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.     

The complexity of recursive constraint satisfaction problems

We investigate the complexity of finding solutions to infinite recursive constraint satisfaction problems. We show that, in general, the problem of finding a solution to an infinite recursive constraint satisfaction problem is equivalent to the problem of finding an infinite path through a recursive tree. We also identify natural classes of infinite recursive constraint satisfaction problems where the problem of finding a solution to the infinite recursive constraint satisfaction problem is equivalent to the problem of finding an infinite path through finitely branching recursive trees or recursive binary trees. There are a large number of results in the literature on the complexity of the problem of finding an infinite path through a recursive tree. Our main result allows us to automatically transfer such results to give equivalent results about the complexity of the problem of finding a solution to a recursive constraint satisfaction problem.     

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.     

A new heuristic algorithm solving the linear ordering problem

The linear ordering problem is an NP-hard combinatorial problem with a large number of applications. Contrary to another very popular problem from the same category, the traveling salesman problem, relatively little space in the literature has been devoted to the linear ordering problem so far. This is particularly true for the question of developing good heuristic algorithms solving this problem.In the paper we propose a new heuristic algorithm solving the linear ordering problem. In this algorithm we made use of the sorting through insertion pattern as well as of the operation of permutation reversal. The surprisingly positive effect of the reversal operation, justified in part theoretically and confirmed in computational examples, seems to be the result of a unique property of the problem, called in the paper the symmetry of the linear ordering problem. This property consists in the fact that if a given permutation is an optimal solution of the problem with the criterion function being maximized, then the reversed permutation is a solution of the problem with the same criterion function being minimized.     

Production planning with discounting and stochastic demands

A general continuous review production planning problem with stochastic demand is considered. Conditions under which the stochastic problem may be correctly solved using an equivalent deterministic problem are developed. This deterministic problem is known to have the same solution as the stochastic problem. Moreover, conditions are established under which the deterministic equivalent problem differs from a commonly used deterministic approximation to the problem only in the interest rate used in discounting. Thus, solving the stochastic problem is no more difficult than solving a commonly used approximation of the problem.     

A Dynamical Systems Analysis of Semidefinite Programming with Application to Quadratic Optimization with Pure Quadratic Equality Constraints

This paper considers the problem of minimizing a quadratic cost subject to purely quadratic equality constraints. This problem is tackled by first relating it to a standard semidefinite programming problem. The approach taken leads to a dynamical systems analysis of semidefinite programming and the formulation of a gradient descent flow which can be used to solve semidefinite programming problems. Though the reformulation of the initial problem as a semidefinite pro- gramming problem does not in general lead directly to a solution of the original problem, the initial problem is solved by using a modified flow incorporating a penalty function. Accepted 10 March 1998     

Complexities and algorithms for synchronized scheduling of parallel machine assembly and air transportation in consumer electronics supply chain

In this paper, we study the problem of synchronized scheduling of assembly and air transportation to achieve accurate delivery with minimized cost in consumer electronics supply chain. This problem was motivated by a major PC manufacturer in consumer electronics industry. The overall problem is decomposed into two sub-problems, which consist of an air transportation allocation problem and an assembly scheduling problem. The air transportation allocation problem is formulated as an integer linear programming problem with the objective of minimizing transportation cost and delivery earliness tardiness penalties. The assembly scheduling problem seeks to determine a schedule ensuring that the orders are completed on time and catch the flights such that the waiting penalties between assembly and transportation is minimized. The problem is formulated as a parallel machine scheduling problem with earliness penalties. The computational complexities of the two sub-problems are investigated. The air transportation allocation problem with split delivery is shown to be solvable. The parallel machine assembly scheduling problem is shown to be NP-complete. Simulated annealing based heuristic algorithms are presented to solve the parallel machine problem.     

A note on solving nonlinear minimax problems via a differentiable penalty function

The note demonstrates that modeling a nonlinear minimax problem as a nonlinear programming problem and applying a classical differentiable penalty function to attempt to solve the problem can lead to convergence to a stationary point of the penalty function which is not a feasible point of the nonlinear programming problem. This occurred naturally in an application from statistical reliability theory. The note resolves the problem through modification of both the problem formulation and the iterative penalty function method.     

An algorithm to solve the proportional network flow problem

The proportional network flow problem is a generalization of the equal flow problem on a generalized network in which the flow on arcs in given sets must all be proportional. This problem appears in several natural contexts, including processing networks and manufacturing networks. This paper describes a transformation on the underlying network that reduces the problem to the equal flow problem; this transformation is used to show that algorithms that solve the equal flow problem can be directly applied to the proportional network flow problem as well, with no increase in asymptotic running time. Additionally, computational results are presented for the proportional network flow problem demonstrating equivalent performance to the same algorithm for the equal flow problem.     

A semidefinite programming approach for solving fractional optimal control problems

This paper presents a numerical scheme for solving fractional optimal control. The fractional derivative in this problem is in the Riemann–Liouville sense. The proposed method, based upon the method of moments, converts the fractional optimal control problem to a semidefinite optimization problem; namely, the nonlinear optimal control problem is converted to a convex optimization problem. The Grunwald–Letnikov formula is also used as an approximation for fractional derivative. The solution of fractional optimal control problem is found by solving the semidefinite optimization problem. Finally, numerical examples are presented to show the performance of the method.