Convexification of a noninferior frontier

The convexity of a noninferior frontier can be achieved in an appropriate equivalent objective space for general nonconvex multi-objective optimization problems. Specifically, this paper proves that applying thep-power to the objective functions can act as a convexification scheme for a noninferior frontier.The author appreciates the discussions with Jian-Bo Yang and the comments from Yacov Y. Haimes.     

Convexification of a Noninferior Frontier

In a recent paper by Li (Ref. 1), a scheme was proposed to convexify an efficient frontier for a vector optimization problem by rescaling each component of the vector objective functions by its p-power. For sufficiently large p, it was shown that the transformed efficient frontier is cone-convex; hence, the usual linear scalarization (or supporting hyperplane) method can be used to find the efficient solutions. An outstanding question remains: What is the minimum value of p such that the efficient frontier can be convexified? In this note, we answer the above question by deriving some theoretical lower bounds for p.

     

Maximizing a Linear Fractional Function on a Pareto Efficient Frontier

We consider the problem of maximizing a linear fractional function on the Pareto efficient frontier of two other linear fractional functions. We present a finite pivoting-type algorithm that solves the maximization problem while computing simultaneously the efficient frontier. Application to multistage efficiency analysis is discussed. An example demonstrating the computational procedure is included.     

Generating pareto-optimal alternatives by a nonfeasible hierarchical method

A hierarchical algorithm for generating Pareto-optimal alternatives for convex multicriteria problems is derived. At the upper level, values for Lagrange multipliers of the coupling constraints are first given. Then at the subsystems, Pareto-optimal values are determined for the subsystem objectives, whereby an additional term or an additional objective is included due to the Lagrange multipliers. In the subsystem optimizations, the coupling equations between the subsystems are not satisfied; therefore, the method is called nonfeasible. Finally, the upper level checks which of the subsystem solutions satisfy the coupling constraints; these solutions are Pareto-optimal solutions for the overall system.     

综合国力估量的前缘分析方法

把前缘分析方法引入国力估量,展示了这种方法在国力计算、比较测度和结构分析等方面的有效应用.     

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.     

A Branch-and-Bound Based Method for Solving Monotone Optimization Problems

Monotone optimization problems are an important class of global optimization problems with various applications. In this paper, we propose a new exact method for monotone optimization problems. The method is of branch-and-bound framework that combines three basic strategies: partition, convexification and local search. The partition scheme is used to construct a union of subboxes that covers the boundary of the feasible region. The convexification outer approximation is then applied to each subbox to obtain an upper bound of the objective function on the subbox. The performance of the method can be further improved by incorporating the method with local search procedure. Illustrative examples describe how the method works. Computational results for small randomly generated problems are reported. Dedicated to Professor Alex Rubinov on the occasion of his 65th birthday. The authors appreciate very much the discussions with Professor Alex Rubinov and his suggestion of using local search. Research supported by the National Natural Science Foundation of China under Grants 10571116 and 10261001, and Guangxi University Scientific Research Foundation (No. X051022).     

极大熵方法与指数罚函数

就非线性极大极小问题，阐明了极大熵方法与指数罚方法的关系．通过分析相关Ｈｅｓｓｉａｎ阵的条件数，对二者进行了对比．     

Value-Estimation Function Method for Constrained Global Optimization

A novel value-estimation function method for global optimization problems with inequality constraints is proposed in this paper. The value-estimation function formulation is an auxiliary unconstrained optimization problem with a univariate parameter that represents an estimated optimal value of the objective function of the original optimization problem. A solution is optimal to the original problem if and only if it is also optimal to the auxiliary unconstrained optimization with the parameter set at the optimal objective value of the original problem, which turns out to be the unique root of a basic value-estimation function. A logarithmic-exponential value-estimation function formulation is further developed to acquire computational tractability and efficiency. The optimal objective value of the original problem as well as the optimal solution are sought iteratively by applying either a generalized Newton method or a bisection method to the logarithmic-exponential value-estimation function formulation. The convergence properties of the solution algorithms guarantee the identification of an approximate optimal solution of the original problem, up to any predetermined degree of accuracy, within a finite number of iterations.     

广义模糊一致矩阵的性质及指数排序方法

在模糊一致矩阵的基础上,引入了广义模糊一致矩阵.研究了广义模糊一致矩阵的性质及指数排序方法,给出了判定模糊矩阵是广义模糊一致矩阵的充要条件,得到了广义模糊一致矩阵的求排序向量的指数排序方法,以及该指数排序方法是强条件下保序的结论.     

Multi-Path Approach for Reliability-Redundancy Allocation Using a Scaling Method

The reliability-redundancy allocation problem is an optimization problem that achieves better system reliability by determining levels of component redundancies and reliabilities simultaneously. The problem is classified with the hardest problems in the reliability optimization field because the decision variables are mixed-integer and the system reliability function is nonlinear, non-separable, and non-convex. Thus, iterative heuristics are highly recommended for solving the problem due to their reasonable solution quality and relatively short computation time. At present, most iterative heuristics use sensitivity factors to select an appropriate variable which significantly improves the system reliability. The sensitivity factor represents the impact amount of each variable to the system reliability at a designated iteration. However, these heuristics are inefficient in terms of solution quality and computation time because the sensitivity factor calculations are performed only at integer variables. It results in degradation of the exploration and growth in the number of subsequent continuous nonlinear programming (NLP) subproblems. To overcome the drawbacks of existing iterative heuristics, we propose a new scaling method based on the multi-path iterative heuristics introduced by Ha (2004). The scaling method is able to compute sensitivity factors for all decision variables and results in a decreased number of NLP subproblems. In addition, the approximation heuristic for NLP subproblems helps to avoid redundant computation of NLP subproblems caused by outlined solution candidates. Numerical experimental results show that the proposed heuristic is superior to the best existing heuristic in terms of solution quality and computation time.     

