Some duality relationships for the generalized Neyman-Pearson problem

In 1963, Kuhn presented a dual problem to a relatively well-known location problem, variously referred to as the generalized Fermat problem and the Steiner-Weber problem. The purpose of this paper is to point out how Kuhn's results can be adapted to provide a dual to the generalized Neyman-Pearson problem, a problem of fundamental interest in statistics, which has applications in control theory and a number of other areas. The Neyman-Pearson problem, termed the dual problem, is a constrained maximization problem and may be considered to be a calculus-of-variations analog to the bounded-variable problem of linear programming. When the dual problem has equality constraints, the primal problem is an unconstrained minimization problem. Duality results are also obtained for the case where the dual problem has inequality constraints.This work was partially supported by the National Science Foundation, Grant Nos. NSF-GK-1571 and NSF-GK-3038. The authors would like to acknowledge the very useful comments of one of the referees, which led to more direct and general proofs of Properties 2.3 and 2.6.     

三种新变形的全一问题

该文研究三种新变形的全一问题及最小全一问题. 原始的全一问题可被形象的称为顶点点亮顶点问题, 而这三类新问题则分别被称为顶点点亮边问题,边点亮顶点问题,边点亮边问题. 顶点点亮顶点问题已经得到了广泛的研究. 比如,解的存在性问题和求解的有效算法已经被解决,一般图上的最小顶点点亮顶点问题已经被证明是NP- 完备的,树、单圈图和双圈图上的最小顶点点亮顶点问题的线性时间最优算法也已被给出等. 该文对于顶点点亮边问题,证明一个图有解当且仅当它是二部图,因此只可能有两组解和最优解. 对于边点亮顶点问题,证明一个图有解当且仅当它包含偶数个顶点,并通过将其最优问题多项式变换成最小权的完美匹配问题,得出一般图上的最小边点亮顶点问题可在多项式时间内求解. 边点亮边问题可归约成线图上的顶点点亮顶点问题.     

一些约束规划问题的近似全局最优解

首先将一个具有多个约束的规划问题转化为一个只有一个约束的规划问题,然后通过利用这个单约束的规划问题,对原来的多约束规划问题提出了一些凸化、凹化的方法,这样这些多约束的规划问题可以被转化为一些凹规划、反凸规划问题．最后,还证明了得到的凹规划和反凸规划的全局最优解就是原问题的近似全局最优解．     

Convexification, Concavification and Monotonization in Global Optimization

We show in this paper that via certain convexification, concavification and monotonization schemes a nonconvex optimization problem over a simplex can be always converted into an equivalent better-structured nonconvex optimization problem, e.g., a concave optimization problem or a D.C. programming problem, thus facilitating the search of a global optimum by using the existing methods in concave minimization and D.C. programming. We first prove that a monotone optimization problem (with a monotone objective function and monotone constraints) can be transformed into a concave minimization problem over a convex set or a D.C. programming problem via pth power transformation. We then prove that a class of nonconvex minimization problems can be always reduced to a monotone optimization problem, thus a concave minimization problem or a D.C. programming problem.     

关于二层规划的逼近问题

下层问题以上层决策变量作为参数,而上层是以下层问题的最优值作为响应 的一类最优化问题——二层规划问题。我们给出了由一系列此类二层规划去逼近原二层规划的逼近法,得到了这种逼近的一些有趣的结果.     

现代物流技术中装卸工问题的拟多项式时间可解情况

装卸工问题是从现代物流技术中提出的一个实际问题，这个问题的雏形早在上个世纪60年代中国科学院数学研究所就提出和研究过。现代物流业的迅速发展，促成和推动装卸工问题的提出和研究。装卸工问题是一个新的NP困难的组合优化问题，本文研究限制情形下的装卸工问题，并证明是拟多项式时间可解的。     

The zero-one knapsack problem with equality constraint

The zero-one knapsack problem is a linear zero-one programming problem with a single inequality constraint. This problem has been extensively studied and many applications and efficient algorithms have been published. In this paper we consider a similar problem, one with an equality instead of the inequality constraint. By replacing the equality by two inequalities one of which is placed in the economic function, a Lagrangean relaxation of the problem is obtained. The relation between the relaxed problem and the original problem is examined and it is shown how the optimal value of the relaxed problem varies with increasing values of the Lagrangean multiplier. Using these results an algorithm for solving the problem is proposed.The paper concludes with a discussion of computational experience.     

