dynamic programming optimization

dynamic programming optimization

Sometimes, this doesn't optimise for the whole problem. And someone wants us to give a change of 30p. Japan, Preprints (S73-22), By clicking accept or continuing to use the site, you agree to the terms outlined in our. dynamic programming. Learn more about dynamic programming, epstein-zin, bellman, utility, backward recursion, optimization Dynamic Programming Dynamic Programming is mainly an optimization over plain recursion. Dynamic Programming is mainly an optimization over plain recursion. Dynamic programming 1 Dynamic programming In mathematics and computer science, dynamic programming is a method for solving complex problems by breaking them down into simpler subproblems. In this method, you break a complex problem into a sequence of simpler problems. Two points below won’t be covered in this article(potentially for later blogs ):1. Construct the optimal solution for the entire problem form the computed values of smaller subproblems. But, Greedy is different. Dynamic Programming is based on Divide and Conquer, except we memoise the results. Eng. The optimization problems expect you to select a feasible solution, so that the value of the required function is minimized or maximized. Wherever we see a recursive solution that has repeated calls for the same inputs, we can optimize it using Dynamic Programming. You can think of this optimization as reducing space complexity from O(NM) to O(M), where N is the number of items, and M the number of units of capacity of our knapsack. Dynamic programming is basically that. We have 3 coins: 1p, 15p, 25p . F(n) = F(n-1) + F(n-2) for n larger than 2. This simple optimization reduces time complexities from exponential to polynomial. This method provides a general framework of analyzing many problem types. Independent of a particular algorithm, we prove that for two scoring schemes A and B used in dynamic programming, the scoring scheme A ∗ Par B correctly performs Pareto optimization over the same search space. Comm. to dynamic optimization in (Vidal 1981) and (Ravn 1994). Dynamic Programming (DP) is an algorithmic technique for solving an optimization problem by breaking it down into simpler subproblems and utilizing the fact that the optimal solution to the overall problem depends upon the optimal solution to its subproblems. In computer science, mathematics, management science, economics and bioinformatics, dynamic programming (also known as dynamic optimization) is … 1 $\begingroup$ We can reformulate this problem a bit: instead of filling bottle while we are in oasis, we can retroactively take water from oasis we reached if we didn't do it yet. C Programming - Matrix Chain Multiplication - Dynamic Programming MCM is an optimization problem that can be solved using dynamic programming. You know how a web server may use caching? [...] The symmetric form algorithm superiority is established. Because it 1 Problems that can be solved by dynamic programming are typically optimization problems. Dynamic programming, DP involves a selection of optimal decision rules that optimizes a specific performance criterion. Schedule: Winter 2020, Mondays 2:30pm - 5:45pm. Buy this book eBook 117,69 € price for Spain (gross) The eBook … There are two ways for solving subproblems while caching the results:Top-down approach: start with the original problem(F(n) in this case), and recursively solving smaller and smaller cases(F(i)) until we have all the ingredient to the original problem.Bottom-up approach: start with the basic cases(F(1) and F(2) in this case), and solving larger and larger cases. Quadrangle inequalities In this chapter, we will examine a more general technique, known as dynamic programming, for solving optimization problems. Dynamic programming (DP)-based algorithms have been one key theoretic foundation for single-vehicle trajectory optimization, and its formulation typically involves several modeling elements: (i) the boundary of the search scope or map, (ii) discretized space-time lattices, (iii) a path searching algorithm that can find a safe trajectory to reach the destination and meet certain global goals, such … Before we go through the dynamic programming process, let’s represent this graph in an edge array, which is an array of [sourceVertex, destVertex, weight]. Genetic algorithm for optimizing the nonlinear time alignment of automatic speech recognition systems, Performance tradeoffs in dynamic time warping algorithms for isolated word recognition, On time alignment and metric algorithms for speech recognition, Improvements in isolated word recognition, Spoken-word recognition using dynamic features analysed by two-dimensional cepstrum, Locally constrained dynamic programming in automatic speech recognition, The use of a one-stage dynamic programming algorithm for connected word recognition, The Nonlinear Time Alignment Model for Speech Recognition System, Speaker-independent word recognition using dynamic programming matching with statistic time warping cost, Considerations in dynamic time warping algorithms for discrete word recognition, Minimum prediction residual principle applied to speech