Convergence and Application of a Decomposition Method Using Duality Bounds for Nonconvex Global Optimization

The subject of this article is a class of global optimization problems, in which the variables can be divided into two groups such that, in each group, the functions involved have the same structure (e.g. linear, convex or concave, etc.). Based on the decomposition idea of Benders (Ref. 1), a corresponding master problem is defined on the space of one of the two groups of variables. The objective function of this master problem is in fact the optimal value function of a nonlinear parametric optimization problem. To solve the resulting master problem, a branch-and-bound scheme is proposed, in which the estimation of the lower bounds is performed by applying the well-known weak duality theorem in Lagrange duality. The results of this article concentrate on two subjects: investigating the convergence of the general algorithm and solving dual problems of some special classes of nonconvex optimization problems. Based on results in sensitivity and stability theory and in parametric optimization, conditions for the convergence are established by investigating the so-called dual properness property and the upper semicontinuity of the objective function of the master problem. The general algorithm is then discussed in detail for some nonconvex problems including concave minimization problems with a special structure, general quadratic problems, optimization problems on the efficient set, and linear multiplicative programming problems.     

Exponential stability of a network of elastic and thermoelastic materials

In this paper, we study the stability of a network of some elastic and thermoelastic edges. We establish the exponential decay rate of the whole system. Copyright © 2012 John Wiley & Sons, Ltd.     

分段连续型随机微分方程指数Euler方法的稳定性

给出了线性分段连续型随机微分方程指数Euler方法的均方指数稳定性.经典的对稳定性理论分析,通常应用的是Lyapunov泛函理论,然而,应用该方程本身的特点和矩阵范数的定义给出了该方程精确解的均方稳定性.以往对于该方程应用隐式Euler方法得到对于任意步长数值解的均方稳定性,而应用显式Euler方法得到了相同的结果.最...     

A Globally Convergent Polak-Ribière-Polyak Conjugate Gradient Method with Armijo-Type Line Search

In this paper, we propose a globally convergent Polak-Ribière-Polyak (PRP)conjugate gradient method for nonconvex minimization of differentiable functions by employing an Armijo-type line search which is simpler and less demanding than those defined in [4,10]. A favorite property of this method is that we can choose the initial stepsize as the one-dimensional minimizer of a quadratic model Φ(t):= f(xk)+tgTkdk+1/2t2dTkQkdk, where Qk is a positive definite matrix that carries some second order information of the objective function f. So, this line search may make the stepsize tk more easily accepted. Preliminary numerical results show that this method is efficient.     

Exponential sums over primes in short intervals

In this paper we establish one new estimate on exponential sums over primes in short intervals. As an application of this result, we sharpen Hua's result by proving that each sufficiently large integer N congruent to 5 modulo 24 can be written as N = p12 p22 p32 p42 p52, with |pj-(N/5)~(1/2)|≤U = N1/2-1/20 ε, where pj are primes. This result is as good as what one can obtain from the generalized Riemann hypothesis.     

Solving a Class of Brouwer Fixed-point Problems via a Modified Aggregate Constraint Homotopy Method

In this paper, we provide an aggregate function homotopy interior point method to solve a class of Brouwer fixed-point problems. Compared with the homotopy method （proposed by Yu and Lin, Appl. Math. Comput., 74（1996）, 65）, the main adavantages of this method are as foUows： on the one hand, it can solve the Brouwer fixed-point problems in a broader class of nonconvex subsets Ω in R^n （in this paper, we let Ω={x∈ R^n ： gi（x） ≤0, i= 1,... , m}）; on the other hand, it can also deal with the subsets Ω with larger amount of constraints more effectively.     

用改进的凝聚约束同伦方法求解一类Brouwer不动点问题

In this paper, we provide an aggregate function homotopy interior point method to solve a class of Brouwer fixed-point problems. Compared with the homotopy method (proposed by Yu and Lin, Appl. Math. Comput., 74(1996), 65), the main adavantages of this method are as follows: on the one hand, it can solve the Brouwer fixed-point problems in a broader class of nonconvex subsets Ω in Rn (in this paper, we let Ω = [x ∈ Rn: gi(x) ≤ 0, i = 1,… ,m]); on the other hand, it can also deal with the subsets Ω with larger amount of constraints more effectively.     

舍选法几何解释及曲边梯形概率密度随机数生成算法

本文首先从几何概率角度看舍选法的直观意义,找出改进舍选法的途径;然后讨论梯形和曲边梯形概率密度随机数生成算法,给出若干简单概率密度随机数生成实例。     

Convergence and Mean-Square Stability of Exponential Euler Method for Semi-Linear Stochastic Delay Integro-Differential Equations

In this paper, the numerical methods for semi-linear stochastic delay integro-differential equations are studied. The uniqueness, existence and stability of analytic solutions of semi-linear stochastic delay integro-differential equations are studied and some suitable conditions for the mean-square stability of the analytic solutions are also obtained. Then the numerical approximation of exponential Euler method for semi-linear stochastic delay integro-differential equations is constructed and the convergence and the stability of the numerical method are studied. It is proved that the exponential Euler method is convergent with strong order $\frac{1}{2}$ and can keep the mean-square exponential stability of the analytical solutions under some restrictions on the step size. In addition, numerical experiments are presented to confirm the theoretical results.