一类非单调线性互补问题的高阶仿射尺度算法

In this paper, a new interior point algorithm-high-order atone scaling for a class of nonmonotonic linear complementary problems is developed. On the basis of idea of primal-dual affine scaling method for linear programming , the search direction of our algorithm is obtained by a linear system of equation at each step . We show that, by appropriately choosing the step size, the algorithm has polynomial time complexity. We also give the numberical results of the algorithm for two test problems.     

An affine scaling method with an infeasible starting point: Convergence analysis under nondegeneracy assumption

In this paper, we propose an infeasible-interior-point algorithm for linear programning based on the affine scaling algorithm by Dikin. The search direction of the algorithm is composed of two directions, one for satisfying feasibility and the other for aiming at optimality. Both directions are affine scaling directions of certain linear programming problems. Global convergence of the algorithm is proved under a reasonable nondegeneracy assumption. A summary of analogous global convergence results without any nondegeneracy assumption obtained in [17] is also given.     

一类非单调线性互补问题的高阶Dikin型仿射尺度算法

对于一类非单调线性互补问题提出了一个新算法：高阶Dikin型仿射尺度算法，算法的每步迭代．基于线性规划Dikin原始-对偶算法思想来求解一个线性方程组得到迭代方向，再适当选取步长，得到了算法的多项式复杂性。     

Limiting behavior of the affine scaling continuous trajectories for linear programming problems

We consider the continuous trajectories of the vector field induced by the primal affine scaling algorithm as applied to linear programming problems in standard form. By characterizing these trajectories as solutions of certain parametrized logarithmic barrier families of problems, we show that these trajectories tend to an optimal solution which in general depends on the starting point. By considering the trajectories that arise from the Lagrangian multipliers of the above mentioned logarithmic barrier families of problems, we show that the trajectories of the dual estimates associated with the affine scaling trajectories converge to the so called centered optimal solution of the dual problem. We also present results related to asymptotic direction of the affine scaling trajectories. We briefly discuss how to apply our results to linear programs formulated in formats different from the standard form. Finally, we extend the results to the primal-dual affine scaling algorithm.     

Global convergence of the affine scaling algorithm for primal degenerate strictly convex quadratic programming problems

In this paper we deal with global convergence of the affine scaling algorithm for strictly convex QP problems satisfying a dual nondegeneracy condition. By means of the local Karmarkar potential function which was successfully applied to demonstrate global convergence of the affine scaling algorithm for LP, we show global convergence of the algorithm when the step-size 1/8 is adopted without requiring any primal nondegeneracy condition.This paper was presented at the 14th International Symposium on Mathematical Programming, held August 5–9, 1991, in Amsterdam, The Netherlands.     

Superlinearly convergent affine scaling interior trust-region method for linear constrained LC<Superscript>1</Superscript> minimization

We extend the classical affine scaling interior trust region algorithm for the linear constrained smooth minimization problem to the nonsmooth case where the gradient of objective function is only locally Lipschitzian. We propose and analyze a new affine scaling trust-region method in association with nonmonotonic interior backtracking line search technique for solving the linear constrained LC1 optimization where the second-order derivative of the objective function is explicitly required to be locally Lipschitzian. The general trust region subproblem in the proposed algorithm is defined by minimizing an augmented affine scaling quadratic model which requires both first and second order information of the objective function subject only to an affine scaling ellipsoidal constraint in a null subspace of the augmented equality constraints. The global convergence and fast local convergence rate of the proposed algorithm are established under some reasonable conditions where twice smoothness of the objective function is not required. Applications of the algorithm to some nonsmooth optimization problems are discussed.     

Affine scaling algorithm fails for semidefinite programming

In this paper, we introduce an affine scaling algorithm for semidefinite programming (SDP), and give an example of a semidefinite program such that the affine scaling algorithm converges to a non-optimal point. Both our program and its dual have interior feasible solutions and unique optimal solutions which satisfy strict complementarity, and they are non-degenerate everywhere.     

Hopfield neural networks in large-scale linear optimization problems

Hopfield neural networks and affine scaling interior point methods are combined in a hybrid approach for solving linear optimization problems. The Hopfield networks perform the early stages of the optimization procedures, providing enhanced feasible starting points for both primal and dual affine scaling interior point methods, thus facilitating the steps towards optimality. The hybrid approach is applied to a set of real world linear programming problems. The results show the potential of the integrated approach, indicating that the combination of neural networks and affine scaling interior point methods can be a good alternative to obtain solutions for large-scale optimization problems.     

