Optimal consumption models in economic growth

We study the optimal consumption problem in the one-sector model of economic growth under uncertainty. We show the existence of a classical solution of the Hamilton-Jacobi-Bellman equation associated with the stochastic optimization problem, and then give an optimal consumption policy in terms of its solution.     

Singular Arcs in the Optimal Evasion Against a Proportional Navigation Vehicle

The two-dimensional optimal evasion problem against a proportional navigation pursuer is analyzed using a nonlinear model. The velocities of both players have constant modulus, but change in direction. The problem is to determine the time-minimum trajectory (disengagement) or time-maximum trajectory (evasion) of the evader while moving from the assigned initial conditions to the final conditions. A maximum principle procedure allows one to reduce the optimal control problem to the phase portrait analysis of a system of two differential equations. The qualitative features of the optimal process are determined.     

Optimal control and long-run dynamics for a spatial economic growth model with physical capital accumulation and pollution diffusion

In this work we analyze the large-time behavior of a spatially structured economic growth model coupling physical capital accumulation and pollution diffusion. This model extends other results in the literature along different directions. Alongside the classical Cobb–Douglas production function, a convex–concave production function is considered. We add a negative feedback to the production function in order to describe the (negative) influence of pollution on output, and therefore on capital accumulation. We also present an optimal control problem for the above model.     

An extension of the extended basic period approach for economic lot scheduling problems

This paper addresses the problem of choosing cyclical production patterns for several products which are produced on a common production facility. The extended basic period approach of Elmaghraby is extended by allowing more than two basic periods. The problem of dimensionality of the dynamic programming formulation is dealt with by aggregating the capacity data in different basic periods.The author is indebted to Dr. S. E. Elmaghraby for inspiring him to work on the economic lot scheduling problem and for many valuable suggestions. The author is also grateful to Mr. Z. Naman for carrying out the computer programming.     

Discounted dynamic programming with unbounded returns: Application to economic models

In this paper, we study discounted Markov decision processes on an uncountable state space. We allow a utility (reward) function to be unbounded both from above and below. A new feature in our approach is an easily verifiable rate of growth condition introduced for a positive part of the utility function. This assumption, in turn, enables us to prove the convergence of a value iteration algorithm to a solution to the Bellman equation. Moreover, by virtue of the optimality equation we show the existence of an optimal stationary policy.     

Optimal regulation policies for a multipurpose reservoir with seasonal input and return function

Optimal regulation policies for a multipurpose reservoir are characterized analytically for the case where the input and return functions are subject to seasonal variations and the return function is piecewise linear, concave, and separable in reservoir volume and discharge. Efficient upper and lower bounds on optimal policies are derived for two important special cases. Necessary and sufficient conditions are obtained for a finite-time horizon optimal policy to be also optimal for the infinite-time horizon.This research was supported in part by the Office of Water Resources Research, Grant No. A-043-CAL, through the Water Resources Center, University of California.     

Optimal control of a facility with periodic interrupted demand

The system investigated consists of a stochastic periodic stream of raw material, a continuous processing operation with controllable deterministic service rates, and a storage facility. The arrival stream is periodically interrupted and divided into alternating on-off intervals of fixed length. The processing facility is allowed to operate during the off-interval. Superimposed on this system is a cost structure composed of processing and holding costs. Such operations may be found in manufacturing as well as service systems (for example, dry cleaners, machine shops, repair and maintenance shops, printers, information processing centers, etc). A service rate control rule that minimizes the infinite-horizon discounted expected total cost is found. Existence and uniqueness of long-term optimal cost and policy functions is shown. Since the optimal policy cannot be expressed explicitly, an approximate solution was obtained. An error bound on the optimal cost associated with this solution is exhibited. The approximate solution is characterized by a service rate control rule that is a linear function of the level of inventory at the start of each on-interval and a piecewise linear function of inventory at the start of each off-interval. The optimal discounted expected total cost is quadratic in the inventory level at the start of each interval. Computational results indicate relative cost errors in the order of 2–3 percent.This research was performed at the Sanitary Engineering Research Laboratory and Operations Research Center of the University of California, Berkeley. It was made possible by US Public Health Research Grant UI-00547 from the Environmental Control Administration-Bureau of Solid Waste Management and by National Science Foundation Grant GK-1684.The author thanks Professor C. R. Glassey for not only suggesting this research, but for his constant encouragement and suggestions throughout its duration. He also thanks Professors W. S. Jewell and P. H. McGauhey whose comments on the draft were very helpful.     

Optimal control of a dam under seasonal electricity prices

This study concerns the optimal control of a hydroelectric dam under seasonal electricity prices: high in winter, low in summer. The goal is to maximize the expected discounted infinite-horizon return through a policy that determines the amount of electricity to be produced in each time period, depending on the water level at the beginning of the period under consideration. The prices are assumed to be deterministic, and the flows into the reservoir are seasonal, stochastic, but independent from one period to another. The electric power generated is proportional to the amount of water flow through the turbines. There exist seepage and evaporation losses.It is shown that in the simplest price structure, the optimal policy is entirely determined by a single critical water level in each period of time, at which one starts producing. An example shows that the discretization of the reservoir levels can destroy this property. A method is proposed to avoid this difficulty. Another way of defining a policy is through goal-levels. This approach is shown to give higher returns than the standard approach.     

Optimal control synthesis in therapy of solid tumor growth

A mathematical model of tumor growth therapy is considered. The total amount of a drug is bounded and fixed. The problem is to choose an optimal therapeutic strategy, i.e., to choose an amount of the drug permanently affecting the tumor that minimizes the number of tumor cells by a given time. The problem is solved by the dynamic programming method. Exact and approximate solutions to the corresponding Hamilton-Jacobi-Bellman equation are found. An error estimate is proved. Numerical results are presented.     

