Stabilizing employment in a fluctuating resource economy

We consider a resource management problem in which the management objective is to minimize fluctuations in resource economics. Stabilizing management policies consist of memoryless state feedback control strategies for a class of discrete-time resource models which contain unknown but bounded fluctuations. The underlying theory is based on conditions developed for Lyapunov-type stability of sets. The design of the stabilizing policies is illustrated by a simulation example from resource economics. Specifically, employment of fishermen is stabilized by using a subsidizing and taxing policy in an open-access common-property fishery in which the value of the resource and the resource level fluctuate, and where entry and exit dynamics are determined by fluctuating revenues obtained from the fishery.This work was supported by NSF and AFOSR under Grant No. ECS-86-02524.The support by the Yrjö Jahnsson Foundation is greatly acknowledged.     

Positive Periodic Solutions of a Class of Non-autonomous Single Species Population Model with Delays and Feedback Control

With the help of a continuation theorem based on Gaines and Mawhin＇s coincidence degree, several verifiable criteria are established for the global existence of positive periodic solutions of a class of non-autonomous single species population model with delays （both state-dependent delays and continuous delays） and feedback control. After that, by constructing a suitable Lyapunov functional, sufficient conditions which guarantee the existence of a unique globally asymptotic stable positive periodic solution of a kind of nonlinear feedback control ecosystem are obtained. Our results extend and improve the existing results, and have further applications in population dynamics.     

Stabilizing management of fluctuating resources

We consider a resource management problem in which the management objective is to minimize fluctuations in resource stocks. Stabilizing management policies constitute the designing of memoryless state feedback control strategies for a discrete time resource model which contains unknown but bounded fluctuations. We also show that the problem of maximizing sustainable yield in an uncertain fishery can be considered as the problem of stabilizing of the stock level.The paper corresponds to an invited talk at the 14th Symposium on Operations Research, Ulm, September 6–8, 1989.The support by Yrjö Jahnsson Foundation is gratefully acknowledged.This work was supported by NSF and AFOSR under grant ECS 8602524.     

Optimal harvesting policy of an inland fishery resource under incomplete information

A new mathematical model for finding the optimal harvesting policy of an inland fishery resource under incomplete information is proposed in this paper. The model is based on a stochastic control formalism in a regime‐switching environment. The incompleteness of information is due to uncertainties involved in the body growth rate of the fishery resource: a key biological parameter. Finding the most cost‐effective harvesting policy of the fishery resource ultimately reduces to solving a terminal and boundary value problem of a Hamilton‐Jacobi‐Bellman equation: a nonlinear and degenerate parabolic partial differential equation. A simple finite difference scheme for solving the equation is then presented, which turns out to be convergent and generates numerical solutions that comply with certain theoretical upper and lower bounds. The model is finally applied to the management of Plecoglossus altivelis, a major inland fishery resource in Japan. The regime switching in this case is due to the temporal dynamics of benthic algae, the main food of the fish. Model parameter values are identified from field measurement results in 2017. Our computational results clearly show the dependence of the optimal harvesting policy on the river environmental and biological conditions. The proposed model would serve as a mathematical tool for fishery resource management under uncertainties.     

Optimal consumption,training, working time,and leisure over the life cycle

In this paper, we analyze the joint determination of optimal consumption and allocation of time between learning (accumulation of human capital), working for wages (used for consumption and accumulation of financial capital), and leisure. Using Hopf bifurcation theory, we are able to show that cyclical training, working, leisuring, and consumption are optimal under certain constellations of parameters.This research was supported by the Austrian Science Foundation under Contract No. P6601.     

Regulation of a prey-predator fishery by taxation

Bioeconomic exploitation of a prey-predator fishery is discussed by using a reasonable catch-rate function instead of the usual 'catch-per-unit-effort hypothesis'. The model is based on the assumption that only the prey species is harvested. A regulatory agency controls exploitation of the fishery by imposing a tax per unit biomass of the landed fish. The fishing effort is taken as a dynamic variable depending on the capital invested in the fishery. The dynamical system consisting of the growth equations of the species and also the fishing effort is formulated. The existence and stability of steady states are discussed. The optimal tax policy is studied with the help of Pontryagin's maximum principle. Finally, some numerical examples are carried out to illustrate the results of the model.     

Spectral method for Zakharov equation with periodic boundary conditions

This paper describes the spectral method for numerically solving Zakharov equation with periodic boundary conditions. This method is spectral method for spatial variable and difference method for time variable. We make error estimation of approximate solution and prove the convergence of spectral method. We had given the convergence rate. Also, we prove the stability of approximate method for initial values.Project supported by the Science Foundation of the Chinese Academy of Sciences.     

On the matrix completion problem for multivariate filter bank construction

We survey the main techniques for the construction of multivariate filter banks and present new results about special matrices of order four and eight suitable for their construction. Qiuhui Chen: Supported in part by NSFC under grant 10201034 and project-sponsored by SRF for ROCS, SEM. Charles A. Micchelli: Supported in part by the US National Science Foundation under grant CCR-0407476. Yuesheng Xu: All correspondence to this author. Supported in part by the US National Science Foundation under grant CCR-0407476, by the Natural Science Foundation of China under grant 10371122 and by the Chinese Academy of Sciences under the program “One Hundred Distinguished Chinese Young Scientists”.     