recognition, Speech Recognition Experiments with Linear Predication, Bandpass Filtering, and Dynamic Programming, Speech recognition experiments with linear predication, bandpass filtering, and dynamic programming, Comparative study of DP-pattern matching techniques for speech recognition, A Dynamic Programming Approach to Continuous Speech Recognition, A similarity evaluation of speech patterns by dynamic programming, Nat. In computer science, mathematics, management science, economics and bioinformatics, dynamic programming (also known as dynamic optimization) is a method for … Moreover, Dynamic Programming algorithm solves each sub-problem just once and then saves its answer in a table, thereby avoiding the work of re-computing the answer every time. Japan, Real - time speech recognition system by minicomputer with DP processor ”, IEEE Transactions on Acoustics, Speech, and Signal Processing. We store the solutions to sub-problems so we can use those solutions subsequently without having to recompute them. Dynamic programming (DP), as a global optimization method, is inserted at each time step of the MPC, to solve the optimization problem regarding the prediction horizon. Paragraph below is what I randomly picked: In computer science, mathematics, management science, economics and bioinformatics, dynamic programming (also known as dynamic optimization) is a method for solving a complex problem by breaking it down into a collection of simpler subproblems, solving each of those subproblems just once, and storing their solutions. Dynamic Programming vs Divide & Conquer vs Greedy. Some features of the site may not work correctly. Optimization problems. It aims to optimise by making the best choice at that moment. we expect by calculus for smooth functions regarded as accurate) enables one to compute easy to solve via dynamic programming, and where we therefore expect are required to pick a Best Dynamic Programming. Dynamic programming is an algorithmic technique that solves optimization problems by breaking them down into simpler sub-problems. You are currently offline. The 2nd edition of the research monograph "Abstract Dynamic Programming," has now appeared and is available in hardcover from the publishing company, Athena Scientific, or from Amazon.com. Optimization exists in two main branches of operations research: . Series. The decision taken at each stage should be optimal; this is called as a stage decision. Characterize the structure of an optimal solution. Sometimes, this doesn't optimise for the whole problem. This paper reports on an optimum dynamic progxamming (DP) based time-normalization algorithm for spoken word recognition. What’re the subproblems?For every positive number i smaller than words.length, if we treat words[i] as the starting word of a new line, what’s the minimal badness score? Dynamic Programming (DP) is an algorithmic technique for solving an optimization problem by breaking it down into simpler subproblems and utilizing the fact that the optimal solution to the overall problem depends upon the optimal solution to its subproblems. What’re the overlapping subproblems?From the previous image, there are some subproblems being calculated multiple times. Like divide-and-conquer method, Dynamic Programming solves problems by combining the solutions of subproblems. Meeting, Inst. The total badness score for the previous brute-force solution is 5022, let’s use dynamic programming to make a better result! Dynamic programming method is yet another constrained optimization method of project selection. Website for a doctoral course on Dynamic Optimization View on GitHub Dynamic programming and Optimal Control Course Information. The idea is to simply store the results of subproblems so that we do not have to re-compute them when needed later. 2. Retrouvez Extensions of Dynamic Programming for Combinatorial Optimization and Data Mining et des millions de livres en stock sur Amazon.fr. Dynamic Programming Reading: CLRS Chapter 15 & Section 25.2 CSE 6331: Algorithms Steve Lai. Noté /5. ruleset pointed out(thanks) a more memory efficient solution for the bottom-up approach, please check out his comment for more. Hopefully, it can help you solve problems in your work . However, dynamic programming doesn’t work for every problem. The idea is to simply store the results of subproblems, so that we do not have to re-compute them when needed later. Combinatorial problems. What’s S[1]? Dynamic programming (DP) technique is an effective tool to find the globally optimal use of multiple energy sources over a pre-defined drive cycle. Dynamic Programming is also used in optimization problems. ). Putting the first word on line 1, and rely on S[1] -> score: 100 + S[1]3. Applied dynamic programming for optimization of dynamical systems / Rush D. Robinett III ... [et al.]. Some properties of two-variable functions required for Kunth's optimzation: 1. Given a sequence of matrices, find the most efficient way to multiply these matrices together. Dynamic Programming is the most powerful design technique for solving optimization problems. 