A Continuous Method for Convex Programming Problems

In this paper, we present a continuous method for convex programming (CP) problems. Our approach converts first the convex problem into a monotone variational inequality (VI) problem. Then, a continuous method, which includes both a merit function and an ordinary differential equation (ODE), is introduced for the resulting variational inequality problem. The convergence of the ODE solution is proved for any starting point. There is no Lipschitz condition required in our proof. We show also that this limit point is an optimal solution for the original convex problem. Promising numerical results are presented.This research was supported in part by Grants FRG/01-02/I-39 and FRG/01-02/II-06 of Hong Kong Baptist University and Grant HKBU2059/02P from the Research Grant Council of Hong Kong.The author thanks Professor Bingsheng He for many helpful suggestions and discussions. The author is also grateful for the comments and suggestions of two anonymous referees. In particular, the author is indebted to one referee who drew his attention to References 15, 17, 18.     

Proximal Point Algorithms for General Variational Inequalities

This paper presents a unified framework of proximal point algorithms (PPAs) for solving general variational inequalities (GVIs). Some existing PPAs for classical variational inequalities, including both the exact and inexact versions, are extended to solving GVIs. Consequently, several new PPA-based algorithms are proposed. M. Li was supported by NSFC Grant 10571083 and SRFDP Grant 200802861031. L.Z. Liao was supported in part by grants from Hong Kong Baptist University and the Research Grant Council of Hong Kong. X.M. Yuan was supported in part by FRG/08-09/II-40 from Hong Kong Baptist University and NSFC Grant 10701055.     

Asymptotics for Kernel Estimation of Slicing Average Third-Moment Estimation

To estimate central dimension-reduction space in multivariate nonparametric rcgression, Sliced Inverse Regression （SIR）, Sliced Average Variance Estimation （SAVE） and Slicing Average Third-moment Estimation （SAT） have been developed, Since slicing estimation has very different asymptotic behavior for SIR, and SAVE, the relevant study has been madc case by case, when the kernel estimators of SIH and SAVE share similar asymptotic properties. In this paper, we also investigate kernel estimation of SAT. We. prove the asymptotic normality, and show that, compared with tile existing results, the kernel Slnoothing for SIR, SAVE and SAT has very similar asymptotic behavior,     

An Exact Solution Method for Reliability Optimization in Complex Systems

Systems reliability plays an important role in systems design, operation and management. Systems reliability can be improved by adding redundant components or increasing the reliability levels of subsystems. Determination of the optimal amount of redundancy and reliability levels among various subsystems under limited resource constraints leads to a mixed-integer nonlinear programming problem. The continuous relaxation of this problem in a complex system is a nonconvex nonseparable optimization problem with certain monotone properties. In this paper, we propose a convexification method to solve this class of continuous relaxation problems. Combined with a branch-and-bound method, our solution scheme provides an efficient way to find an exact optimal solution to integer reliability optimization in complex systems. This research was partially supported by the Research Grants Council of Hong Kong, grants CUHK4056/98E, CUHK4214/01E and 2050252, and the National Natural Science Foundation of China under Grants 79970107 and 10271073.     

Continuous methods for extreme and interior eigenvalue problems

In this paper, continuous methods are introduced to compute both the extreme and interior eigenvalues and their corresponding eigenvectors for real symmetric matrices. The main idea is to convert the extreme and interior eigenvalue problems into some optimization problems. Then a continuous method which includes both a merit function and an ordinary differential equation (ODE) is introduced for each resulting optimization problem. The convergence of each ODE solution is proved for any starting point. The limit of each ODE solution for any starting point is fully studied. Both the extreme and the interior eigenvalues and their corresponding eigenvectors can be easily obtained under a very mild condition. Promising numerical results are also presented.     

Exact Algorithm for Concave Knapsack Problems: Linear Underestimation and Partition Method

Integer programming problems with a concave cost function are often encountered in optimization models involving economics of scale. In this paper, we propose an efficient exact algorithm for solving concave knapsack problems. The algorithm consists of an iterative process between finding lower and upper bounds by linearly underestimating the objective function and performing domain cut and partition by exploring the special structure of the problem. The lower bound is improved iteratively via cutting and partitioning the domain. This iteration process converges to the optimality in a finite number of steps. Promising computational results are reported for large-scale concave knapsack problems with up to 1200 integer variables. Comparison results with other existing methods in the literature are also presented. *Research supported by the National Natural Science Foundation of China under Grants 79970107 and 10271073,and the Research Grants Council of Hong Kong under Grant CUHK 4214/01E.     

