences of solutions to these approximation problems. To do this, we employ results and the language of Variational Analysis. The novelty of this approach lies in the fact that it allows us to obtain not only existence results but also stability ones. We consider that our results can be used for developing numerical algorithms for solving multivalued complementarity problems.     

Minimum mean-squared deviation method for stochastic complementarity problems

In this paper, we propose a new reformulation for stochastic complementarity problems (SCPs). The new formulation is based on the minimum mean-squared deviation rule in statistics. Under mild conditions, we prove the existence of the solution of the new reformulation for SCP. Furthermore, we present a smoothing sample average approximation method for solving the problems. The convergence properties of the optimal solutions of the approximation problems are studied under mild conditions. Finally, some numerical results are listed as well.