Necessary conditions for infinite-horizon discounted two-stage optimal control problems

In many situations in Economics, one would like to analyse the optimal timing of switching between alternative and consecutive regimes. A natural framework for such an analysis seems to be one of multi-stage optimal control problems, where the switching instants between regimes are endogenously determined. The existing literature considers only the case of two-stage optimal control problems with finite horizon. However, in many cases, it seems more appropriate to consider a problem with an infinite horizon. By deriving the appropriate necessary conditions for such a problem we generalise the methodology in analysing endogenous regime switches.     

Delivery lags and buffer stocks in the theory of investment by the firm

This paper contains a dynamic model of a firm holding a stock of unfilled orders for output while facing a fixed delivery lag attached to acquiring new capital goods. The model leads to a two-stage optimal control problem where the initial time interval corresponds to the Marshallian short-run and the second interval is the period during which deliveries of capital goods may arrive. Optimality criteria are provided for both time intervals and the behavior of the firm is characterized along both adjustment paths. The short-run decisions of the firm are shown to be tied to long-run decisions planned by the firm.     

两阶段法求解混装模式下的加热炉调度

针对目前加热炉调度模型少有考虑混装模式下加热炉调度优化的不足,建立了连铸一热轧混装一体化模式下的加热炉生产调度优化模型,并提出了基于贪婪算法和模拟退火算法的两阶段求解方法。生产数据测试表明该模型和算法能有效解决加热炉调度问题。     

Scientific planning of urban cordon sanitaire for desired queuing time

The outbreak of the COVID-19 pandemic disrupted ofur normal life. Many cities enforced a cordon sanitaire as a countermeasure to protect densely inhabited areas. Travelers can only cross the cordon after being checked. To minimize the waiting time in the queue, this paper proposes a method to determine the scientific planning of urban cordon sanitaire for desired queuing time, which is a significant problem that has not been explored. A novel two-stage optimization model is proposed where the first stage is the transportation system equilibrium problem to predict traffic inflow, and the second stage is the queuing network design problem to determine the allocation of test stations. This method aims to minimize the total health infrastructure investment for the desired maximum queuing time. Note that queuing theory is used to represent the queuing phenomenon at each urban entrance. A heuristic algorithm is designed to solve the proposed model where the Method of Successive Averages (MSA) is adopted for the first stage, and the Genetic Algorithm (GA) with elite strategy is adopted for the second stage. An experimental study with sensitivity analysis is conducted to demonstrate the effectiveness of the proposed methods. The results show that the methods can find a good heuristic optimal solution. This research is helpful for policymakers to determine the optimal investment and planning of cordon sanitaire for disease prevention and control, as well as other criminal activities such as drunk driving, terrorists, and smuggling.     

On a use of stratified sampling in optimal estimation

Summary Postulating an intra-class correlation structure for a finite population vector of variate — values optimal stratified and unstratified strategies for estimating the total are identified and relative efficiencies noted. Higher efficiency of stratified over unstratified sampling is demonstrated for usual estimators in ppswr case, in particular. In the uncorrelated case appropriately chosen two-stage strategies are observed to fare at par with optimal stratified ones.     

Regional economic models and continuous programming

This paper is concerned with an optimal investment allocation problem in a simple N-regional economic model. The problem is described as a class of optimal control problem, and formulated into a continuous linear programming problem. Both the primal and dual problems are considered. The procedure finds an optimal regional allocation of investment derived in terms of continuous programming.     

An algorithm for the solution of stochastic optimal control problems for large nonlinear econometric models

This paper considers the problem of solving an optimal control problem for large dynamic economic models which are both nonlinear and stochastic. It proposes a technique which combines conventional deterministic optimal control algorithms with the procedure of stochastic simulation, which calculates a numerical approximation to the distribution of the models endogenous variables. The new technique is computationally feasible for even large nonlinear models and, as an illustration of this, the Bank of England's large quarterly forecasting model is used in an example.     

A simplified treatment of the theory of optimal regulation of Brownian motion

Consider a stochastic control problem where the state variable follows a Brownian motion. The flow reward is a function of the state, which can be regulated with a lump-sum and linear cost of adjustment. Using a discrete approximation, a simple exposition of the control problem is developed. The value matching conditions that hold for any given control parameters, and the smooth pasting conditions that hold for the optimal control are derived. Some extensions and economic applications of the method are also discussed.     

Optimal allocation in J-stage sampling with stratification from finite population

Summary  The known results of optimal allocation in two-stage simple random sampling with stratification from finite population are generalized for J-stages. The optimalization is treated by minimizing the cost function for given variance as well as by minimizing the variance for given cost. It turns out that for both cases the optimal values differ only for the first-stage units and are the same for all the subsequent. The general formula for minimum variance of the unbiased estimator of the population mean per element is given.     

Optimal allocation in J-stage sampling with stratification from finite population

Summary The known results of optimal allocation in two-stage simple random sampling with stratification from finite population are generalized for J-stages. The optimalization is treated by minimizing the cost function for given variance as well as by minimizing the variance for given cost. It turns out that for both cases the optimal values differ only for the first-stage units and are the same for all the subsequent. The general formula for minimum variance of the unbiased estimator of the population mean per element is given.     