A Note on Mascarenhas' Counterexample about Global Convergence of the Affine Scaling Algorithm

Mascarenhas gave an instance of linear programming problems to show that the long-step affine scaling algorithm can fail to converge to an optimal solution with the step-size λ=0.999 . In this note, we give a simple and clear geometrical explanation for this phenomenon in terms of the Newton barrier flow induced by projecting the homogeneous affine scaling vector field conically onto a hyperplane where the objective function is constant. Based on this interpretation, we show that the algorithm can fail for "any" λ greater than about 0.91 (a more precise value is 0.91071), which is considerably shorter than λ = 0.95 and 0.99 recommended for efficient implementations. Accepted 17 February 1998     

An affine scaling interior trust-region method combining with line search filter technique for optimization subject to bounds on variables

This paper proposes and analyzes an affine scaling trust-region method with line search filter technique for solving nonlinear optimization problems subject to bounds on variables. At the current iteration, the trial step is generated by the general trust-region subproblem which is defined by minimizing a quadratic function subject only to an affine scaling ellipsoidal constraint. Both trust-region strategy and line search filter technique will switch to trail backtracking step which is strictly feasible. Meanwhile, the proposed method does not depend on any external restoration procedure used in line search filter technique. A new backtracking relevance condition is given which is weaker than the switching condition to obtain the global convergence of the algorithm. The global convergence and fast local convergence rate of this algorithm are established under reasonable assumptions. Preliminary numerical results are reported indicating the practical viability and show the effectiveness of the proposed algorithm.     

An Affine Scaling Interior Trust Region Method via Optimal Path for Solving Monotone Variational Inequality Problem with Linear Constraints

Based on a differentiable merit function proposed by Taji et al. in ＂Math. Prog. Stud., 58, 1993, 369-383＂, the authors propose an affine scaling interior trust region strategy via optimal path to modify Newton method for the strictly monotone variational inequality problem subject to linear equality and inequality constraints. By using the eigensystem decomposition and affine scaling mapping, the authors form an affine scaling optimal curvilinear path very easily in order to approximately solve the trust region subproblem. Theoretical analysis is given which shows that the proposed algorithm is globally convergent and has a local quadratic convergence rate under some reasonable conditions.     

Superlinear convergence of affine scaling interior point newton method for linear inequality constrained minimization without strict complementarity

In this paper we extend and improve the classical affine scaling interior-point Newton method for solving nonlinear optimization subject to linear inequality constraints in the absence of the strict complementarity assumption. Introducing a computationally efficient technique and employing an identification function for the definition of the new affine scaling matrix, we propose and analyze a new affine scaling interior-point Newton method which improves the Coleman and Li affine sealing matrix in [2] for solving the linear inequlity constrained optimization. Local superlinear and quadratical convergence of the proposed algorithm is established under the strong second order sufficiency condition without assuming strict complementarity of the solution.     

线性不等式约束的广义非线性互补问题的仿射内点信赖域方法

提供了一种新的非单调内点回代线搜索技术的仿射内点信赖域方法解线性不等式约束的广义非线性互补问题(GCP).基于广义互补问题构成的半光滑方程组的广义Jacobian矩阵,算法使用l2范数作为半光滑方程组的势函数,形成的信赖域子问题为一个带椭球约束的线性化的二次模型.利用广义牛顿方程计算试探迭代步,通过内点映射回代技术确保迭代点是严格内点,保证了算法的整体收敛性.在合理的条件下,证明了信赖域算法在接近最优点时可转化为广义拟牛顿步,进而具有局部超线性收敛速率.非单调技术将克服高度非线性情况加速收敛进展.最后,数值结果表明了算法的有效性.     

线性不等式约束的广义非线性互补问题的仿射内点信赖域方法

提供了一种新的非单调内点回代线搜索技术的仿射内点信赖域方法解线性不等式约束的广义非线性互补问题(GCP).基于广义互补问题构成的半光滑方程组的广义Jacobian矩阵,算法使用l_2范数作为半光滑方程组的势函数,形成的信赖域子问题为一个带椭球约束的线性化的二次模型.利用广义牛顿方程计算试探迭代步,通过内点映射回代技术确保迭代点是严格内点,保证了算法的整体收敛性.在合理的条件下,证明了信赖域算法在接近最优点时可转化为广义拟牛顿步,进而具有局部超线性收敛速率.非单调技术将克服高度非线性情况加速收敛进展.最后,数值结果表明了算法的有效性.     