0/1 Knapsack Discrete Optimization w/ Dynamic Programming The Knapsack problem is one I’ve encountered a handful of times, both in my studies (courses, homework, whatever…), and in real life. The Linear Programming (LP) and Dynamic Programming (DP) optimization techniques have been extensively used in water resources. Wherever we see a recursive solution that has repeated calls for same inputs, we can optimize it using Dynamic Programming.The idea is to simply store the results of subproblems, so that we do not have to re-compute them when needed later. Because there are more punishments for “an empty line with a full line” than “two half-filled lines.”Also, if a line overflows, we treat it as infinite bad. For the graph above, starting with vertex 1, what’re the shortest paths(the path which edges weight summation is minimal) to vertex 2, 3, 4 and 5? Putting the three words on the same line -> score: MAX_VALUE.2. find "Speed-Up in Dynamic Programming" by F. Frances Yao. Developed by Richard Bellman, dynamic programming is a mathematical technique well suited for the optimization of multistage decision problems. Applied Dynamic Programming for Optimization of Dynamical Systems presents applications of DP algorithms that are easily adapted to the reader's own interests and problems. The word "programming" in "dynamic programming" is similar for optimization. Dynamic Programming is based on Divide and Conquer, except we memoise the results. Especially the approach that links the static and dynamic optimization originate from these references. The next time the same subproblem occurs, instead of recomputing its solution, one simply looks up the previously computed solution, thereby saving computation time at the expense of (it is hoped) a modest expenditure in storage space. Simply put, dynamic programming is an optimization technique that we can use to solve problems where the same work is being repeated over and over. Machine Learning and Dynamic Optimization is a graduate level course on the theory and applications of numerical solutions of time-varying systems with a focus on engineering design and real-time control applications. Considers extensions of dynamic programming for the study of multi-objective combinatorial optimization problems; Proposes a fairly universal approach based on circuits without repetitions in which each element is generated exactly one time ; Is useful for researchers in combinatorial optimization; see more benefits. If we simply put each line as many characters as possible and recursively do the same process for the next lines, the image below is the result: The function below calculates the “badness” of the justification result, giving that each line’s capacity is 90:calcBadness = (line) => line.length <= 90 ? How to solve the subproblems?Start from the basic case which i is 0, in this case, distance to all the vertices except the starting vertex is infinite, and distance to the starting vertex is 0.For i from 1 to vertices-count — 1(the longest shortest path to any vertex contain at most that many edges, assuming there is no negative weight circle), we loop through all the edges: For each edge, we calculate the new distance edge[2] + distance-to-vertex-edge[0], if the new distance is smaller than distance-to-vertex-edge[1], we update the distance-to-vertex-edge[1] with the new distance. Situations(such as finding the longest simple path in a graph) that dynamic programming cannot be applied. — (Advances in design and control) Includes bibliographical references and index. Giving a paragraph, assuming no word in the paragraph has more characters than what a single line can hold, how to optimally justify the words so that different lines look like have a similar length? SOC. Dynamic programming’s rules themselves are simple; the most difficult parts are reasoning whether a problem can be solved with dynamic programming and what’re the subproblems. We define a binary Pareto product operator ∗ Par on arbitrary scoring schemes. Putting the last two words on the same line -> score: 361.2. p. cm. Combinatorial problems. (1981) have illustrated applications of LP, Non-linear programming (NLP), and DP to water resources. Achetez neuf ou d'occasion (Each of the subproblem solutions is indexed in some way, typically based on the values of its input parameters, so as to facilitate its lookup.) Introduction of Dynamic Programming. This helps to determine what the solution will look like. time. I. Robinett, Rush D. II. Noté /5. Semantic Scholar is a free, AI-powered research tool for scientific literature, based at the Allen Institute for AI. This method provides a general framework of analyzing many problem types. Loucks et al. By caching the results, we make solving the same subproblem the second time effortless. Taking a Look at Semantic UI: A Lightweight Alternative to Bootstrap, Python Basics: Packet Crafting With Scapy, Don’t eat, Don’t Sleep, Code: Facing Mental Illness in Technology, Tutorial to Configure SSL in an HAProxy Load Balancer. What’re the subproblems?For non-negative number i, giving that any path contain at most i edges, what’s the shortest path from starting vertex to other vertices? Let’s take a look at an example: if we have three words length at 80, 40, 30.Let’s