Dynamic Programming
... that for the exogenous state space variable. (X; ) and (Z; ) are measurable spaces and (S; )=(X xZ; x ) is the set of possible states of the system. ...
... that for the exogenous state space variable. (X; ) and (Z; ) are measurable spaces and (S; )=(X xZ; x ) is the set of possible states of the system. ...
Chapter 7 An Introduction to Linear Programming Learning Objectives
... Obtain an overview of the kinds of problems linear programming has been used to solve. ...
... Obtain an overview of the kinds of problems linear programming has been used to solve. ...
due 4/01/2016 in class
... Obtain an integer vector x from x∗ by rounding each component to the nearest integer. Is x an optimal solution to the integer program P? If it is not, find an optimal solution to the integer program P. Hint: Since the objective function of the integer linear program is just x1 , you can find an opti ...
... Obtain an integer vector x from x∗ by rounding each component to the nearest integer. Is x an optimal solution to the integer program P? If it is not, find an optimal solution to the integer program P. Hint: Since the objective function of the integer linear program is just x1 , you can find an opti ...
2008
... 2. The Pythagorean Integer Triple Problem All numbers to be considered in this problem are positive integers. A Pythagorean Triple (PT) is an ordered set of three numbers (a,b,c) such that a2 +b2 = c2; e.g. (3,4,5) and (4,3,5) are each a PT but (5,3,4) is not a PT. . (a) Prove that if (a,b,c) is a P ...
... 2. The Pythagorean Integer Triple Problem All numbers to be considered in this problem are positive integers. A Pythagorean Triple (PT) is an ordered set of three numbers (a,b,c) such that a2 +b2 = c2; e.g. (3,4,5) and (4,3,5) are each a PT but (5,3,4) is not a PT. . (a) Prove that if (a,b,c) is a P ...
Finding Empirical and Molecular Formulas (1A)
... 1. Convert all masses to moles of each element. If %-mass are given, assume 100 g of the sample (so that % = massing) and then convert to number of moles using the respective atomic masses. 2. Find which number of moles is the smallest and divide all the different numbers of moles by that smallest n ...
... 1. Convert all masses to moles of each element. If %-mass are given, assume 100 g of the sample (so that % = massing) and then convert to number of moles using the respective atomic masses. 2. Find which number of moles is the smallest and divide all the different numbers of moles by that smallest n ...
An alternative quadratic formula
... The correctness of the formula is readily checked by expanding the square and comparing to (4). By looking at (9), we read off that, for real coefficients u, v 2 ∈ R, (8) has • a double root (namely 2u) iff u = ±v, • two distinct real roots iff u2 > v 2 , • two real roots of opposite sign iff v 2 < ...
... The correctness of the formula is readily checked by expanding the square and comparing to (4). By looking at (9), we read off that, for real coefficients u, v 2 ∈ R, (8) has • a double root (namely 2u) iff u = ±v, • two distinct real roots iff u2 > v 2 , • two real roots of opposite sign iff v 2 < ...
Steps for solving an Empirical Formula Problem
... Empirical Formula and Molecular Formula Problem Solving Steps for solving an Empirical Formula Problem 1. If given % data, assume 100g of sample. If given grams, then use grams and proceed to #2. 2. Convert g to moles for each element in the compound 3. Divide each of these mole amounts by smallest ...
... Empirical Formula and Molecular Formula Problem Solving Steps for solving an Empirical Formula Problem 1. If given % data, assume 100g of sample. If given grams, then use grams and proceed to #2. 2. Convert g to moles for each element in the compound 3. Divide each of these mole amounts by smallest ...
Dynamic programming
In mathematics, computer science, economics, and bioinformatics, dynamic programming is a method for solving a complex problem by breaking it down into a collection of simpler subproblems. It is applicable to problems exhibiting the properties of overlapping subproblems and optimal substructure (described below). When applicable, the method takes far less time than other methods that don't take advantage of the subproblem overlap (like depth-first search).In order to solve a given problem using a dynamic programming approach, we need to solve different parts of the problem (subproblems), then combine the solutions of the subproblems to reach an overall solution. Often when using a more naive method, many of the subproblems are generated and solved many times. The dynamic programming approach seeks to solve each subproblem only once, thus reducing the number of computations: once the solution to a given subproblem has been computed, it is stored or ""memoized"": the next time the same solution is needed, it is simply looked up. This approach is especially useful when the number of repeating subproblems grows exponentially as a function of the size of the input.Dynamic programming algorithms are used for optimization (for example, finding the shortest path between two points, or the fastest way to multiply many matrices). A dynamic programming algorithm will examine the previously solved subproblems and will combine their solutions to give the best solution for the given problem. The alternatives are many, such as using a greedy algorithm, which picks the locally optimal choice at each branch in the road. The locally optimal choice may be a poor choice for the overall solution. While a greedy algorithm does not guarantee an optimal solution, it is often faster to calculate. Fortunately, some greedy algorithms (such as minimum spanning trees) are proven to lead to the optimal solution.For example, let's say that you have to get from point A to point B as fast as possible, in a given city, during rush hour. A dynamic programming algorithm will look at finding the shortest paths to points close to A, and use those solutions to eventually find the shortest path to B. On the other hand, a greedy algorithm will start you driving immediately and will pick the road that looks the fastest at every intersection. As you can imagine, this strategy might not lead to the fastest arrival time, since you might take some ""easy"" streets and then find yourself hopelessly stuck in a traffic jam.Sometimes, applying memoization to a naive basic recursive solution already results in a dynamic programming solution with asymptotically optimal time complexity; however, the optimal solution to some problems requires more sophisticated dynamic programming algorithms. Some of these may be recursive as well but parametrized differently from the naive solution. Others can be more complicated and cannot be implemented as a recursive function with memoization. Examples of these are the two solutions to the Egg Dropping puzzle below.