Recursive methods offer a powerful approach in dynamic macroeconomics. This book contains both an introduction to recursive tools, including standard applications such as asset pricing, and advanced material, including analyses of reputational mechanisms and contract design. The tools are presented with enough technical sophistication to get the reader started working on practical problems. When numerical simulations are called for, the book provides suggestions for how to proceed, as well as references for further reading. The applications cover many substantive issues in macroeconomics, such as equilibrium asset prices, market incompleteness, wealth distribution, fiscal-monetary theories of inflation, government debt, optimal labour and capital taxation, time consistency and credible government policies, optimal social insurance, economic growth and labour market dynamics.

Recursive Macroeconomic Theory

We introduce several solution concepts for multicriteria optimization problems, give a characterization of approximately efficient elements and discuss a general scalarization procedure. Furthermore, we derive necessary and sufficient optimality conditions, a minimal point theorem, a vector-valued variational principle of Ekeland’s type, Lagrangean multiplier rules and duality statements. An overview on vector variational inequalities and vector equilibria is given. Moreover, we discuss the results for special classes of vector optimization problems (vector-valued location and approximation problems, multicriteria fractional programming and optimal control problems).

Theory of Vector Optimization

http://www.yaroslavvb.com/papers/blondel-survey.pdf

Survey A survey of computational complexity results in systems and control

Power grids, flexible manufacturing, cellular communications: interconnectedness has consequences. This remarkable book gives the tools and philosophy you need to build network models detailed enough to capture essential dynamics but simple enough to expose the structure of effective control solutions and to clarify analysis. Core chapters assume only exposure to stochastic processes and linear algebra at the undergraduate level; later chapters are for advanced graduate students and researchers/practitioners. This gradual development bridges classical theory with the state-of-the-art. The workload model that is the basis of traditional analysis of the single queue becomes a foundation for workload relaxations used in the treatment of complex networks. Lyapunov functions and dynamic programming equations lead to the celebrated MaxWeight policy along with many generalizations. Other topics include methods for synthesizing hedging and safety stocks, stability theory for networks, and techniques for accelerated simulation. Examples and figures throughout make ideas concrete. Solutions to end-of-chapter exercises available on a companion website.

http://www.meyn.ece.ufl.edu/archive/spm_files/Presentations/Atlanta05.pdf

Control Techniques for Complex Networks

Tabu search (TS) has provided advances for solving difficult optimization problems in many domains. At the same time, fundamental TS strategies are often not applied as effectively as they might be, and their underlying rationale is often not completely understood. We examine basic concepts and principles of tabu search, emphasizing those that have sometimes led to applying the label “adaptive memory programming” to this class of methods.

/pdf/tabu-search-and-adaptive-memory-programming-advances-2ls9z4f5ay.pdf

Tabu Search and Adaptive Memory Programming — Advances, Applications and Challenges

The policy iteration method of dynamic programming is studied in an abstract setting. It is shown to be equivalent to the Newton-Kantorovich iteration procedure applied to the functional equation of dynamic programming. This equivalence is used to obtain the rate of convergence and error bounds for the sequence of values generated by policy iteration. These results are discussed in the context of the finite state Markovian decision problem with compact action space. An example is analyzed in detail.

On the Convergence of Policy Iteration in Stationary Dynamic Programming

This paper obtains bounds, in terms of the first two moments of the input, on the expected wait in an A/G/k queue with stationary input. To this end, two single-server systems are constructed. The wait in queue for the first single-server system is stochastically larger than the wait in the given multiserver system, and the expected wait in the second single-server system is used to obtain a lower bound on the expected wait in the A/G/k system. The paper also develops, as a consequence of the lower bounds, some results concerning the optimum number of servers, given a fixed work capacity.

Some Inequalities for Parallel-Server Queues

When a customer requests a discount fare, the airline must decide whether to sell the seat at the requested discount or to hold the seat in hope that a customer will arrive later who will pay more. We model this situation for a single-leg flight with multiple fare classes and customers who arrive according to a semi-Markov process (possibly nonhomogeneous). These customers can request multiple seats (batch requests) and can be overbooked. Under certain conditions, we show that the value function decreases as departure approaches. If each customer only requests a single seat or if the requests can be partially satisfied, then we show that there are optimal booking curves which decrease as departure approaches. We also provide counterexamples to show that this structural property of the optimal policy need not hold for more general arrival processes if the requests can be for more than one seat and must be accepted or rejected as a whole.

Dynamic Airline Revenue Management with Multiple Semi-Markov Demand

A duality theory for the problem of finding the properly efficient values of a multiple objective convex program is obtained using arguments and concepts analogous to those used by Rockafellar for scalar valued problems. A key result is that xI minimizes a convex, vector valued function f if and only if there is a “zero-like” subgradient of f at xI. Consequently, if a vector valued convex program with a given perturbation is represented by a convex bifunction, then the properly efficient values of the program are equal to the perturbation function evaluated at zero-like perturbations. A dual program is defined in terms of the adjoint bifunction. Certain closure conditions insure the absence of a duality gap. The dual variables are matrices with the interpretation that the i, j-th element is the value of the ith resource in terms of the jth objective.

A concept of optimality called Isermann efficiency is introduced and compared with efficiency and proper efficiency. Isermann efficiency generalizes a concept from Isermann's work on multiple objective linear programs.

Duality for Multiple Objective Convex Programs

The well known formula L = λW relates the time-average number in queue to the expected wait in queue of a customer. This paper specializes a more general formula, denoted by H = λG, in order to obtain relations between moments of L and W other than the first. The basic queue considered is G/G/k with stationary input. The special case where the arrival times form a renewal process and the more special case where they are a Poisson process are also discussed.

Shelby Brumelle

Papers

On the Convergence of Policy Iteration in Stationary Dynamic Programming

Some Inequalities for Parallel-Server Queues

Dynamic Airline Revenue Management with Multiple Semi-Markov Demand

Duality for Multiple Objective Convex Programs

A Generalization of L = λW to Moments of Queue Length and Waiting Times