Analysis, Control and Optimization of Complex Dynamic Systems (Gerad 25th Anniversary)

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Dynamical Systems Introduction

Citing Literature. Supporting Information As a service to our authors and readers, this journal provides supporting information supplied by the authors. Filename Description admasupS1. Volume 26 , Issue 1 January 8, Pages Related Information. Close Figure Viewer. Browse All Figures Return to Figure. Experimental work shows that it can produce a much higher rate of convergence than the knowledge gradient with independent beliefs, in addition to outperforming other more classical information collection mechanisms. The knowledge gradient policy is introduced here as a method for solving the ranking and selection problem, which is an off-line version of the multiarmed bandit problem.

Imagine that you have M choices M is not too large where you have a normally distributed belief about the value of each choice. You have a budget of N measurements to evaluate each choice to refine your distribution of belief. After your N measurements, you have to choose what appears to be the best based on your current belief.

The knowledge gradient policy guides this search by always choosing to measure the choice which would produce the highest value if you only have one more measurement the knowledge gradient can be viewed as a method of steepest ascent. The paper shows that this policy is myopically optimal by construction , but is also asymptotically optimal, making it the only stationary policy that is both myopically and asymptotically optimal.

Analysis, Control and Optimization of Complex Dynamic Systems

The paper provides bounds for finite measurement budgets, and provides experimental work that shows that it works as well as, and often better, than other standard learning policies. Dayanik, W. Powell, and K. One of the most famous problems in information collection is the multiarmed bandit problem, where make a choice out of a discrete set of choices , observe a reward, and use this observation to update estimates of the future value of rewards.

This problem can be solved by choosing the option with the highest index known as the Gittins index. In this paper, we present an index problem for the case where not all the choices are available each time. As a result, it is sometimes important to make an observation just because the observation is available to be made. Approximate dynamic programming back. Much of our work falls in the intersection of stochastic programming and dynamic programming.

The dynamic programming literature primarily deals with problems with low dimensional state and action spaces, which allow the use of discrete dynamic programming techniques. The stochastic programming literature, on the other hands, deals with the same sorts of higher dimensional vectors that are found in deterministic math programming.

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However, the stochastic programming community generally does not exploit state variables, and does not use the concepts and vocabulary of dynamic programming. Our contributions to the area of approximate dynamic programming can be grouped into three broad categories: general contributions , transportation and logistics , which we have broadened into general resource allocation, discrete routing and scheduling problems , and batch service problems.

A series of short introductory articles are also available. Nascimento, W. This paper proves convergence for an ADP algorithm using approximate value iteration TD 0 , for problems that feature vector-valued decisions e. The problem arises in settings where resources are distributed from a central storage facility. The algorithm is well suited to continuous problems which requires that the function that captures the value of future inventory be finely discretized, since the algorithm adaptively generates break points for a piecewise linear approximation.

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The strategy does not require exploration, which is common in reinforcement learning. Powell, J. We review the literature on approximate dynamic programming, with the goal of better understanding the theory behind practical algorithms for solving dynamic programs with continuous and vector-valued states and actions, and complex information processes. We build on the literature that has addressed the well-known problem of multidimensional and possibly continuous states, and the extensive literature on model-free dynamic programming which also assumes that the expectation in Bellman's equation cannot be computed.

We then describe some recent research by the authors on approximate policy iteration algorithms that offer convergence guarantees with technical assumptions for both parametric and nonparametric architectures for the value function. These two short chapters provide yet another brief introduction to the modeling and algorithmic framework of ADP.

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The first chapter actually has nothing to do with ADP it grew out of the second chapter. Instead, it describes the five fundamental components of any stochastic, dynamic system. There is also a section that discusses "policies", which is often used by specific subcommunities in a narrow way.

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  6. I describe nine specific examples of policies. I think this helps put ADP in the broader context of stochastic optimization. The second chapter provides a brief introduction to algorithms for approximate dynamic programming. In the tight constraints of these chapters for Wiley's Encyclopedia, it is not possible to do a topic like this justice in 20 pages, but if you need a quick peek into ADP, this is one sample.

    The exploration-exploitation problem in dynamic programming is well-known, and yet most algorithms resort to heuristic exploration policies such as epsilon-greedy. We use a Bayesian model of the value of being in each state with correlated beliefs, which reflects the common fact that visiting one state teaches us something about visiting other states. We use the knowledge gradient algorithm with correlated beliefs to capture the value of the information gained by visiting a state. Using both a simple newsvendor problem and a more complex problem of making wind commitments in the presence of stochastic prices, we show that this method produces significantly better results than epsilon-greedy for both Bayesian and non-Bayesian beliefs.

    These are shown for both offline and online implementations. This paper addresses four problem classes, defined by two attributes: the number of entities being managed single or many , and the complexity of the attributes of an entity simple or complex. All the problems are stochastic, dynamic optimization problems. Single, simple-entity problems can be solved using classical methods from discrete state, discrete action dynamic programs.

    The AI community often works on problems with a single, complexity entity e.

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    The OR community tends to work on problems with many simple entities. This paper briefly describes how advances in approximate dynamic programming performed within each of these communities can be brought together to solve problems with multiple, complex entities. Nascimento, J.

    I have worked for a number of years using piecewise linear function approximations for a broad range of complex resource allocation problems. A few years ago we proved convergence of this algorithmic strategy for two-stage problems click here for a copy. In this latest paper, we have our first convergence proof for a multistage problem. This paper is more than a convergence proof for this particular problem class - it lays out a proof technique, which combines our work on concave approximations with theory laid out by Bertsekas and Tsitsiklis in their Neuro-Dynamic Programming book.

    Ma, J. This conference proceedings paper provides a sketch of a proof of convergence for an ADP algorithm designed for problems with continuous and vector-valued states and actions. In addition, it also assumes that the expected in Bellman's equation cannot be computed. The proof assumes that the value function can be expressed as a finite combination of known basis functions. The proof is for a form of approximate policy iteration. George, A. There are a number of problems in approximate dynamic programming where we have to use coarse approximations in the early iterations, but we would like to transition to finer approximations as we collect more information.

    This paper studies the statistics of aggregation, and proposes a weighting scheme that weights approximations at different levels of aggregation based on the inverse of the variance of the estimate and an estimate of the bias.

    Analysis, Control and Optimization of Complex Dynamic Systems

    This weighting scheme is known to be optimal if we are weighting independent statistics, but this is not the case here. What is surprising is that the weighting scheme works so well. We demonstrate this, and provide some important theoretical evidence why it works. One of the first challenges anyone will face when using approximate dynamic programming is the choice of stepsizes. Use the wrong stepsize formula, and a perfectly good algorithm will appear not to work.

    Deterministic stepsize formulas can be frustrating since they have parameters that have to be tuned difficult if you are estimating thousands of values at the same time. This paper reviews a number of popular stepsize formulas, provides a classic result for optimal stepsizes with stationary data, and derives a new optimal stepsize formula for nonstationary data. This result assumes we know the noise and bias knowing the bias is equivalent to knowing the answer. A formula is provided when these quantities are unknown. Our result is compared to other deterministic formulas as well as stochastic stepsize rules which are proven to be convergent.

    The numerical work suggests that the new optimal stepsize formula OSA is very robust. It often is the best, and never works poorly. Approximate dynamic programming in transportation and logistics:.