Optimizing venture capital investments in a jump diffusion model

We study two practical optimization problems in relation to venture capital investments and/or Research and Development (R&D) investments. In the first problem, given the amount of the initial investment and the cash flow structure at the initial public offering (IPO), the venture capitalist wants to maximize overall discounted cash flows after subtracting subsequent investments, which keep the invested company solvent. We describe this problem as a mixture of singular stochastic control and optimal stopping problems. The second problem is concerned with optimal dividend policy. Rather than selling the company at an IPO, the investor may want to harvest technological achievements in the form of dividend when it is appropriate. The optimal control policy in this problem is a mixture of singular and impulse controls. E. Bayraktar was supported in part by the National Science Foundation, under grant DMS-0604491.     

On translation invariant operators which preserve the B-spline recurrence

It was observed in [4] that the Hilbert transform of the univariate B-spline preserves the B-spline recurrence. Motivated by this observation, we characterize translation invariant operators that preserve the multivariate B-spline recurrence and analogous results are also provided for the multivariate cube spline. Charles A. Micchelli was supported in part by the US National Science of Foundation under grant CCR-0407476. Yuesheng Xu was supported in part by the US National Science Foundation under grant CCR-0407476, by the Natural Science Foundation of China under grant 10371122, by the Chinese Academy of Sciences under the program “One Hundred Distinguished Chinese Scientists” and by Ministry of Education, People’s Republic of China, under the Changjian Scholarship through Zhongshan University.     

Epi-convergent discretization of the generalized Bolza problem in dynamic optimization

The paper is devoted to well-posed discrete approximations of the so-called generalized Bolza problem of minimizing variational functionals defined via extended-real-valued functions. This problem covers more conventional Bolza-type problems in the calculus of variations and optimal control of differential inclusions as well of parameterized differential equations. Our main goal is find efficient conditions ensuring an appropriate epi-convergence of discrete approximations, which plays a significant role in both the qualitative theory and numerical algorithms of optimization and optimal control. The paper seems to be the first attempt to study epi-convergent discretizations of the generalized Bolza problem; it establishes several rather general results in this direction. Research of B. S. Mordukhovich was partially supported by the USA National Science Foundation under grants DMS-0304989 and DMS-0603846 and by the Australian Research Council under grant DP-0451168. Research of T. Pennanen was supported by the Finnish Academy of Sciences under contract No. 3385.     

Relaxation methods for problems with strictly convex separable costs and linear constraints

We consider the minimization problem with strictly convex, possibly nondifferentiable, separable cost and linear constraints. The dual of this problem is an unconstrained minimization problem with differentiable cost which is well suited for solution by parallel methods based on Gauss-Seidel relaxation. We show that these methods yield the optimal primal solution and, under additional assumptions, an optimal dual solution. To do this it is necessary to extend the classical Gauss-Seidel convergence results because the dual cost may not be strictly convex, and may have unbounded level sets. Work supported by the National Science Foundation under grant NSF-ECS-3217668.     

Multilevel augmentation methods for differential equations

We develop multilevel augmentation methods for solving differential equations. We first establish a theoretical framework for convergence analysis of the boundary value problems of differential equations, and then construct multiscale orthonormal bases in H₀^m(0,1) spaces. Finally, the multilevel augmentation methods in conjunction with the multiscale orthonormal bases are applied to two-point boundary value problems of both second-order and fourth-order differential equations. Theoretical analysis and numerical tests show that these methods are computationally stable, efficient and accurate. Dedicated to Professor Charles A. Micchelli on the occasion of his 60th birthday with friendship and esteem. Mathematics subject classifications (2000) 65J15, 65R20. Zhongying Chen: Supported in part by the Natural Science Foundation of China under grants 10371137 and 10201034, the Foundation of Doctoral Program of National Higher Education of China under grant 20030558008, Guangdong Provincial Natural Science Foundation of China under grant 1011170 and the Foundation of Zhongshan University Advanced Research Center. Yuesheng Xu: Corresponding author. Supported in part by the US National Science Foundation under grants 9973427 and 0312113, by NASA under grant NCC5-399, by the Natural Science Foundation of China under grant 10371122 and by the Chinese Academy of Sciences under the program of “One Hundred Distinguished Young Scientists”.     

Maximum principle of optimal control for functional differential systems

We show that the maximum principle holds for optimal periodic control problems governed by functional differential equations under a Lipschitz condition on the value functional. Generalizations to other boundary conditions are also considered.This research was partially supported by NSF Grant No. DMS-84-01719.The first author was partially supported by the Science Fund of the Chinese Academy of Sciences, Beijing, China.     