Pendants-median支撑树及其一个相关问题:复杂性和算法

The complexity status of Pendants-median spanning tree problem is an open problem. Using the complexity of the X3C problem, the paper proves that Pendants-median spanning tree problem is NP-complete. Global-median spanning tree problem is a related problem. Using the complexity of 3SAT, the paper proves that this problem is also NP-complete, and a polynomial -time algorithm to this problem is given, whose time complexity is O(n^3).     

限制情况下装卸工问题的最优解

装卸工问题是从现代物流技术中提出的一个实际问题,这个问题的雏形早在上个世纪60年代中国科学院数学研究所就提出和研究过.现代物流技术迅速发展,促成和推动装卸工问题的提出和研究.装卸工问题是一个新的NP困难的组合优化问题,首先介绍装卸工问题及限制情况下装卸工问题的数学模型,然后分析限制情况下的装卸工问题的性质,最后给出该问题的所有最优解.     

Single machine stochastic JIT scheduling problem subject to machine breakdowns

In this paper we research the single machine stochastic JIT scheduling problem subject to the machine breakdowns for preemptive-resume and preemptive-repeat.The objective function of the problem is the sum of squared deviations of the job-expected completion times from the due date.For preemptive-resume,we show that the optimal sequence of the SSDE problem is V-shaped with respect to expected processing times.And a dynamic programming algorithm with the pseudopolynomial time complexity is given.We discuss the difference between the SSDE problem and the ESSD problem and show that the optimal solution of the SSDE problem is a good approximate optimal solution of the ESSD problem,and the optimal solution of the SSDE problem is an optimal solution of the ESSD problem under some conditions.For preemptive-repeat,the stochastic JIT scheduling problem has not been solved since the variances of the completion times cannot be computed.We replace the ESSD problem by the SSDE problem.We show that the optimal sequence of the SSDE problem is V-shaped with respect to the expected occupying times.And a dynamic programming algorithm with the pseudopolynomial time complexity is given.A new thought is advanced for the research of the preemptive-repeat stochastic JIT scheduling problem.     

Mixed equilibrium problems with Z*-bifunctions and least element problems in Banach lattices

The purpose of this paper is to study the relations among a mixed equilibrium problem, a least element problem and a minimization problem in Banach lattices. We propose the concept of Z*-bifunctions as well as the concept of a feasible set for the mixed equilibrium problem. We prove that the feasible set of the mixed equilibrium problem is a sublattice provided that the associated bifunction is a strictly α-monotone Z*-bifunction. We establish the equivalence of the mixed equilibrium problem, the least element problem and the minimization problem under strict α-monotonicity and Z*-bifunction conditions.     

The Vehicle Routing-Allocation Problem: A unifying framework

Summary In this paper the Vehicle Routing-Allocation Problem (VRAP) is presented. In VRAP not all customers need be visited by the vehicles. However customers not visited either have to be allocated to some customer on one of the vehicle tours or left isolated. We concentrate our discussion on the Single Vehicle Routing-Allocation Problem (SVRAP). An integer linear programming formulation of SVRAP is presented and we show how SVRAP provides a unifying framework for understanding a number of the papers and problems presented in the literature. Specifically the covering tour problem, the covering salesman problem, the median tour problem, the maximal covering tour problem, the travelling salesman problem, the generalised travelling salesman problem, the selective travelling salesman problem, the prize collecting travelling salesman problem, the maximum covering/shortest path problem, the maximum population/shortest path problem, the shortest covering path problem, the median shortest path problem, the minimum covering/shortest path problem and the hierarchical network design problem are special cases/variants of SVRAP.     

A study of mixed discrete bilevel programs using semidefinite and semi-infinite programming

In this paper, we study the bilevel programming problem with discrete polynomial lower level problem. We start by transforming the problem into a bilevel problem comprising a semidefinite program (SDP for short) in the lower level problem. Then, we are able to deduce some conditions of existence of solutions for the original problem. After that, we again change the bilevel problem with SDP in the lower level problem into a semi-infinite program. With the aid of the exchange technique, for simple bilevel programs, an algorithm for computing a global optimal solution is suggested, the convergence is shown, and a numerical example is given.     

