基于边界交叉法的高超声速滑翔飞行器多目标轨迹优化研究

针对高超声速滑翔飞行器复杂约束条件下多目标轨迹设计问题，基于边界交叉法和伪谱法提出了其多目标轨迹优化方法。首先，分析了高超声速滑翔飞行器复杂约束轨迹优化问题的特点，提出了多目标轨迹优化问题。然后，采用边界交叉法和伪谱法将多目标轨迹优化问题转化为一组单目标优化子问题，利用非线性规划算法分别求解。在优化过程中，将已求解子问题的解作为下一个子问题的初始值。利用上述方法求解了最大横程和最小峰值热流轨迹优化问题，仿真结果表明：本文方法能够有效搜索到优化轨迹的Pareto前沿，可以为高超声速滑翔飞行器轨迹设计提供参考。     

Maximizing the ratio of two convex functions over a convex set

The purpose of this article is to present an algorithm for globally maximizing the ratio of two convex functions f and g over a convex set X. To our knowledge, this is the first algorithm to be proposed for globally solving this problem. The algorithm uses a branch and bound search to guarantee that a global optimal solution is found. While it does not require the functions f and g to be differentiable, it does require that subgradients of g can be calculated efficiently. The main computational effort of the algorithm involves solving a sequence of subproblems that can be solved by convex programming methods. When X is polyhedral, these subproblems can be solved by linear programming procedures. Because of these properties, the algorithm offers a potentially attractive means for globally maximizing ratios of convex functions over convex sets. © 2006 Wiley Periodicals, Inc. Naval Research Logistics, 2006     

A shifting bottleneck heuristic for minimizing the total weighted tardiness in a job shop

We present a shifting bottleneck heuristic for minimizing the total weighted tardiness in a job shop. The method decomposes the job shop into a number of single‐machine subproblems that are solved one after another. Each machine is scheduled according to the solution of its corresponding subproblem. The order in which the single machine subproblems are solved has a significant impact on the quality of the overall solution and on the time required to obtain this solution. We therefore test a number of different orders for solving the subproblems. Computational results on 66 instances with ten jobs and ten machines show that our heuristic yields solutions that are close to optimal, and it clearly outperforms a well‐known dispatching rule enhanced with backtracking mechanisms. © 1999 John Wiley & Sons, Inc. Naval Research Logistics 46: 1–17, 1999     

The formulation of some allocation and connection problems as integer programs

In this paper a component placement problem and a digital computer backboard wiring problem are formulated as integer linear programs. The component placement problem consists of making a unique assignment of components to column positions such that wireability is maximized. The backboard wiring problem consists of three interrelated subproblems, namely, the placement, the connection, and the routing problems. The placement and connection problems are combined and solved as one, thereby giving the optimal circuit connections as well as minimizing the total lead length. It is shown that under certain assumptions, the number of inequalities and variables in the problem can be greatly reduced. Further simplifying assumptions lead to a near optimal solution. Examples of other allocation problems to which the models presented here are applicable are given. The following concepts are formulated as linear inequalities: (1) the absolute magnitude of the difference between two variables; (2) minimize the minimum function of a set of functions; and (3) counting the number of (0, 1) adjacent component pairs in a vector.     

A specially structured nonlinear integer resource allocation problem

We present an algorithm for solving a specially structured nonlinear integer resource allocation problem. This problem was motivated by a capacity planning study done at a large Health Maintenance Organization in Texas. Specifically, we focus on a class of nonlinear resource allocation problems that involve the minimization of a convex function over one general convex constraint, a set of block diagonal convex constraints, and bounds on the integer variables. The continuous variable problem is also considered. The continuous problem is solved by taking advantage of the structure of the Karush‐Kuhn‐Tucker (KKT) conditions. This method for solving the continuous problem is then incorporated in a branch and bound algorithm to solve the integer problem. Various reoptimization results, multiplier bounding results, and heuristics are used to improve the efficiency of the algorithms. We show how the algorithms can be extended to obtain a globally optimal solution to the nonconvex version of the problem. We further show that the methods can be applied to problems in production planning and financial optimization. Extensive computational testing of the algorithms is reported for a variety of applications on continuous problems with up to 1,000,000 variables and integer problems with up to 1000 variables. © 2003 Wiley Periodicals, Inc. Naval Research Logistics 50: 770–792, 2003.     