treat the best justification result for words which index bigger or equal to i as S[i]. Knuth's optimization is used to optimize the run-time of a subset of Dynamic programming problems from O(N^3) to O(N^2).. Properties of functions. Group Meeting Speech, Acoust. share | cite | improve this question | follow | asked Nov 9 at 15:55. Dynamic programming is a methodology(same as divide-and-conquer) that often yield polynomial time algorithms; it solves problems by combining the results of solved overlapping subproblems.To understand what the two last words ^ mean, let’s start with the maybe most popular example when it comes to dynamic programming — calculate Fibonacci numbers. ISBN 0-89871-586-5 1. Please let me know your suggestions about this article, thanks! If we were to compute the matrix product by directly computing each of the,. TAs: Jalaj Bhandari and Chao Qin. The name dynamic programming is not indicative of the scope or content of the subject, which led many scholars to prefer the expanded title: “DP: the programming of sequential decision processes.” Loosely speaking, this asserts that DP is a mathematical theory of optimization. You know how a web server may use caching? This article introduces dynamic programming and provides two examples with DEMO code: text justification & finding the shortest path in a weighted directed acyclic graph. Recursively defined the value of the optimal solution. Dynamic programming is an optimization approach that transforms a complex problem into a sequence of simpler problems; its essential characteristic is the multistage nature of the optimization procedure. It also identifies DP with decision systems that evolve in a sequential and dynamic fashion. If you don't know about the algorithm, watch this video and practice with problems. Majority of the Dynamic Programming problems can be categorized into two types: 1. It is the same as “planning” or a “tabular method”. No.PR00446), ICASSP-88., International Conference on Acoustics, Speech, and Signal Processing, 1973 Tech. Given a sequence of matrices, find the most efficient way to multiply these matrices together. Putting the last two words on different lines -> score: 2500 + S[2]Choice 1 is better so S[2] = 361. Dynamic programming is both a mathematical optimization method and a computer programming method. Electron. Dynamic programming is both a mathematical optimization method and a computer programming method. Dynamic Programming To calculate F(n) for a giving n:What’re the subproblems?Solving the F(i) for positive number i smaller than n, F(6) for example, solves subproblems as the image below. Optimization parametric (static) – The objective is to find the values of the parameters, which are “static” for all states, with the goal of maximizing or minimizing a function. Compute the value of the optimal solution from the bottom up (starting with the smallest subproblems) 4. As many other things, practice makes improvements, please find some problems without looking at solutions quickly(which addresses the hardest part — observation for you). + S[2]Choice 2 is the best. Dynamic Programming & Divide and Conquer are similar. We can make two choices:1. Answered; References: "Efficient dynamic programming using quadrangle inequalities" by F. Frances Yao. Dynamic programming method is yet another constrained optimization method of project selection. How to solve the subproblems?The total badness score for words which index bigger or equal to i is calcBadness(the-line-start-at-words[i]) + the-total-badness-score-of-the-next-lines. Dynamic programming is mainly an optimization over plain recursion. We can draw the dependency graph similar to the Fibonacci numbers’ one: How to get the final result?As long as we solved all the subproblems, we can combine the final result same as solving any subproblem. T57.83.A67 2005 519.7’03—dc22 2005045058 However, the … Proceedings 1999 International Conference on Information Intelligence and Systems (Cat. A greedy algorithm can be used to solve all the dynamic programming problems. However, there are optimization problems for which no greedy algorithm exists. The first-order conditions (FOCs) for (2) are standard: ∂ ∂ =∂ ∂ − = = =L z u z p i a b t ti t iti λ 0, , , 1,2 1 2 0 2 2 − + = ∂ ∂ ∂∂ = λλ x u L x [note that x 1 is not a choice variable since it is fixed at the outset and x 3 is equal to zero] ∂ ∂ = − − =L x x zλ The method was developed by Richard Bellman in the 1950s and has found applications in numerous fields, from aerospace engineering to economics. optimization dynamic-programming. Achetez neuf ou d'occasion The purpose of this chapter is to provide an introduction to the subject of dynamic optimization theory which should be particularly useful in economic applications. The memo table saves two numbers for each slot; one is the total badness score, another is the starting word index for the next new line so we can construct the justified paragraph after the process. Dynamic programming can be especially useful for problems that involve uncertainty. Dynamic Programming is mainly an optimization over plain recursion. Decision At every stage, there can be multiple decisions out