线性比式和规划问题的输出空间分支定界算法

本文旨在针对线性比式和规划这一NP-Hard非线性规划问题提出新的全局优化算法.首先,通过引入p个辅助变量把原问题等价的转化为一个非线性规划问题,这个非线性规划问题的目标函数是乘积和的形式并给原问题增加了p个新的非线性约束,再通过构造凸凹包络的技巧对等价问题的目标函数和约束条件进行相应的线性放缩,构成等价问题的一个下界线性松弛规划问题,从而提出了一个求解原问题的分支定界算法,并证明了算法的收敛性.最后,通过数值结果比较表明所提出的算法是可行有效的.     

Iterative approximation of a common solution of a split equilibrium problem,a variational inequality problem and a fixed point problem

In this paper, we introduce an iterative method to approximate a common solution of a split equilibrium problem, a variational inequality problem and a fixed point problem for a nonexpansive mapping in real Hilbert spaces. We prove that the sequences generated by the iterative scheme converge strongly to a common solution of the split equilibrium problem, the variational inequality problem and the fixed point problem for a nonexpansive mapping. The results presented in this paper extend and generalize many previously known results in this research area.     

Preemptive and nonpreemptive multi-objective programming: Relationship and counterexamples

In this paper, we formally establish connections between two standard approaches proposed for resolving multi-objective programs, namely, the nonpreemptive and the preemptive methods. We demonstrate in the linear case that, if the preemptive problem has an optimal solution, then there exists a set of weights for the nonpreemptive problem, such that any optimal solution to the nonpreemptive problem is optimal to the preemptive problem. Conversely, and more importantly, any optimal solution to the preemptive problem is optimal to the nonpreemptive problem. A similar result is established for arbitrary multi-objective functions being optimized over a finite discrete set. Thus, the preemptive problem is subsumed within the nonpreemptive problem in these cases. Although we actually construct a set of equivalent weights, we do not advocate our technique as a computational device for solving the preemptive problem. However, a previous attempt (Ref. 1), which does prescribe a set of equivalent weights to solve a preemptive problem as a linear program, is shown to be erroneous. Moreover, our constructive proof exhibits the features of the problem which govern the determination of such equivalent weights.     

一类非单调互补问题的Canonical对偶问题研究

对于一类具有广泛应用背景的非单调互补问题,我们构建了这类问题的Canonical对偶问题。其对偶问题可以写成和原问题类似的互补问题。我们给出了对偶问题和原问题解之间的对偶关系,并且将对偶问题转化成一个一维优化问题,这不但可以方便的求解这类问题,也为研究这类问题性质提供了一个非常直观的研究工具。最后,本文给出了几个算例来演示对偶问题的性质。     

非线性-线性二层规划问题的罚函数方法

利用下层问题的K-T最优性条件将下层为线性规划的一类非线性二层规划转化成相应的单层规划,同时取下层问题的互补条件为罚项,构造了该类非线性二层规划的罚问题.通过对相应罚问题性质的分析,得到了该类非线性二层规划问题的最优性条件,同时设计了该类二层规划问题的求解方法.数值结果表明该方法是可行、有效的.     

Application of the finite integral transform method to solving a mixed problem with integro-differential conditions for a nonclassical equation

We consider a mixed problem with integro-differential boundary conditions for a nonclassical equation. Under certain conditions, we apply a finite integral transform to this problem and obtain a parametric problem. We introduce the notion of proper boundary conditions of the parametric problem, which is wider than the notion of regularity. By applying the inverse integral transform to the solution of the parametric problem, we obtain an analytic representation of the solution of the original mixed problem.     

A Lagrangian relaxation-based heuristic for the vehicle routing with full container load

We address a problem of vehicle routing that arises in picking up and delivering full container load from/to an intermodal terminal. The substantial cost and time savings are expected by efficient linkage between pickup and delivery tasks, if the time of tasks and the suitability of containers for cargo allow. As this problem is NP-hard, we develop a subgradient heuristic based on a Lagrangian relaxation which enables us to identify a near optimal solution. The heuristic consists of two sub-problems: the classical assignment problem and the generalized assignment problem. As generalized assignment problem is also NP-hard, we employ an efficient solution procedure for a bin packing based problem, which replaces the generalized assignment problem. The heuristic procedure is tested on a wide variety of problem examples. The test results demonstrate that the procedure developed here can efficiently solve large instances of the problem.