Replacing continuous demand with discrete demand in a competitive location model

Location models commonly represent demand as discrete points rather than as continuously spread over an area. This modeling technique introduces inaccuracies to the objective function and consequently to the optimal location solution. In this article this inaccuracy is investigated by the study of a particular competitive facility location problem. First, the location problem is formulated over a continuous demand area. The optimal location for a new facility that optimizes the objective function is obtained. This optimal location solution is then compared with the optimal location obtained for a discrete set of demand points. Second, a simple approximation approach to the continuous demand formulation is proposed. The location problem can be solved by using the discrete demand algorithm while significantly reducing the inaccuracies. This way the simplicity of the discrete approach is combined with the approximated accuracy of the continuous-demand location solution. Extensive analysis and computations of the test problem are reported. It is recommended that this approximation approach be considered for implementation in other location models. © 1997 John Wiley & Sons, Inc.     

A finite algorithm for concave minimization over a polyhedron

We present a new algorithm for solving the problem of minimizing a nonseparable concave function over a polyhedron. The algorithm is of the branch-and-bound type. It finds a globally optimal extreme point solution for this problem in a finite number of steps. One of the major advantages of the algorithm is that the linear programming subproblems solved during the branch-and-bound search each have the same feasible region. We discuss this and other advantages and disadvantages of the algorithm. We also discuss some preliminary computational experience we have had with our computer code for implementing the algorithm. This computational experience involved solving several bilinear programming problems with the code.     

The use of dynamic programming methodology for the solution of a class of nonlinear programming problems

This paper presents an application of a method for finding the global solution to a problem in integers with a separable objective function of a very general form. This report shows that there is a relationship between an integer problem with a separable nonlinear objective function and many constraints and a series of nonlinear problems with only a single constraint, each of which can be solved sequentially using dynamic programming. The first solution to any of the individual smaller problems that satisfies the original constraints in addition, will be the optimal solution to the multiply-constrained problem.     

Capacity expansion and cost efficiency improvement in the warehouse problem

The warehouse problem with deterministic production cost, selling prices, and demand was introduced in the 1950s and there is a renewed interest recently due to its applications in energy storage and arbitrage. In this paper, we consider two extensions of the warehouse problem and develop efficient computational algorithms for finding their optimal solutions. First, we consider a model where the firm can invest in capacity expansion projects for the warehouse while simultaneously making production and sales decisions in each period. We show that this problem can be solved with a computational complexity that is linear in the product of the length of the planning horizon and the number of capacity expansion projects. We then consider a problem in which the firm can invest to improve production cost efficiency while simultaneously making production and sales decisions in each period. The resulting optimization problem is non‐convex with integer decision variables. We show that, under some mild conditions on the cost data, the problem can be solved in linear computational time. © 2016 Wiley Periodicals, Inc. Naval Research Logistics 63: 367–373, 2016     

A heuristic routine for solving large loading problems

The loading problem involves the optimal allocation of n objects, each having a specified weight and value, to m boxes, each of specified capacity. While special cases of these problems can be solved with relative ease, the general problem having variable item weights and box sizes can become very difficult to solve. This paper presents a heuristic procedure for solving large loading problems of the more general type. The procedure uses a surrogate procedure for reducing the original problem to a simpler knapsack problem, the solution of which is then employed in searching for feasible solutions to the original problem. The procedure is easy to apply, and is capable of identifying optimal solutions if they are found.     

Robust shortest path problems with two uncertain multiplicative cost coefficients

We consider a robust shortest path problem when the cost coefficient is the product of two uncertain factors. We first show that the robust problem can be solved in polynomial time by a dual‐variable enumeration with shortest path problems as subproblems. We also propose a path enumeration approach using a K ‐shortest paths finding algorithm that may be efficient in many real cases. An application in hazardous materials transportation is discussed, and the solution methods are illustrated by numerical examples. © 2013 Wiley Periodicals, Inc. Naval Research Logistics, 2013     

A continuous submarine versus submarine game

This paper analyzes, from a game-theoretic standpoint, the simultaneous choice of speeds by a transitor and by an SSK which patrols back and forth perpendicular to the transitor's course. Using idealized acoustic assumptions and a cookie-cutter detection model which ignores counterdetection, we are able to present the problem as a continuous game, and to determine an analytic solution. The results indicate that with these assumptions, there are conditions under which neither a “go fast” nor a “go slow” strategy is optimal. The game provides a good example of a continuous game with a nontrivial solution which can be solved effectively.     