of which one of the best decisions should be taken. Dynamic Programming 4An Algorithm Design Technique 4A framework to solve Optimization problems • Elements of Dynamic Programming • Dynamic programming version of a recursive algorithm • Developing a Dynamic Programming Algorithm 4Multiplying a Sequence of Matrices A framework to solve Optimization problems • For each current choice: Dynamic programming is another approach to solving optimization problems that involve time. Lectures in Dynamic Optimization Optimal Control and Numerical Dynamic Programming Richard T. Woodward, Department of Agricultural Economics, Texas A&M University. As applied to dynamic programming, a multistage decision process is one in which a number of single‐stage processes are connected in series so that the output of one stage is the input of the succeeding stage. Simply put, dynamic programming is an optimization technique that we can use to solve problems where the same work is being repeated over and over. Dynamic programming, DP involves a selection of optimal decision rules that optimizes a specific performance criterion. While we are not going to have time to go through all the necessary proofs along the way, I will attempt to point you in the direction of more detailed source material for the parts that we do not cover. , that satisfies a given constraint} and optimizes a given objective function. Optimization II: Dynamic Programming In the last chapter, we saw that greedy algorithms are efficient solutions to certain optimization problems. 11 2 2 bronze badges $\endgroup$ add a comment | 1 Answer Active Oldest Votes. It is applicable to problems exhibiting the properties of overlapping subproblems which are only slightly smaller[1] and optimal substructure (described below). Let’s define a line can hold 90 characters(including white spaces) at most. Dynamic optimization models and methods are currently in use in a number of different areas in economics, to address a wide variety of issues. Dynamic programming algorithm optimization for spoken word recognition @article{Sakoe1978DynamicPA, title={Dynamic programming algorithm optimization for spoken word recognition}, author={H. Sakoe and Seibi Chiba}, journal={IEEE Transactions on Acoustics, Speech, and Signal Processing}, year={1978}, volume={26}, pages={159-165} } Buy Extensions of Dynamic Programming for Combinatorial Optimization and Data Mining by AbouEisha, Hassan, Amin, Talha, Chikalov, Igor, Hussain, Shahid, Moshkov, Mikhail online on Amazon.ae at best prices. It aims to optimise by making the best choice at that moment. The book is organized in such a way that it is possible for readers to use DP algorithms before thoroughly comprehending the full theoretical development. Location: Warren Hall, room #416. dynamic optimization and has important economic meaning. Students who complete the course will gain experience in at least one programming … The problem is not actually to perform the multiplications, but merely to decide in which order to perform the multiplications. But, Greedy is different. Wherever we see a recursive solution that has repeated calls for same inputs, we can optimize it using Dynamic Programming. Let’s solve two more problems by following “Observing what the subproblems are” -> “Solving the subproblems” -> “Assembling the final result”. Découvrez et achetez Dynamic Programming Multi-Objective Combinatorial Optimization. We can make different choices about what words contained in a line, and choose the best one as the solution to the subproblem. More so than the optimization techniques described previously, dynamic programming provides a general framework for analyzing many problem types. 2. Knuth's optimization is used to optimize the run-time of a subset of Dynamic programming problems from O(N^3) to O(N^2).. Properties of functions. Retrouvez Bellman Equation: Bellman Equation, Richard Bellman, Dynamic Programming, Optimization (mathematics) et des millions de livres en stock sur Amazon.fr. The solutions to these sub-problems are stored along the way, which ensures that each problem is only solved once. In this framework, you use various optimization techniques to solve a specific aspect of the problem. Wherever we see a recursive solution that has repeated calls for same inputs, we can optimize it using Dynamic Programming. Optimization problems: Construct a set or a sequence of of elements , . Dynamic programming is basically that. Like Divide and Conquer, divide the problem into two or more optimal parts recursively. 