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.     

A smoothing Newton method for second-order cone optimization based on a new smoothing function

A new smoothing function is given in this paper by smoothing the symmetric perturbed Fischer-Burmeister function. Based on this new smoothing function, we present a smoothing Newton method for solving the second-order cone optimization (SOCO). The method solves only one linear system of equations and performs only one line search at each iteration. Without requiring strict complementarity assumption at the SOCO solution, the proposed algorithm is shown to be globally and locally quadratically convergent. Numerical results demonstrate that our algorithm is promising and comparable to interior-point methods.     

Error bounds and strong upper semicontinuity for monotone affine variational inequalities

Global error bounds for possibly degenerate or nondegenerate monotone affine variational inequality problems are given. The error bounds are on an arbitrary point and are in terms of the distance between the given point and a solution to a convex quadratic program. For the monotone linear complementarity problem the convex program is that of minimizing a quadratic function on the nonnegative orthant. These bounds may form the basis of an iterative quadratic programming procedure for solving affine variational inequality problems. A strong upper semicontinuity result is also obtained which may be useful for finitely terminating any convergent algorithm by periodically solving a linear program.This material is based on research supported by Air Force Office of Scientific Research Grant AFOSR-89-0410 and National Science Foundation Grants CCR-9101801 and CCR-9157632.     

New Sequential Quadratically-Constrained Quadratic Programming Method of Feasible Directions and Its Convergence Rate

This paper discusses optimization problems with nonlinear inequality constraints and presents a new sequential quadratically-constrained quadratic programming (NSQCQP) method of feasible directions for solving such problems. At each iteration. the NSQCQP method solves only one subproblem which consists of a convex quadratic objective function, convex quadratic equality constraints, as well as a perturbation variable and yields a feasible direction of descent (improved direction). The following results on the NSQCQP are obtained: the subproblem solved at each iteration is feasible and solvable: the NSQCQP is globally convergent under the Mangasarian-Fromovitz constraint qualification (MFCQ); the improved direction can avoid the Maratos effect without the assumption of strict complementarity; the NSQCQP is superlinearly and quasiquadratically convergent under some weak assumptions without thestrict complementarity assumption and the linear independence constraint qualification (LICQ). Research supported by the National Natural Science Foundation of China Project 10261001 and Guangxi Science Foundation Projects 0236001 and 0249003. The author thanks two anonymous referees for valuable comments and suggestions on the original version of this paper.     

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.     

A general parametric analysis approach and its implication to sensitivity analysis in interior point methods

Adler and Monteiro (1992) developed a parametric analysis approach that is naturally related to the geometry of the linear program. This approach is based on the availability of primal and dual optimal solutions satisfying strong complementarity. In this paper, we develop an alternative geometric approach for parametric analysis which does not require the strong complementarity condition. This parametric analysis approach is used to develop range and marginal analysis techniques which are suitable for interior point methods. Two approaches are developed, namely the LU factorization approach and the affine scaling approach. Presented at the ORSA/TIMS, Nashville, TN, USA, May 1991. Supported by the National Science Foundation (NSF) under Grant No. DDM-9109404 and Grant No. DMI-9496178. This work was done while the author was a faculty member of the Systems and Industrial Engineering Department at The University of Arizona. Supported in part by the GTE Laboratories and the National Science Foundation (NSF) under Grant No. CCR-9019469.     

Inexact Newton methods for the nonlinear complementarity problem

An exact Newton method for solving a nonlinear complementarity problem consists of solving a sequence of linear complementarity subproblems. For problems of large size, solving the subproblems exactly can be very expensive. In this paper we study inexact Newton methods for solving the nonlinear, complementarity problem. In such an inexact method, the subproblems are solved only up to a certain degree of accuracy. The necessary accuracies that are needed to preserve the nice features of the exact Newton method are established and analyzed. We also discuss some extensions as well as an application. This research was based on work supported by the National Science Foundation under grant ECS-8407240.     

Newton Methods for Quasidifferentiable Equations and Their Convergence

The Newton method and the inexact Newton method for solving quasidifferentiable equations via the quasidifferential are investigated. The notion of Q-semismoothness for a quasidifferentiable function is proposed. The superlinear convergence of the Newton method proposed by Zhang and Xia is proved under the Q-semismooth assumption. An inexact Newton method is developed and its linear convergence is shown.Project sponsored by Shanghai Education Committee Grant 04EA01 and by Shanghai Government Grant T0502.     