Lower-Order Penalization Approach to Nonlinear Semidefinite Programming

In this paper, we reformulate a nonlinear semidefinite programming problem into an optimization problem with a matrix equality constraint. We apply a lower-order penalization approach to the reformulated problem. Necessary and sufficient conditions that guarantee the global (local) exactness of the lower-order penalty functions are derived. Convergence results of the optimal values and optimal solutions of the penalty problems to those of the original semidefinite program are established. Since the penalty functions may not be smooth or even locally Lipschitz, we invoke the Ekeland variational principle to derive necessary optimality conditions for the penalty problems. Under certain conditions, we show that any limit point of a sequence of stationary points of the penalty problems is a KKT stationary point of the original semidefinite program. Communicated by Y. Zhang This work was supported by a Postdoctoral Fellowship of Hong Kong Polytechnic University and by the Research Grants Council of Hong Kong.     

An Inexact Newton Method Derived from Efficiency Analysis

We consider solving an unconstrained optimization problem by Newton-PCG like methods in which the preconditioned conjugate gradient method is applied to solve the Newton equations. The main question to be investigated is how efficient Newton-PCG like methods can be from theoretical point of view. An algorithmic model with several parameters is established. Furthermore, a lower bound of the efficiency measure of the algorithmic model is derived as a function of the parameters. By maximizing this lower bound function, the parameters are specified and therefore an implementable algorithm is obtained. The efficiency of the implementable algorithm is compared with Newtons method by theoretical analysis and numerical experiments. The results show that this algorithm is competitive.Mathematics Subject Classification: 90C30, 65K05.This work was supported by the National Science Foundation of China Grant No. 10371131, and Hong Kong Competitive Earmarked Research Grant CityU 1066/00P from Hong Kong University Grant Council     

Dynamically Consistent Cooperative Solution in a Differential Game of Transboundary Industrial Pollution

This paper presents a cooperative differential game of transboundary industrial pollution. A noted feature of the game model is that the industrial sectors remain competitive among themselves while the governments cooperate in pollution abatement. It is the first time that time consistent solutions are derived in a cooperative differential game on pollution control with industries and governments being separate entities. A stochastic version of the model is presented and a subgame-consistent cooperative solution is provided. This is the first study of pollution management in a stochastic differential game framework. This research was supported by the Research Grant Council of Hong Kong Grant HKBU2103/04H and Hong Kong Baptist University Grant FRG/05-06/II22.     

Subgame Consistent Solutions of a Cooperative Stochastic Differential Game with Nontransferable Payoffs

Subgame consistency is a fundamental element in the solution of cooperative stochastic differential games. In particular, it ensures that the extension of the solution policy to a later starting time and any possible state brought about by the prior optimal behavior of the players would remain optimal. Recently, mechanisms for the derivation of subgame consistent solutions in stochastic cooperative differential games with transferable payoffs have been found. In this paper, subgame consistent solutions are derived for a class of cooperative stochastic differential games with nontransferable payoffs. The previously intractable subgame consistent solution for games with nontransferable payoffs is rendered tractable.This research was supported by the Research Grant Council of Hong Kong, Grant HKBU2056/99H and by Hong Kong Baptist University, Grant FRG/02-03/II16.Communicated by G. Leitmann     

Convergence Analysis of a Class of Penalty Methods for Vector Optimization Problems with Cone Constraints

In this paper, we consider convergence properties of a class of penalization methods for a general vector optimization problem with cone constraints in infinite dimensional spaces. Under certain assumptions, we show that any efficient point of the cone constrained vector optimization problem can be approached by a sequence of efficient points of the penalty problems. We also show, on the other hand, that any limit point of a sequence of approximate efficient solutions to the penalty problems is a weekly efficient solution of the original cone constrained vector optimization problem. Finally, when the constrained space is of finite dimension, we show that any limit point of a sequence of stationary points of the penalty problems is a KKT stationary point of the original cone constrained vector optimization problem if Mangasarian–Fromovitz constraint qualification holds at the limit point.This work is supported by the Postdoctoral Fellowship of Hong Kong Polytechnic University.     