Dynamics and optimal control in a spatially structured economic growth model with pollution diffusion and environmental taxation

In the first part of this paper we present a spatially structured dynamic economic growth model which takes into account the level of pollution and a possible taxation based on the amount of produced pollution. In the second part we analyze an optimal harvesting control problem with an objective function composed of three terms, namely the intertemporal utility of the decision maker, the space–time average of the level of pollution in the habitat, and the disutility due to the imposition of taxation.     

关于“一类最优指派问题的动态规划模型”的注记

考虑一类较一般的最优指派问题 :欲指派 m个人做 n项工作 (m≥n) ,要求每个人只做一项工作 ,第j项工作可以由 bj个人共同去做 ,其中 bj是待求未知数 ,满足 dj≤ bj≤ ej(即 ej,dj为第 j项工作所需人数的上下限 )及 ∑nj=1bj=m(即每个人都有工作 ) ,dj,ej为已知常数 ,j =1 ,… ,n.第 i人做第 j项工作的效益为 cij≥ 0 ,i =1 ,… ,m;j =1 ,… ,n.本文建立求解上述最优指派问题 (使总的效益最大 )的动态规划模型 ,并将文 [1]作为本文的特例 .     

An application of traveling wave analysis in economic growth model

A temporal–spatial economic growth model is established in this paper. As a useful tool, traveling wave analysis is used to analyze technological growth and diffusion. Numerical simulation shows that this model has perfect performance.     

Optimal cost and policy for a Markovian replacement problem

We consider the computation of the optimal cost and policy associated with a two-dimensional Markov replacement problem with partial observations, for two special cases of observation quality. Relying on structural results available for the optimal policy associated with these two particular models, we show that, in both cases, the infinitehorizon, optimal discounted cost function is piecewise linear, and provide formulas for computing the cost and the policy. Several examples illustrate the usefulness of the results.This research was supported by the Air Force Office of Scientific Research Grant AFOSR-86-0029, by the National Science Foundation Grant ECS-86-17860, by the Advanced Technology Program of the State of Texas, and by the Air Force Office of Scientific Research (AFSC) Contract F49620-89-C-0044.     

Heston随机波动率市场中带VaR约束的最优投资策略

本文研究了Heston随机波动率市场下, 基于VaR约束下的动态最优投资组合问题。
假设Heston随机波动率市场由一个无风险资产和一个风险资产构成,投资者的目标为最大化其终端的期望效用。与此同时, 投资者将动态地评估其待选的投资组合的VaR风险,并将其控制在一个可接受的范围之内。本文在合理的假设下,使用动态规划的方法,来求解该问题的最优投资策略。在特定的参数范围内,利用数值方法计算出近似的最优投资策略和相应值函数, 并对结果进行了分析。     

基于动态VaR约束与随机波动率模型的最优投资策略

研究Stein-Stein随机波动率模型下带动态VaR约束的最优投资组合选择问题. 假设投资者的目标是最大化终端财富的期望幂效用,可投资于无风险资产和一种风险资产, 风险资产的价格过程由Stein-Stein随机波动率模型刻画. 同时, 投资者期望能在投资过程中利用动态VaR约束控制所面对的风险.运用Bellman动态规划方法和Lagrange乘子法, 得到了该约束问题最优策略的解析式及特殊情形下最优值函数的解析式; 并通过理论分析和数值算例, 阐述了动态VaR约束与随机波动率对最优投资策略的影响.     

Optimal control of a class of systems with continuous lags: Dynamic programming approach and economic interpretations

This paper derives a maximum principle for dynamic systems with continuous lags, i.e., systems governed by integrodifferential equations, using dynamic programming. As a result, the adjoint variables can be provided with useful economic interpretations.This research was supported by NSERC Grant No. A4619.     

Optimal compactification of a floorplan and its relation to other optimization problems-a dynamic programming approach

We examine an algorithm for the compactification of an arrangement of rectangles in the plane as it is used for floorplans in the automated design of electronic circuits (also called sizing of floorplans). We reformulate this problem as a multistage decision problem and show that the algorithm is in fact the optimal solution obtained by the backward induction procedure of dynamic programing. The model allows generalisations to non-geometrical applications in scheduling and reliability.     

延迟退休和经济增长

§ 1　IntroductionPopulation aging is a common feature in developed economies.How to deal with it?Most governments turn to postponing retirement.What impact does it have on macroeco-nomic performance?For instance,if a longer working time decreases the ratio of retiredpeople to working people,does itincrease the aggregate saving rate?Doesittherefore accel-erate economic growth?We study these issues in a general equilibrium model of life cyclesavings combined with endogenous growth.An intuitiv…     

Optimal policy for a dynamic,non-stationary,stochastic inventory problem with capacity commitment

This paper studies a single-product, dynamic, non-stationary, stochastic inventory problem with capacity commitment, in which a buyer purchases a fixed capacity from a supplier at the beginning of a planning horizon and the buyer’s total cumulative order quantity over the planning horizon is constrained with the capacity. The objective of the buyer is to choose the capacity at the beginning of the planning horizon and the order quantity in each period to minimize the expected total cost over the planning horizon. We characterize the structure of the minimum sum of the expected ordering, storage and shortage costs in a period and thereafter and the optimal ordering policy for a given capacity. Based on the structure, we identify conditions under which a myopic ordering policy is optimal and derive an equation for the optimal capacity commitment. We then use the optimal capacity and the myopic ordering policy to evaluate the effect of the various parameters on the minimum expected total cost over the planning horizon.