A new inexact SQP algorithm for nonlinear systems of mixed equalities and inequalities

Traditional inexact SQP algorithm can only solve equality constrained optimization (Byrd et al. Math. Program. 122, 273–299 2010). In this paper, we propose a new inexact SQP algorithm with affine scaling technique for nonlinear systems of mixed equalities and inequalities, which arise in complementarity and variational inequalities. The nonlinear systems are transformed into a special nonlinear optimization with equality and bound constraints, and then we give a new inexact SQP algorithm for solving it. The new algorithm equipped with affine scaling technique does not require a quadratic programming subproblem with inequality constraints. The search direction is computed by solving one linear system approximately using iterative linear algebra techniques. Under mild assumptions, we discuss the global convergence. The preliminary numerical results show the effectiveness of the proposed algorithm.     

A B-differentiable equation-based,globally and locally quadratically convergent algorithm for nonlinear programs,complementarity and variational inequality problems

This paper presents a globally convergent, locally quadratically convergent algorithm for solving general nonlinear programs, nonlinear complementarity and variational inequality problems. The algorithm is based on a unified formulation of these three mathematical programming problems as a certain system of B-differentiable equations, and is a modification of the damped Newton method described in Pang (1990) for solving such systems of nonsmooth equations. The algorithm resembles several existing methods for solving these classes of mathematical programs, but has some special features of its own; in particular, it possesses the combined advantage of fast quadratic rate of convergence of a basic Newton method and the desirable global convergence induced by one-dimensional Armijo line searches. In the context of a nonlinear program, the algorithm is of the sequential quadratic programming type with two distinct characteristics: (i) it makes no use of a penalty function; and (ii) it circumvents the Maratos effect. In the context of the variational inequality/complementarity problem, the algorithm provides a Newton-type descent method that is guaranteed globally convergent without requiring the F-differentiability assumption of the defining B-differentiable equations.This work was based on research supported by the National Science Foundation under Grant No. ECS-8717968.     

A self-adaptive trust region method for the extended linear complementarity problems

By using some NCP functions, we reformulate the extended linear complementarity problem as a nonsmooth equation. Then we propose a self-adaptive trust region algorithm for solving this nonsmooth equation. The novelty of this method is that the trust region radius is controlled by the objective function value which can be adjusted automatically according to the algorithm. The global convergence is obtained under mild conditions and the local superlinear convergence rate is also established under strict complementarity conditions. This work is supported by National Natural Science Foundation of China (No. 10671126) and Shanghai Leading Academic Discipline Project (S30501).     

Further applications of a splitting algorithm to decomposition in variational inequalities and convex programming

A classical method for solving the variational inequality problem is the projection algorithm. We show that existing convergence results for this algorithm follow from one given by Gabay for a splitting algorithm for finding a zero of the sum of two maximal monotone operators. Moreover, we extend the projection algorithm to solveany monotone affine variational inequality problem. When applied to linear complementarity problems, we obtain a matrix splitting algorithm that is simple and, for linear/quadratic programs, massively parallelizable. Unlike existing matrix splitting algorithms, this algorithm converges under no additional assumption on the problem. When applied to generalized linear/quadratic programs, we obtain a decomposition method that, unlike existing decomposition methods, can simultaneously dualize the linear constraints and diagonalize the cost function. This method gives rise to highly parallelizable algorithms for solving a problem of deterministic control in discrete time and for computing the orthogonal projection onto the intersection of convex sets.This research is partially supported by the U.S. Army Research Office, contract DAAL03-86-K-0171 (Center for Intelligent Control Systems), and by the National Science Foundation under grant NSF-ECS-8519058.Thanks are due to Professor J.-S. Pang for his helpful comments.     

A unifying geometric solution framework and complexity analysis for variational inequalities

In this paper, we propose a concept of polynomiality for variational inequality problems and show how to find a near optimal solution of variational inequality problems in a polynomial number of iterations. To establish this result, we build upon insights from several algorithms for linear and nonlinear programs (the ellipsoid algorithm, the method of centers of gravity, the method of inscribed ellipsoids, and Vaidya's algorithm) to develop a unifying geometric framework for solving variational inequality problems. The analysis rests upon the assumption of strong-f-monotonicity, which is weaker than strict and strong monotonicity. Since linear programs satisfy this assumption, the general framework applies to linear programs.Preparation of this paper was supported, in part, by NSF Grant 9312971-DDM from the National Science Foundation.     

