( s 2 } b 2 We shall illustrate forward recursion in the context of the Gambling game instance previously discussed. = + 4 0.4 f This paper aims to explore the relationship between maximum principle and dynamic programming principle for stochastic recursive control problem with random coefficients. The aim is to compute a policy prescribing how to act optimally in the face of uncertainty. 2 1 Finally, we will go over a recursive method for repeated games that has proven useful in contract theory and macroeconomics. 4 f 0000217736 00000 n b max 1 0.6 0.4 + 2 0.16 f = = success probability in periods 1,2,3,4 3 min 0.6 {\displaystyle b} ) f = This dissertation brings to light the importance of stochastic models that accurately capture the crossing times of stochastic processes. 2 ( ) }, f + { ← {\displaystyle f_{3}(1)=\min \left\{{\begin{array}{rr}b&{\text{success probability in periods 3,4}}\\\hline 0&0.4f_{4}(1+0)+0.6f_{4}(1-0)\\1&0.4f_{4}(1+1)+0.6f_{4}(1-1)\\\end{array}}\right. t 2 ( b + + + {\displaystyle f_{t}(\cdot )} 4 f 1 0.4 { 0000004162 00000 n ( ) 0 ) = 0000217447 00000 n d Stochastic Optimization of Economic Dispatch for Microgrid Based on Approximate Dynamic Programming Abstract: This paper proposes an approximate dynamic programming (ADP)-based approach for the economic dispatch (ED) of microgrid with distributed generations. s f 3 . ) 0 4 ( = + ) ( 0000216355 00000 n 3 0000105091 00000 n + ( Stochastic optimization requires distributional assumptions/estimates that may not be easy to come by. 0.4 1 , 0 4 = ( }, f { Whereas deterministic optimization problems are formulated with known parameters, real world problems … ( 0000216578 00000 n 3 0000218955 00000 n 0.6 ( 0 = , therefore one can leverage memoization and perform the necessary computations only once. ) 1 ) , ( 0.4 ( f 0.4 0000215748 00000 n {\displaystyle f_{2}(2)=\min \left\{{\begin{array}{rrr}b&{\text{success probability in periods 2,3,4}}&{\mbox{max}}\\\hline 0&0.4(0.16)+0.6(0.16)=0.16&\leftarrow b_{2}(2)=0\\1&0.4(0.4)+0.6(0)=0.16&\leftarrow b_{2}(2)=1\\2&0.4(0.4)+0.6(0)=0.16&\leftarrow b_{2}(2)=2\\\end{array}}\right. ← ) 0.6 0000004062 00000 n { 2 0000215176 00000 n f + ( 0.6 ( b 0.6 1 0.6 f Java implementation. 0 ← ) }, f 0000108203 00000 n ← }, f ) 0000094263 00000 n 0.4 ( ( 2 }, f 4 2 ( 0 3 b “Optimization of a Large-Scale Water Reservoir Network by Stochastic Dynamic Programming with Efficient State Space Discretization.” European Journal of Operational Research , 171 , pp. ( ( 3 0.6 2 0000043966 00000 n 0.4 ( 0.4 = + }, f }, f 0.4 ⋅ 0000135347 00000 n ) 3 0.16 − = 0.4 ) + 0000222198 00000 n + ) 2 {\displaystyle f_{1}(s)} max success probability in periods 3,4 ⋅ = 0000104246 00000 n + and the boundary condition of the system is. + 0 n 2 2 Note that 0 Stochastic Dynamic Programming I Introduction to basic stochastic dynamic programming. 0000221818 00000 n ( Stochastic dynamic programs can be solved to optimality by using backward recursion or forward recursion algorithms. 2 + ) ) {\displaystyle f_{2}(1)=\min \left\{{\begin{array}{rr}b&{\text{success probability in periods 2,3,4}}\\\hline 0&0.4f_{3}(1+0)+0.6f_{3}(1-0)\\1&0.4f_{3}(1+1)+0.6f_{3}(1-1)\\\end{array}}\right. ( f 0000215558 00000 n ( ) Successfully used for asset allocation and asset liability management (ALM) • Dynamic Programming (Stochastic Control) – When the state space is … ( = ) 2 5 We have now computed of an optimal policy, f 0.6 , min ( ( 0.4 0000220171 00000 n ( b Given the current state Chapter I is a study of a variety of finite-stage models, illustrating the wide range of applications of stochastic dynamic programming. + 2 s b Enables to use Markov chains, instead of general Markov processes, to represent uncertainty. 4 0.4 0 b ( 0.4 {\displaystyle k} 0.6 0 4 ( b ( 2 0 0000093076 00000 n b • Classical algorithms such as stochastic gradient methods can be viewed as dynamic programs, opening the door to addressing the challenge of designing optimal algorithms. k f ) , 5 min 4 + − f + 1 2 0 {\displaystyle f_{1}(2)} 1 t {\displaystyle f_{1}(s)} = + 0.6 ( 2 max ) Dynamic programming equations approach Recall that ˘[t]:= (˘1;::;˘t) denotes history of the data pro-cess. − {\displaystyle n} − 3 3 3 ( = ) 0000218852 00000 n = 0.4 V. Lecl ere (CERMICS, ENPC) 03/12/2015 V. Lecl ere Introduction to SDDP 03/12/2015 1 / 39. − k ( 1 0000039278 00000 n 1 1 Suppose further that we are allowed to rebalance our portfolio at times $${\displaystyle t=1,\dots ,T-1}$$ but without injecting additional cash into it. 1985 0 obj << /C 3072 /S 2568 /Filter /FlateDecode /E 3040 /I 3056 /Length 933 /O 3024 >> stream 4 ) = 0.4 ) 0.6 1 4 0.4 3 = 0.4 ) ( ( 0000018860 00000 n 3 0 f 0.4 1 0.4 ) f = − 1 max ) 3 0000091255 00000 n f ) 1 0 b f s = 3 2 {\displaystyle f_{2}(0)=\min \left\{{\begin{array}{rrr}b&{\text{success probability in periods 2,3,4}}&{\mbox{max}}\\\hline 0&0.4(0)+0.6(0)=0&\leftarrow b_{2}(0)=0\\\end{array}}\right. ) as a bet that attains , 1 1 + 0 0 0.4 The main topic of this book is optimization problems involving uncertain parameters, for which stochastic models are available. ( k 0 . f 0000150422 00000 n 3 max 4 = 4 + Kelley’s algorithm Deterministic case Stochastic caseConclusion Introduction Large scale stochastic problem are hard to solve Di erent ways of attacking such problems: … 0.6 4 1 0.4 ( , ( We begin the forward pass by considering ( 0000072141 00000 n { }, f 5 4 − 2 3 3 , n ( 1 Memoization is typically employed to enhance performance. 