Affine scaling interior Levenberg–Marquardt method for bound-constrained semismooth equations under local error bound conditions

We develop and analyze a new affine scaling Levenberg–Marquardt method with nonmonotonic interior backtracking line search technique for solving bound-constrained semismooth equations under local error bound conditions. The affine scaling Levenberg–Marquardt equation is based on a minimization of the squared Euclidean norm of linear model adding a quadratic affine scaling matrix to find a solution that belongs to the bounded constraints on variable. The global convergence results are developed in a very general setting of computing trial directions by a semismooth Levenberg–Marquardt method where a backtracking line search technique projects trial steps onto the feasible interior set. We establish that close to the solution set the affine scaling interior Levenberg–Marquardt algorithm is shown to converge locally Q-superlinearly depending on the quality of the semismooth and Levenberg–Marquardt parameter under an error bound assumption that is much weaker than the standard nonsingularity condition, that is, BD-regular condition under nonsmooth case. A nonmonotonic criterion should bring about speed up the convergence progress in the contours of objective function with large curvature.     

A trust region affine scaling method for bound constrained optimization

We study a new trust region affine scaling method for general bound constrained optimization problems. At each iteration, we compute two trial steps. We compute one along some direction obtained by solving an appropriate quadratic model in an ellipsoidal region. This region is defined by an affine scaling technique. It depends on both the distances of current iterate to boundaries and the trust region radius. For convergence and avoiding iterations trapped around nonstationary points, an auxiliary step is defined along some newly defined approximate projected gradient. By choosing the one which achieves more reduction of the quadratic model from the two above steps as the trial step to generate next iterate, we prove that the iterates generated by the new algorithm are not bounded away from stationary points. And also assuming that the second-order sufficient condition holds at some nondegenerate stationary point, we prove the Q-linear convergence of the objective function values. Preliminary numerical experience for problems with bound constraints from the CUTEr collection is also reported.     

Monotone Variable–Metric Algorithm for Linearly Constrained Nonlinear Programming

A new method for linearly constrained nonlinear programming is proposed. This method follows affine scaling paths defined by systems of ordinary differential equations and it is fully parallelizable. The convergence of the method is proved for a nondegenerate problem with pseudoconvex objective function. In practice, the algorithm works also under more general assumptions on the objective function. Numerical results obtained with this computational method on several test problems are shown.     

Affine scaling inexact generalized Newton algorithm with interior backtracking technique for solving bound-constrained semismooth equations

We develop and analyze an affine scaling inexact generalized Newton algorithm in association with nonmonotone interior backtracking line technique for solving systems of semismooth equations subject to bounds on variables. By combining inexact affine scaling generalized Newton with interior backtracking line search technique, each iterate switches to inexact generalized Newton backtracking step to strict interior point feasibility. The global convergence results are developed in a very general setting of computing trial steps by the affine scaling generalized Newton-like method that is augmented by an interior backtracking line search technique projection onto the feasible set. Under some reasonable conditions we establish that close to a regular solution the inexact generalized Newton method is shown to converge locally p-order q-superlinearly. We characterize the order of local convergence based on convergence behavior of the quality of the approximate subdifferentials and indicate how to choose an inexact forcing sequence which preserves the rapid convergence of the proposed algorithm. A nonmonotonic criterion should bring about speeding up the convergence progress in some ill-conditioned cases.     

Global and local convergence of a new affine scaling trust region algorithm for linearly constrained optimization

Chen and Zhang [Sci.China,Ser.A,45,1390–1397(2002)] introduced an affine scaling trust region algorithm for linearly constrained optimization and analyzed its global convergence.In this paper,we derive a new affine scaling trust region algorithm with dwindling filter for linearly constrained optimization.Different from Chen and Zhang's work,the trial points generated by the new algorithm are accepted if they improve the objective function or improve the first order necessary optimality conditions.Under mild conditions,we discuss both the global and local convergence of the new algorithm.Preliminary numerical results are reported.     

Improved complexity using higher-order correctors for primal-dual Dikin affine scaling

In this paper we show that the primal-dual Dikin affine scaling algorithm for linear programming of Jansen. Roos and Terlaky enhances an asymptotical $O(\sqrt n L)$ complexity by using corrector steps. We also show that the result remains valid when the method is applied to positive semi-definite linear complementarity problems.