A ‘funnel’ turnpike theorem for optimal growth problems with discounting

A turnpike theorem for the optimal control problem with discounting is given. The optimal trajectory is shown to lie in an exponentially bounded region of the optimal steady-state. This region, referred to as a funnel, is determined by the discount rate of the problem. The funnel theorem reduces to the classical turnpike theorem when the discount rate is zero.     

A two-stage sampling by unequal size samples from two normal populations with unknown variances

A two-stage procedure based on the statistic introduced by Scheffé (1943) to solve the Behrens-Fisher problem is considered in estimating the difference between the mean values of two normal distributions having unequal and unknown variances. Samples of unequal size are considered at each stage. A numerical example is given where the two-stage procedure is compared with the Satterthwaite (1946) approximate method.     

Optimal regime switching and threshold effects

We consider a general control problem with two types of optimal regime switch. The first one concerns technological and/or institutional regimes indexed by a finite number of discrete parameter values, and the second features regimes relying on given threshold values for given state variables. We propose a general optimal control framework allowing to derive the first-order optimality conditions and in particular to characterize the geometry of the shadow prices at optimal switching times (if any). We apply this new optimal control material to address the problem of the optimal management of natural resources under ecological irreversibility, and with the possibility to switch to a backstop technology.     

完全信息下的最优保费控制问题研究

应用最大值原理研究了一类完全信息下最优保费控制问题,考虑了折现因子为函数且具有终端条件限制的情况,建立了完全信息下的正倒向随机控制系统。获得了唯一的递归的最优保费策略。     

Optimal investment with taxes: an existence result

We study the deterministic control problem of maximizing utility from consumption of an agent who seeks to optimally allocate his wealth between consumption and investment in a financial asset subject to taxes on benefits with first-in–first-out priority rule on sales. Short sales are prohibited and consumption is restricted to be non-negative. Such a problem has been introduced in a previous paper by the same authors where the first-order conditions have been derived. In this paper, we establish an existence result for this non-classical optimal control problem.     

Reliable design of a congested disaster relief network: A two-stage stochastic-robust optimization approach

The preparedness of humanitarian relief networks can be enhanced by pre-positioning resources in strategic locations and using them when disasters strike, a strategy that gives rise to a two-stage planning problem. This paper presents a novel two-stage stochastic-robust optimization approach for integrated planning of pre- and post-disaster positioning and allocation of relief resources, while taking into consideration the uncertainty about demand for relief services and disruptions in the relief facilities and the transportation network. The proposed approach enables planners to effectively use limited historical data and imperfect experts’ opinions to obtain robust solutions while avoiding the over-conservatism of classical robust optimization methods. The objective sought is to minimize the expected total time victims need to receive assistance, including both access time to facilities and waiting/service time in them. Congestion in relief facilities is accounted for by modeling them as queuing systems and penalizing waiting time. A decomposition method based on column-and-constraint generation is implemented to solve the problem, whereas the nonlinear terms corresponding to queuing in the second-stage problem are handled using a direct search procedure. Applicability of the proposed approach is demonstrated through a real case study and the numerical results are analyzed to draw managerial insights.     

Optimal control on an infinite time horizon: The turnpike approach

The optimal control problem with infinite horizon is defined. Under assumptions motivated by the theory of optimal economic growth the controllability of the system, the turnpike property, the asymptotic behavior and the existence of optimal trajectories are fully explored.     

禁忌搜索算法在车辆调度决策中的应用

论文探讨了交通线路可能被敌方毁坏情况下的车辆路径优化问题，建立了不确定的两阶段整数规划数学模型，分析了路径方案的最大和最小效用值。在线路方案评价值的计算中只计算最小值和最大值，简化了方案的目标值计算。设计了求解模型的双层禁忌搜索算法。最后通过应用示例验证了模型和算法的有效性。     

Optimal strategies in linear multisector models: Value function and optimality conditions

In this paper we study an optimal control problem with mixed constraints related to a multisector linear model with endogenous growth. The main aim is to establish a set of necessary and a set of sufficient conditions which are the basis for studying the qualitative properties of optimal trajectories. The presence of possibly degenerate mixed constraints, the unboundedness and non-strict convexity of the Hamiltonian, make the problem difficult to deal with. We develop first the dynamic programming approach, proving that the value function is a bilateral viscosity solution to the associated Hamilton–Jacobi–Bellman (HJB) equation. Then, using our results, we give a set of sufficient and a set of necessary optimality conditions which involve so-called co-state inclusion: this can be interpreted as the existence of a dual path of prices supporting the optimal path.     

Optimal impulse control for cash management¶with quadratic holding-penalty costs

This paper studies optimal control for an infinite horizon cash management problem where the cash fund fluctuates as a Brownian motion. Holding-penalty costs are assumed to be a quadratic function of the cash level and there are fixed and proportional transaction costs. Using the “impulse technique”, we prove that optimal control exists and takes the form of a control band policy. Received: 4 January 2001 / Accepted: 5 June 2001