The Lagrangian Globalization Method for Nonsmooth Constrained Equations

The difficulty suffered in optimization-based algorithms for the solution of nonlinear equations lies in that the traditional methods for solving the optimization problem have been mainly concerned with finding a stationary point or a local minimizer of the underlying optimization problem, which is not necessarily a solution of the equations. One method to overcome this difficulty is the Lagrangian globalization (LG for simplicity) method. This paper extends the LG method to nonsmooth equations with bound constraints. The absolute system of equations is introduced. A so-called Projected Generalized-Gradient Direction (PGGD) is constructed and proved to be a descent direction of the reformulated nonsmooth optimization problem. This projected approach keeps the feasibility of the iterates. The convergence of the new algorithm is established by specializing the PGGD. Numerical tests are given. This author's work was done when she was visiting The Hong Kong Polytechnic University. His work is also supported by the Research Grant Council of Hong Kong.     

Two new variants of nonlinear inexact Uzawa algorithms for saddle-point problems

Summary. In this paper, we consider some nonlinear inexact Uzawa methods for iteratively solving linear saddle-point problems. By means of a new technique, we first give an essential improvement on the convergence results of Bramble-Paschiak-Vassilev for a known nonlinear inexact Uzawa algorithm. Then we propose two new algorithms, which can be viewed as a combination of the known nonlinear inexact Uzawa method with the classical steepest descent method and conjugate gradient method respectively. The two new algorithms converge under very practical conditions and do not require any apriori estimates on the minimal and maximal eigenvalues of the preconditioned systems involved, including the preconditioned Schur complement. Numerical results of the algorithms applied for the Stokes problem and a purely linear system of algebraic equations are presented to show the efficiency of the algorithms. Received December 8, 1999 / Revised version received September 8, 2001 / Published online March 8, 2002 RID="*" ID="*" The work of this author was partially supported by a grant from The Institute of Mathematical Sciences, CUHK RID="**" ID="**" The work of this author was partially supported by Hong Kong RGC Grants CUHK 4292/00P and CUHK 4244/01P     

Further Study on Augmented Lagrangian Duality Theory

In this paper, we present a necessary and sufficient condition for a zero duality gap between a primal optimization problem and its generalized augmented Lagrangian dual problems. The condition is mainly expressed in the form of the lower semicontinuity of a perturbation function at the origin. For a constrained optimization problem, a general equivalence is established for zero duality gap properties defined by a general nonlinear Lagrangian dual problem and a generalized augmented Lagrangian dual problem, respectively. For a constrained optimization problem with both equality and inequality constraints, we prove that first-order and second-order necessary optimality conditions of the augmented Lagrangian problems with a convex quadratic augmenting function converge to that of the original constrained program. For a mathematical program with only equality constraints, we show that the second-order necessary conditions of general augmented Lagrangian problems with a convex augmenting function converge to that of the original constrained program.This research is supported by the Research Grants Council of Hong Kong (PolyU B-Q359.)     

Value Estimation Approach to the Iri-Imai Method for Constrained Convex Optimization

In this paper we propose an extension of the so-called Iri-Imai method to solve constrained convex programming problems. The original Iri-Imai method is designed for linear programs and assumes that the optimal objective value of the optimization problem is known in advance. Zhang (Ref. 9) extends the method for constrained convex optimization but the optimum value is still assumed to be known in advance. In our new extension this last requirement on the optimal value is relaxed; instead only a lower bound of the optimal value is needed. Our approach uses a multiplicative barrier function for the problem with a univariate parameter that represents an estimated optimum value of the original optimization problem. An optimal solution to the original problem can be traced down by minimizing the multiplicative barrier function. Due to the convexity of this barrier function the optimal objective value as well as the optimal solution of the original problem are sought iteratively by applying Newtons method to the multiplicative barrier function. A new formulation of the multiplicative barrier function is further developed to acquire computational tractability and efficiency. Numerical results are presented to show the efficiency of the new method.His research supported by Hong Kong RGC Earmarked Grant CUHK4233/01E.Communicated by Z. Q. Luo     

Tracing the Pareto frontier in bi-objective optimization problems by ODE techniques

In this paper we present a deterministic method for tracing the Pareto frontier in non-linear bi-objective optimization problems with equality and inequality constraints. We reformulate the bi-objective optimization problem as a parametric single-objective optimization problem with an additional Normalized Normal Equality Constraint (NNEC) similar to the existing Normal Boundary Intersection (NBI) and the Normalized Normal Constraint method (NNC). By computing the so called Defining Initial Value Problem (DIVP) for segments of the Pareto front and solving a continuation problem with a standard integrator for ordinary differential equations (ODE) we can trace the Pareto front. We call the resulting approach ODE NNEC method and demonstrate numerically that it can yield the entire Pareto frontier to high accuracy. Moreover, due to event detection capabilities available for common ODE integrators, changes in the active constraints can be automatically detected. The features of the current algorithm are illustrated for two case studies whose Matlab® code is available as Electronic Supplementary Material to this article.     

