Introduction two key structural properties of total cost dynamic programming dp models are responsible for most of the mathematical results one can prove about them. Judd hoover institution prepared for ice05 july 20, 2005 lectures. Dynamic programming problem bellmans equation backward induction algorithm 2 the in nite horizon case preliminaries for t. In order to understand the issues involved in dynamic programming, it is instructive to start with the simple example of inventory management. Contraction mapping theorem maximum theorem 3 dynamic programming under certainty bellmans optimality principle. The nstage contraction prop erty is a weakened form of the contraction property. It refers to simplifying a complicated problem by breaking it down into simpler subproblems in a recursive manner. Since dynamic programming has recursively generated the optimal decision rule 0 t, it follows that v0s max e futsd. Dynamic programming is one of the most fundamental building blocks of. We introduce a relaxed version of the bellman operator for qfunctions and prove that it is still a monotone contraction mapping with a unique fixed point.
What is the link between contraction mappings and the. Continuing the dynamic programming recursion, it is straightforward to verify that at each time t the optimal decision rule hr depends only on st and the current time t. The rst is the monotonicity property of the mappings associated with bellmans equation. Problem set 1 asks you to use the foc and the envelope theorem to solve for. We then show how to solve this dynamic programming problem via standard value function iteration. Concerning the multivalued versions of the preceding results. Contraction mapping, inverse and implicit function theorems 1 the contraction mapping theorem denition 1. Rather than assuming observed choices are the result of static utility maximization, observed choices in ddc models are assumed to result from an agents maximization of the present value of utility, generalizing the. If t is a contraction in s,r with modulus b, then 1 there is a unique xed point s 2s, such that s ts 2 iterations of t converge to the xed point rtns0,s bnrs0,s for any s0 2s. Introduction to dynamic programming lecture notes klaus neussery november 30, 2017.
F 2 f 1 is nondecreasing, then every f 1contraction mapping is an f 2contraction mapping. The linear approximation is a contraction when the eigenvalues. Our discussion centers on two fundamental properties that this mapping may have. Several mathematical theorems the contraction mapping theorem also called the banach fixed point theorem, the theorem of the maximum or berges maximum theorem, and blackwells su ciency conditions.
However, the ecm operator does not possess the property of contraction mapping like the regular bellman. Dynamic programming contraction mapping cm theorem. Dynamic optimization cm toulouse school of economics. Therefore, repeated applications of the onpolicy operator converge to a vector q such that q oq. Non linear contraction mapping and its application in. The tricky part in using the contraction mapping theorem is to. In other words, a contraction mapping brings elements of the space s closer to each other. For the representative agent growth model that we started with, the bellman equation is. In the spirit of the linear programming approach to approximate dynamic programming, we exploit the new operator to build a simplified linear program lp for qfunctions. Lazaric markov decision processes and dynamic programming oct 1st, 20 1079. Within each iteration, this dynamic programming approach requires us to solve a collection of static principalagent problems, each of which is. Theorem contraction mapping theorem for any metric space vthat is complete i. Planning by dynamic programming contraction mapping. Then, t is said to be a contraction mapping if there exists a constant l.
Math programming approaches to structural estimation. Prove that the bellman operator maps the space of continuous and bounded functions into. This equation is called the bellman functional equation in dynamic programming. So the result follows from banachs contraction mapping theorem.
In particular, we show how one of the results from the preceding sections can be adapted to. Math programming approaches to structural estimation chelin su. Non linear contraction mapping and its application in dynamic programming himanshu tiwari1, and subhashish biswas1 1 department of mathematics, kalinga university, raipur, chhattisgarh, india. Dynamic programming is most efficient when the problem in question is low dimensional, when the associated policy and value functions are defined on relatively small finite sets or are smooth and easily approximated, and when rewards are bounded, so that the standard contraction mapping arguments apply. A contraction for sovereign debt models princeton university. Convergence of stochastic iterative dynamic programming algorithms 707 jaakkola et al. Classical methods and the c1 contraction mapping method revisited 2. In this article, i will introduce the mathematical background of the classical analysis of reinforcement learning, that is, the bellman operator is essentially a contraction mapping on a complete metric space, and explain how valuebased reinforcement learning, e. Bridging hamiltonjacobi safety analysis and reinforcement.
Contraction mappings in the theory underlying dynamic. In mathematics, a contraction mapping, or contraction or contractor, on a metric space m,d is a function f from m to itself, with the property that there is some nonnegative real number such that for all x and y in m, the smallest such value of k is called the lipschitz constant of f. Our key observation stems from an intuitive interpretation. Convergence of stochastic iterative dynamic programming. Envelope condition method with an application to default. Let s,d be a complete metric space and let t be a contraction mapping. Thus every fcontraction mapping is a continuous mapping. Prove that the bellman equation, viewed as an operator, is a contraction mapping. Dynamic programming with homogeneous functions fernando alvarez department of economics, university of chicago, chicago, illinois 60637, and.
L formulate dynamic programming problems in computationally useful ways l describe key algorithms. Weighted supnorm contractions in dynamic programming. Each of the models cited above satis fies the monotonicity property. Corollary of contraction mapping theorem properties of the value function 1 applying slp to the single sector growth model chapter 4 of stokey lucas prescot demonstrates that we can write a sequential maximization problem as a dynamic program and shows us what kind of assumptions are needed to infer properties about the value function. Anyway, you look at the eigenvalues of the jacobian because near a fixed point you can linearly approximate your map. New common coupled coincidence point theorems for generalized weakly contraction mappings with applications to dynamic programming article pdf available january 2018 with 146 reads how we. Multivalued fcontractions on complete metric space. It also encompasses some models of derman 7, derman and iclein 8, eaton and zadeh 9, and many nstage dynamic programming problems. Just consider k is capital, fk is the production function, fk y is the consumption. A decision rule that depends on the past history of the process only via the current state st and time t is called markovian. For example, let us take a simple interval of the real line as our space. Dynamic programming, bellmens equation, contraction mapping theorem, blackwells sufficiency conditions. Here we show that similar ideas can be applied when contractivity fails or is dif. School of economics, huazhong university of science and technology this version.
We develop an envelope condition method ecm for dynamic programming problems a tractable alternative to expensive conventional value function iteration vfi. Applied dynamic programming by bellman and dreyfus 1962 and dynamic programming and the calculus of variations by dreyfus 1965 provide a good introduction to the main idea of dynamic programming. Why is the contraction mapping theorem useful for dynamic programming. Stokey, lucas jr, and prescott 1989 is the classic economics reference for dynamic programming, but is more advanced than what we will cover. Explain the meaning of bellmans principle of optimality. Why is the maximum theorem of berge useful for dynamic programming. Dynamic programming is typically useful to investigate problems that involve choices to be made over an in nite number of periods. Pdf new common coupled coincidence point theorems for.
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