Large deviations and importance sampling for a tandem network with slow-down

We consider a variant of the two-node tandem Jackson network where the upstream server reduces its service rate when the downstream queue exceeds some prespecified threshold. The rare event of interest is the overflow of the downstream queue. Based on a game/subsolution approach, we rigorously identify the exponential decay rate of the rare event probabilities and construct asymptotically optimal importance sampling schemes. Research of P. Dupuis supported in part by the National Science Foundation (NSF-DMS-0404806 and NSF-DMS-0706003) and the Army Research Office (W911NF-05-1-0289). Research of K. Leder supported in part by the National Science Foundation (NSF-DMS-0404806 and NSF-DMS-0706003). Research of H. Wang supported in part by the National Science Foundation (NSF-DMS-0404806 and NSF-DMS-0706003).     

Spectral conditioning and pseudospectral growth

Using the language of pseudospectra, we study the behavior of matrix eigenvalues under two scales of matrix perturbation. First, we relate Lidskii’s analysis of small perturbations to a recent result of Karow on the growth rate of pseudospectra. Then, considering larger perturbations, we follow recent work of Alam and Bora in characterizing the distance from a given matrix to the set of matrices with multiple eigenvalues in terms of the number of connected components of pseudospectra. J. V. Burke’s research was supported in part by National Science Foundation Grant DMS-0505712. A. S. Lewis’s research was supported in part by National Science Foundation Grant DMS-0504032. M. L. Overton’s research was supported in part by National Science Foundation Grant DMS-0412049.     

A large sample study of randomly weighted bootstrap in linear models

The method of randomly weighted bootstrap is used to derive the approximation ofM-estimates in linear models. It is shown that the approximation is asymptotically valid under some mild conditions. Project partially supported by the National Natural Science Foundation of China (Grant No. 19631040), the Ph. D. Program Foundation of the National Education Commission of China and the Special Foundation of the Chinese Academy of Sciences.     

A MODEL OF TROPICAL MARINE RESERVE‐FISHERY LINKAGES

ABSTRACT. The excessive and unsustainable exploitation of our marine resources has led to the promotion of marine reserves as a fisheries management tool. Marine reserves, areas in which fishing is restricted or prohibited, can offer opportunities for the recovery of exploited stock and fishery enhancement. In this paper we examine the contribution of fully protected tropical marine reserves to fishery enhancement by modeling marine reserve‐fishery linkages. The consequences of reserve establishment on the long‐run equilibrium fish biomass and fishery catch levels are evaluated. In contrast to earlier models this study highlights the roles of both adult (and juvenile) fish migration and larval dispersal between the reserve and fishing grounds by employing a spawner‐recruit model. Uniform larval dispersal, uniform larval retention and complete larval retention combined with zero, moderate and high fish migration scenarios are analyzed in turn. The numerical simulations are based on Mombasa Marine National Park, Kenya, a fully protected coral reef marine reserve comprising approximately 30% of former fishing grounds. Simulation results suggest that the establishment of a fully protected marine reserve will always lead to an increase in total fish biomass. If the fishery is moderately to heavily exploited, total fishery catch will be greater with the reserve in all scenarios of fish and larval movement. If the fishery faces low levels of exploitation, catches can be optimized without a reserve but with controlled fishing effort. With high fish migration from the reserve, catches are optimized with the reserve. The optimal area of the marine reserve depends on the exploitation rate in the neighboring fishing grounds. For example, if exploitation is maintained at 40%, the ‘optimal’ reserve size would be 10%. If the rate increases to 50%, then the reserve needs to be 30% of the management area in order to maximize catches. However, even in lower exploitation fisheries (below 40%), a small reserve (up to 20%) provides significantly higher gains in fish biomass than losses in catch. Marine reserves are a valuable fisheries management tool. To achieve maximum fishery benefits they should be complemented by fishing effort controls.     

THE RATE OF DISCOUNT AND THE PRESERVATION OF NATURAL ENVIRONMENTS

This paper examines the relationship between the rate of discount and the the optimal level of permanently preserved natural environments. A lower rate of discount increases the demand for both preserved environments and capital accumulation. The greater demand for capital indirectly increases the demand for the productive resource inputs found within the natural environments thereby reducing the level of preservation. This indirect effect outweighs the direct effect of the rate of discount on the level of preservation if the elasticity of output with respect to capital is greater that the elasticity of output with respect to the resource input. In general, the asymptotic values of the output elasticities depend upon the rate of discount and so there can be a nonmonotonic relationship between the rate of discount and the optimal level of permanently preserved natural environments.     

Stochastic Optimization Algorithms for Pricing American Put Options Under Regime-Switching Models

This work provides a Markov-modulated stochastic approximation based approach for pricing American put options under a regime-switching geometric Brownian motion market model. The solutions of pricing American options may be characterized by certain threshold values. Here, a class of Markov-modulated stochastic approximation (SA) algorithms is developed to determine the optimal threshold levels. For option pricing in a finite horizon, a SA procedure is carried out for a fixed time T. As T varies, the optimal threshold values obtained via SA trace out a curve, called the threshold frontier. Numerical experiments are reported to demonstrate the effectiveness of the approach. Our approach provides us with a viable computational tool and has advantage in terms of the reduced computational complexity compared with the variational or quasivariational inequality methods for optimal stopping.Communicated by C. T. LeondesThis research was supported in part by the National Science Foundation under Grant DMS-0304928, and in part by the National Natural Science Foundation of China under Grant 60574069.