Parallel solution of large-scale,block-angular linear programs

Many important large-scale optimization problems can be formulated as linear programs with a block-angular structure. This structure lends itself naturally to parallel solutions and is used to great advantage in the solution method described. To demonstrate the efficiency of the method, it has been implemented and computationally tested on both a shared-memory vector multiprocessor (CRAY-2) and a local-memory hypercube (NCUBE/seven) with 64 processors. Computational results for problems with as many as 24,000 rows and 74,000 columns (1,024 blocks and 1.4 million nonzero elements) are presented. A problem of this size was solved on the NCUBE in less than four minutes and the CRAY-2 in 37 seconds.Research supported in part by NSF/CER Grant DCR 8420935, AFOSR 87-0127, and the Minnesota Supercomputer Institute.     

Duality and Penalization in Optimization via an Augmented Lagrangian Function with Applications

This paper aims to establish duality and exact penalization results for the primal problem of minimizing an extended real-valued function in a reflexive Banach space in terms of a valley-at-0 augmented Lagrangian function. It is shown that every weak limit point of a sequence of optimal solutions generated by the valley-at-0 augmented Lagrangian problems is a solution of the original problem. A zero duality gap property and an exact penalization representation between the primal problem and the valley-at-0 augmented Lagrangian dual problem are obtained. These results are then applied to an inequality and equality constrained optimization problem in infinite-dimensional spaces and variational problems in Sobolev spaces, respectively. The first author was supported by the Research Committee of Hong Kong Polytechnic University, by Grant 10571174 from the National Natural Science Foundation of China and Grant 08KJB11009 from the Jiangsu Education Committee of China. The second author was supported by Grant BQ771 from the Research Grants Council of Hong Kong. We are grateful to the referees for useful suggestions which have contributed to the final presentation of the paper.     

The ability of students and teachers to use counter-examples to justify mathematical propositions: a pilot study in South Korea and Hong Kong

Counter-examples, which are a distinct kind of example, have a functional role in inducing logically deductive reasoning skills in the learning process. In this investigation, we compare the ability of students and prospective teachers in South Korea and Hong Kong to use counter-examples to justify mathematical propositions. The results highlight that South Korean students performed better than Hong Kong students at justifying propositions using counter-examples in algebra problems, but both did equally well in geometry problems. In terms of the prospective teachers’ ability to justify propositions using counter-examples in two particular topics, properties of the absolute value function and parallelogram, Hong Kong prospective teachers performed relatively weakly in the absolute value problem but better in the parallelogram problem compared with South Korean prospective teachers. The weaknesses and strengths of students and prospective teachers in generating counter-examples associated with logical reasoning in mathematical contexts in the two regions indicate ways of improving the effectiveness of learning and teaching mathematics through the use of counter-examples.     

Z-eigenvalue methods for a global polynomial optimization problem

As a global polynomial optimization problem, the best rank-one approximation to higher order tensors has extensive engineering and statistical applications. Different from traditional optimization solution methods, in this paper, we propose some Z-eigenvalue methods for solving this problem. We first propose a direct Z-eigenvalue method for this problem when the dimension is two. In multidimensional case, by a conventional descent optimization method, we may find a local minimizer of this problem. Then, by using orthogonal transformations, we convert the underlying supersymmetric tensor to a pseudo-canonical form, which has the same E-eigenvalues and some zero entries. Based upon these, we propose a direct orthogonal transformation Z-eigenvalue method for this problem in the case of order three and dimension three. In the case of order three and higher dimension, we propose a heuristic orthogonal transformation Z-eigenvalue method by improving the local minimum with the lower-dimensional Z-eigenvalue methods, and a heuristic cross-hill Z-eigenvalue method by using the two-dimensional Z-eigenvalue method to find more local minimizers. Numerical experiments show that our methods are efficient and promising. This work is supported by the Research Grant Council of Hong Kong and the Natural Science Foundation of China (Grant No. 10771120).