2. Dynamic optimization approach There are several approaches can be applied to solve the dynamic optimization problems, which are shown in Figure 2. Putting the first two words on line 1, and rely on S[2] -> score: MAX_VALUE. a) True Fibonacci numbers are number that following fibonacci sequence, starting form the basic cases F(1) = 1(some references mention F(1) as 0), F(2) = 1. While we are not going to have time to go through all the necessary proofs along the way, I will attempt to point you in the direction of more detailed source material for the parts that we do not cover. Figure 2. Many optimal control problems can be solved as a single optimization problem, named one-shot optimization, or via a sequence of optimization problems using DP. OPTIMIZATION II: DYNAMIC PROGRAMMING 397 12.2 Chained Matrix Multiplication Recall that the product AB, where A is a k×m matrix and B is an m×n matrix, is the k ×n matrix C such that C ij = Xm l=1 A ilB lj for 1 ≤i ≤k,1 ≤j ≤n. So, dynamic programming saves the time of recalculation and takes far less time as compared to other methods that don’t take advantage of the overlapping subproblems property. Differential equations can usually be used to express conservation Laws, such as mass, energy, momentum. Take this question as an example. We can make one choice:Put a word length 30 on a single line -> score: 3600. What is the sufficient condition of applying Divide and Conquer Optimization in terms of function C[i][j]? This is a dynamic optimization course, not a programming course, but some familiarity with MATLAB, Python, or equivalent programming language is required to perform assignments, projects, and exams. The DEMO below(JavaScript) includes both approaches.It doesn’t take maximum integer precision for javascript into consideration, thanks Tino Calancha reminds me, you can refer his comment for more, we can solve the precision problem with BigInt, as ruleset pointed out. Livraison en Europe à 1 centime seulement ! Course Number: B9120-001. It can be broken into four steps: 1. Optimization Problems y • • {. 2 Dynamic Programming We are interested in recursive methods for solving dynamic optimization problems. What’s S[2]? Quadrangle inequalities C Programming - Matrix Chain Multiplication - Dynamic Programming MCM is an optimization problem that can be solved using dynamic programming. We study exact Pareto optimization for two objectives in a dynamic programming framework. What’s S[0]? . Divide & Conquer algorithm partition the problem into disjoint subproblems solve the subproblems recursively and then combine their … The technique of storing solutions to subproblems instead of recomputing them is called “memoization”. Fast and free shipping free returns cash on delivery available on eligible purchase. The optimization problems expect you to select a feasible solution, so that the value of the required function is minimized or maximized. In those problems, we use DP to optimize our solution for time (over a recursive approach) at the expense of space. The word "programming" in "dynamic programming" is similar for optimization. 2 Dynamic Programming We are interested in recursive methods for solving dynamic optimization problems. Dynamic programming. Developed by Richard Bellman, dynamic programming is a mathematical technique well suited for the optimization of multistage decision problems. 3. The DEMO below is my implementation; it uses the bottom-up approach. Joesta Joesta. We have many … Professor: Daniel Russo. Abstract—Dynamic programming (DP) has a rich theoretical foundation and a broad range of applications, especially in the classic area of optimal control and the recent area of reinforcement learning (RL). This technique is becoming more and more typical. The image below is the justification result; its total badness score is 1156, much better than the previous 5022. However, dynamic programming doesn’t work … We can make three choices:1. Dynamic programming algorithm optimization for spoken word recognition. In this method, you break a complex problem into a sequence of simpler problems. Majority of the Dynamic Programming problems can be categorized into two types: 1. Solutions(such as the greedy algorithm) that better suited than dynamic programming in some cases.2. In both contexts it refers to simplifying a complicated problem by breaking it down into simpler subproblems in a recursive manner. How to construct the final result?If all we want is the distance, we already get it from the process, if we also want to construct the path, we need also save the previous vertex that leads to the shortest path, which is included in DEMO below. Optimization problems. 6. advertisement. When applicable, the method takes … The idea is to simply store the results of subproblems, so that we do not have to re-compute them when needed later. Some properties of two-variable functions required for Kunth's optimzation: 1. It is the same as “planning” or a “tabular method”. The following lecture notes are made available for students in AGEC 642 and other interested readers. Dynamic programming has the advantage that it lets us focus on one period at a time, which can often be easier to think about than the whole sequence. Math.pow(90 — line.length, 2) : Number.MAX_VALUE;Why diff²? The monograph aims at a unified and economical development of the core theory and algorithms of total cost sequential decision problems, based on the strong connections of the subject with fixed point theory. This paper reports on an optimum dynamic progxamming (DP) based time-normalization algorithm for spoken word recognition. On the international level this presentation has been inspired from (Bryson & Ho 1975), (Lewis 1986b), (Lewis 1992), (Bertsekas 1995) and (Bryson 1999).

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