A nonmonotone smoothing Newton method for circular cone programming

The circular cone programming (CCP) problem is to minimize or maximize a linear function over the intersection of an affine space with the Cartesian product of circular cones. In this paper, we study nondegeneracy and strict complementarity for the CCP, and present a nonmonotone smoothing Newton method for solving the CCP. We reformulate the CCP as a second-order cone programming (SOCP) problem using the algebraic relation between the circular cone and the second-order cone. Then based on a one parametric class of smoothing functions for the SOCP, a smoothing Newton method is developed for the CCP by adopting a new nonmonotone line search scheme. Without restrictions regarding its starting point, our algorithm solves one linear system of equations approximately and performs one line search at each iteration. Under mild assumptions, our algorithm is shown to possess global and local quadratic convergence properties. Some preliminary numerical results illustrate that our nonmonotone smoothing Newton method is promising for solving the CCP.     

Convex optimization problems involving finite autocorrelation sequences

 We discuss convex optimization problems in which some of the variables are constrained to be finite autocorrelation sequences. Problems of this form arise in signal processing and communications, and we describe applications in filter design and system identification. Autocorrelation constraints in optimization problems are often approximated by sampling the corresponding power spectral density, which results in a set of linear inequalities. They can also be cast as linear matrix inequalities via the Kalman-Yakubovich-Popov lemma. The linear matrix inequality formulation is exact, and results in convex optimization problems that can be solved using interior-point methods for semidefinite programming. However, it has an important drawback: to represent an autocorrelation sequence of length $n$, it requires the introduction of a large number ($n(n+1)/2$) of auxiliary variables. This results in a high computational cost when general-purpose semidefinite programming solvers are used. We present a more efficient implementation based on duality and on interior-point methods for convex problems with generalized linear inequalities. Received: August 20, 2001 / Accepted: July 16, 2002 Published online: September 27, 2002 RID="★" ID="★" This material is based upon work supported by the National Science Foundation under Grant No. ECS-9733450.     

A multiplier method with automatic limitation of penalty growth

This paper presents a multiplier method for solving optimization problems with equality and inequality constraints. The method realizes all the good features that were foreseen by R. Fletcher for this type of algorithm in the past, but which suffers from none of the drawbacks of the earlier attempts.Research sponsored by the National Science Foundation (RANN) Grant ENV76-04264 the National Science Foundation Grant ENG73-08214-A01 and the Sweden-American Foundation.     

A trust region and affine scaling interior point method for nonconvex minimization with linear inequality constraints

A trust region and affine scaling interior point method (TRAM) is proposed for a general nonlinear minimization with linear inequality constraints [8]. In the proposed approach, a Newton step is derived from the complementarity conditions. Based on this Newton step, a trust region subproblem is formed, and the original objective function is monotonically decreased. Explicit sufficient decrease conditions are proposed for satisfying the first order and second order necessary conditions.?The objective of this paper is to establish global and local convergence properties of the proposed trust region and affine scaling interior point method. It is shown that the proposed explicit decrease conditions are sufficient for satisfy complementarity, dual feasibility and second order necessary conditions respectively. It is also established that a trust region solution is asymptotically in the interior of the proposed trust region subproblem and a properly damped trust region step can achieve quadratic convergence. Received: January 29, 1999 / Accepted: November 22, 1999?Published online February 23, 2000     

Iterative methods for variational and complementarity problems

In this paper, we study both the local and global convergence of various iterative methods for solving the variational inequality and the nonlinear complementarity problems. Included among such methods are the Newton and several successive overrelaxation algorithms. For the most part, the study is concerned with the family of linear approximation methods. These are iterative methods in which a sequence of vectors is generated by solving certain linearized subproblems. Convergence to a solution of the given variational or complementarity problem is established by using three different yet related approaches. The paper also studies a special class of variational inequality problems arising from such applications as computing traffic and economic spatial equilibria. Finally, several convergence results are obtained for some nonlinear approximation methods.This research was based on work supported by the National Science Foundation under grant ECS-7926320.     

Insights into the interior-point methods

In this paper, we study the search directions of three important interior-point algorithms, namely, the primal-affine scaling method (with logarithmic barrier function), the dual-affine scaling method (with logarithmic barrier function), and the primal-dual interior point method. From an algebraic point of view, we show that the search directions of these three algorithms are merely Newton directions along three different paths that lead to a solution of the Karush-Kuhn-Tucker conditions of a given linear programming problem. From a geometric point of view, we show that these directions can be obtained by solving certain well-defined subproblems. Both views provide a general platform for studying the existing interior-point methods and deriving new interior-point algorithms. We illustrate the derivation of new interior-point algorithms by replacing the logarithmic barrier function with an entropic barrier function. The results have been generalized and discussed.This work is partially supported by the North Carolina Supercomputing Center 1990 Cray Grant Program sponsored by Cray Research.