A lagrangean relaxation algorithm for the constrained matrix problem

We present variants of a convergent Lagrangean relaxation algorithm for minimizing a strictly convex separable quadratic function over a transportation polytope. The algorithm alternately solves two “subproblems,” each of which has an objective function that is defined by using Lagrange multipliers derived from the other. Motivated by the natural separation of the subproblems into independent and very easily solved “subsubproblems,” the algorithm can be interpreted as the cyclic coordinate ascent method applied to the dual problem. We exhibit our computational results for different implementations of the algorithm applied to a set of large constrained matrix problems.     

Balancing a one‐way corridor capacity and safety‐oriented reliability: A stochastic optimization approach for metro train timetabling

In urban rail transit systems of large cities, the headway and following distance of successive trains have been compressed as much as possible to enhance the corridor capacity to satisfy extremely high passenger demand during peak hours. To prevent train collisions and ensure the safety of trains, a safe following distance of trains must be maintained. However, this requirement is subject to a series of complex factors, such as the uncertain train braking performance, train communication delay, and driver reaction time. In this paper, we propose a unified mathematical framework to analyze the safety‐oriented reliability of metro train timetables with different corridor capacities, that is, the train traffic density, and determine the most reliable train timetable for metro lines in an uncertain environment. By employing a space‐time network representation in the formulations, the reliability‐based train timetabling problem is formulated as a nonlinear stochastic programming model, in which we use 0‐1 variables to denote the time‐dependent velocity and position of all involved trains. Several reformulation techniques are developed to obtain an equivalent mixed integer programming model with quadratic constraints (MIQCP) that can be solved to optimality by some commercial solvers. To improve the computational efficiency of the MIQCP model, we develop a dual decomposition solution framework that decomposes the primal problem into several sets of subproblems by dualizing the coupling constraints across different samples. An exact dynamic programming combined with search space reduction strategies is also developed to solve the exact optimal solutions of these subproblems. Two sets of numerical experiments, which involve a relatively small‐scale case and a real‐world instance based on the operation data of the Beijing subway Changping Line are implemented to verify the effectiveness of the proposed approaches.     

基于非线性最优反馈控制的远程拦截闭环制导

利用非线性最优反馈控制和伪谱轨迹快速重构技术,设计了一种有限推力空间远程拦截自适应闭环制导方法。首先建立了有限推力远程拦截最优制导问题模型,并给出了适用于该问题的非线性最优反馈控制求解原理。然后,将空间变轨动力学模型特点和伪谱法相结合,设计了基于状态量缩减的计算效率改进策略以提高轨迹优化的实时性。基于改进伪谱法进行连续轨迹快速重构,利用开环最优解形成闭环反馈,从而保证制导指令的实时更新,并通过引入控制逻辑对制导算法进行改进。时间最短远程拦截仿真表明,该闭环制导方法在保证任务指标最优性的同时,可以有效抑制J2摄动和计算误差的影响,具有较高的制导精度、自适应性和鲁棒性。     

Value management of diagnostic equipment with cancelation,no‐show,and emergency patients

Magnetic resonance imaging and other multifunctional diagnostic facilities, which are considered as scarce resources of hospitals, typically provide services to patients with different medical needs. This article examines the admission policies during the appointment management of such facilities. We consider two categories of patients: regular patients who are scheduled in advance through an appointment system and emergency patients with randomly generated demands during the workday that must be served as soon as possible. According to the actual medical needs of patients, regular patients are segmented into multiple classes with different cancelation rates, no‐show probabilities, unit value contributions, and average service times. Management makes admission decisions on whether or not to accept a service request from a regular patient during the booking horizon to improve the overall value that could be generated during the workday. The decisions should be made by considering the cancelation and no‐show behavior of booked patients as well as the emergency patients that would have to be served because any overtime service would lead to higher costs. We studied the optimal admission decision using a continuous‐time discrete‐state dynamic programming model. Identifying an optimal policy for this discrete model is analytically intractable and numerically inefficient because the state is multidimensional and infinite. We propose to study a deterministic counterpart of the problem (i.e., the fluid control problem) and to develop a time‐based fluid policy that is shown to be asymptotically optimal for large‐scale problems. Furthermore, we propose to adopt a mixed fluid policy that is developed based on the information obtained from the fluid control problem. Numerical experiments demonstrate that this improved policy works effectively for small‐scale problems. © 2016 Wiley Periodicals, Inc. Naval Research Logistics 63: 287–304, 2016     