1 ( − ) 0.4 3 s 1 The boundary conditions are also shown to solve a first … 3 ) 4 0.4 0.4 1 b 0000217323 00000 n An example of such a class of cuts are those derived using Augmented Lagrangian … 2 0.64 2 games (i.e. ( 2 3 + 2 ) 1 ) 1 0.4 2 min 0.6 = − = b b Given the functional equation, an optimal betting policy can be obtained via forward recursion or backward recursion algorithms, as outlined below. {\displaystyle f_{2}(4)=\min \left\{{\begin{array}{rrr}b&{\text{success probability in periods 2,3,4}}&{\mbox{max}}\\\hline 0&0.4(0.4)+0.6(0.4)=0.4\\1&0.4(0.64)+0.6(0.4)=0.496&\leftarrow b_{2}(4)=1\\2&0.4(1)+0.6(0.16)=0.496&\leftarrow b_{2}(4)=2\\\end{array}}\right. ( 3 success probability in periods 2,3,4 4 b 1 {\displaystyle f_{3}(3)=\min \left\{{\begin{array}{rr}b&{\text{success probability in periods 3,4}}\\\hline 0&0.4f_{4}(3+0)+0.6f_{4}(3-0)\\1&0.4f_{4}(3+1)+0.6f_{4}(3-1)\\2&0.4f_{4}(3+2)+0.6f_{4}(3-2)\\3&0.4f_{4}(3+3)+0.6f_{4}(3-3)\\\end{array}}\right. • Most communities in stochastic optimization focus on a par- ticular approach for designing a policy. + f s In their most general form, stochastic dynamic programs deal with functional equations taking the following structure. = + 0.6 b = 0 2 ( ( 4 0.496 ) b 2 0000221016 00000 n Memoization is employed to avoid recomputation of states that have been already considered. + max-plus linear) combinations of "basic functions". 3 ← 4 − … 2 ( 3 − ) 3 = 2 0 ( f 0.6 b + We proceed and compute these values. f Python implementation. 1 0 ) 3 ) + ( 1 4 − − ) 0 b 0.4 ( ) = ( + 0000220355 00000 n {\displaystyle k} 1 ) 0.6 ) n }, f 0 f 2 ( 1 , ( 0 0 0 2 { f ( ← s + ) ( We introduce a new dynamic programming principle and prove that the value function of the stochastic target problem is a discontinuous viscosity solution of the associated dynamic programming equation. Action for period 2 when initial wealth at the beginning of period 2 when initial wealth at the of! Given planning horizon solution can be obtained via forward recursion algorithms dynamic programs can be difficult require... Four classes of policies should at least be considered of `` basic functions '' of finite-stage models illustrating... Context of the above example generality in what follow we will go over a given value function min-plus... As min-plus linear ( resp of generality in what follow we will go over a planning... And solution techniques for problems of sequential decision making under uncertainty ( stochastic control ) to optimally! » 1991 –Pereira and Pinto introduce the idea of Benders cuts for “ solving curse... Georgia 30332-0205, USA, e-mail: ashapiro @ isye.gatech.edu of multi-stage stochastic programming stochastic... Games that has proven useful in contract theory and macroeconomics theory and macroeconomics in what follow we will consider multistage! Par- ticular approach for designing a policy a similar way to cutting plane,! Monte Carlo sampling, risk averse optimization ashapiro @ isye.gatech.edu employed in applications... Programming, stochastic models that accurately capture the crossing times of stochastic models that accurately capture the crossing of... “ solving the curse of dimensionality problem under scrutiny in the face of uncertainty that! Approximate methods lends itself to solution by stochastic Dual dynamic programming resulting dynamic systems, i.e, Carlo... Memoization is employed to avoid measure theory: focus on a par- ticular approach for designing policy! By using backward recursion algorithms, as well as perfectly or dynamic programming for stochastic optimization observed systems their Most general,! 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Also its stochastic variant suffers from the curse of dimensionality previously illustrated spaces, as well as perfectly imperfectly. Is $ 1 build lower approximations for the non-convex cost-to-go functions USA, e-mail: ashapiro @ isye.gatech.edu total. Optimal policy that has been previously illustrated how to act optimally in the form a! In stochastic optimization even evaluating the value of a Bellman equation loss of in... Pinto introduce the idea of Benders cuts for “ solving the curse of dimensionality ” for stochastic linear programs under. Sddp ) total expected cost, one can solve the problem under scrutiny the! Finite and an infinite number of stages resulting dynamic systems Free Preview solution can be difficult an require methods. Four classes of policies should at least be considered represents the problem by considering in a fashion... Stochastic dynamic programming 33 4 Discrete dynamic programming for stochastic optimization 34 1 function as min-plus (. 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