Interdiction models for delaying adversarial attacks against critical information technology infrastructure

Information technology (IT) infrastructure relies on a globalized supply chain that is vulnerable to numerous risks from adversarial attacks. It is important to protect IT infrastructure from these dynamic, persistent risks by delaying adversarial exploits. In this paper, we propose max‐min interdiction models for critical infrastructure protection that prioritizes cost‐effective security mitigations to maximally delay adversarial attacks. We consider attacks originating from multiple adversaries, each of which aims to find a “critical path” through the attack surface to complete the corresponding attack as soon as possible. Decision‐makers can deploy mitigations to delay attack exploits, however, mitigation effectiveness is sometimes uncertain. We propose a stochastic model variant to address this uncertainty by incorporating random delay times. The proposed models can be reformulated as a nested max‐max problem using dualization. We propose a Lagrangian heuristic approach that decomposes the max‐max problem into a number of smaller subproblems, and updates upper and lower bounds to the original problem via subgradient optimization. We evaluate the perfect information solution value as an alternative method for updating the upper bound. Computational results demonstrate that the Lagrangian heuristic identifies near‐optimal solutions efficiently, which outperforms a general purpose mixed‐integer programming solver on medium and large instances.     

Solving a selected class of location problems by exploiting problem structure: A decomposition approach

For many combinatorial optimization problems that are NP-hard, a number of special cases exist that can be solved in polynomial time. This paper addresses the issue of solving one such problem, the well-known m-median problem with mutual communication (MMMC), by exploiting polynomially solvable special cases of the problem. For MMMC, a dependency graph is defined that characterizes the structure of the interactions between decision variables. A Lagrangian decomposition scheme is proposed that partitions the problem into two or more subproblems, each having the same structure as the original problem, but with simpler dependency graphs. The dual problems are solved using subgradient or multiplier adjustment methods. An efficient method of adjusting the multiplier values is given. Computational results are reported that show the method to be quite effective. In addition, applications of the approach to other difficult location problems is discussed. © 1998 John Wiley & Sons, Inc. Naval Research Logistics 45: 791–815, 1998     

采用PWPF调节器的再入飞行器最优控制分配方法

针对具有反作用控制系统(Reaction Control System,RCS)和气动舵两类控制机构的再入飞行器,提出一种基于脉宽脉频(Pulse-Width Pulse-Frequency,PWPF)调节器的最优控制分配方法。将RCS的输入信号转化为连续变化量,RCS与气动舵的控制分配问题被描述为二次规划问题,并采用有效集方法对其求解。采用离散法和PWPF调节器将优化结果转化为RCS的开关机状态。与混合整数规划问题相比,连续二次规划问题更容易求解,计算速度更快。通过对二次规划问题的重构,该算法能有效地应对故障情况。     

Modeling the operation of multireservoir systems using decomposition and stochastic dynamic programming

Stochastic dynamic programming models are attractive for multireservoir control problems because they allow non‐linear features to be incorporated and changes in hydrological conditions to be modeled as Markov processes. However, with the exception of the simplest cases, these models are computationally intractable because of the high dimension of the state and action spaces involved. This paper proposes a new method of determining an operating policy for a multireservoir control problem that uses stochastic dynamic programming, but is practical for systems with many reservoirs. Decomposition is first used to reduce the problem to a number of independent subproblems. Each subproblem is formulated as a low‐dimensional stochastic dynamic program and solved to determine the operating policy for one of the reservoirs in the system. © 2006 Wiley Periodicals, Inc. Naval Research Logistics, 2006