Solve the following non-linear program using dynamic programming, by showing all steps.
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Solve the following non-linear program using dynamic
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- Answer the following: This problem exercises the basic concepts of game playing, using tic-tac-toe (noughts and crosses) as an example. We define Xn as the number of rows, columns, or diagonals with exactly n X’s and no O’s. Similarly, On is the number of rows, columns, or diagonals with just n O’s. The utility function assigns +1 to any position with X3=1 and −1 to any position with O3=1. All other terminal positions have utility 0. For nonterminal positions, we use a linear evaluation function defined as Eval(s)=3X2(s)+X1(s)−(3O2(s)+O1(s)). a. Show the whole game tree starting from an empty board down to depth 2 (i.e., one X and one O on the board), taking symmetry into account. b. Mark on your tree the evaluations of all the positions at depth 2. c .Using the minimax algorithm, mark on your tree the backed-up values for the positions at depths 1 and 0, and use those values to choose the best starting move. Provide original solution!This problem exercises the basic concepts of game playing, using tic-tac-toe (noughtsand crosses) as an example. We define Xn as the number of rows, columns, or diagonals with exactly n X’s and no O’s. Similarly, On is the number of rows, columns, or diagonals with just n O’s. The utility function assigns +1 to any position with X3 = 1 and −1 to any position with O3 = 1. All other terminal positions have utility 0. For nonterminal positions, we use a linear evaluation function defined as Eval (s) = 3X2(s)+X1(s)−(3O2(s)+O1(s))."Mark on your tree the evaluations of all the positions at depth 2."Given the following linear programming model; How many artificial variables are needed to solve the exercise? Max Z = 8 X1 + 5 X2 Hold it: 12 X1 + 8X2 <= 96 6 X1 + 12X2 <= 72 X1 <= 2 X1, X2> = 0 Select one: a.None b.three c.two d.one
- Explain the relationship between the Bellman equation and the principle of optimality in dynamic programming.If an optimal solution to a problem can be obtained by greedy, It can also be obtained by dynamic programming. True or False?Consider the problem of making change for n cents using the fewest number of coins. Assume that we live in a country where coins come in k dierent denominations c1, c2, . . . , ck, such that the coin values are positive integers, k ≥ 1, and c1 = 1, i.e., there are pennies, so there is a solution for every value of n. For example, in case of the US coins, k = 4, c1 = 1, c2 = 5, c3 = 10, c4 = 25, i.e., there are pennies, nickels, dimes, and quarters. To give optimal change in the US for n cents, it is sufficient to pick as many quarters as possible, then as many dimes as possible, then as many nickels as possible, and nally give the rest in pennies. Design a bottom-up (non-recursive) O(nk)-time algorithm that makes change for any set of k different coin denominations. Write down the pseudocode and analyze its running time. Argue why your choice of the array and the order in which you fill in the values is the correct one. Notice how it is a lot easier to analyze the running time of…
- Consider the problem of making change for n cents using the fewest number of coins. Assume that we live in a country where coins come in k dierent denominations c1, c2, . . . , ck, such that the coin values are positive integers, k ≥ 1, and c1 = 1, i.e., there are pennies, so there is a solution for every value of n. For example, in case of the US coins, k = 4, c1 = 1, c2 = 5, c3 = 10, c4 = 25, i.e., there are pennies, nickels, dimes, and quarters. To give optimal change in the US for n cents, it is sufficient to pick as many quarters as possible, then as many dimes as possible, then as many nickels as possible, and nally give the rest in pennies. Design a bottom-up (non-recursive) O(nk)-time algorithm that makes change for any set of k different coin denominations. Write down the pseudocode and analyze its running time. Argue why your choice of the array and the order in which you ll in the values is the correct one.Consider the problem of making change for n cents using the fewest number of coins. Assume that we live in a country where coins come in k dierent denominations c1, c2, . . . , ck, such that the coin values are positive integers, k ≥ 1, and c1 = 1, i.e., there are pennies, so there is a solution for every value of n. For example, in case of the US coins, k = 4, c1 = 1, c2 = 5, c3 = 10, c4 = 25, i.e., there are pennies, nickels, dimes, and quarters. To give optimal change in the US for n cents, it is sufficient to pick as many quarters as possible, then as many dimes as possible, then as many nickels as possible, and nally give the rest in pennies. Prove that the coin changing problem exhibits optimal substructure. Design a recursive backtracking (brute-force) algorithm that returns the minimum number of coins needed to make change for n cents for any set of k different coin denominations. Write down the pseudocode and prove that your algorithm is correct.H1. Compute the value of an European Call option at time 0 for the following parameters: strike, K = 120, maturity 1 year, σ = 0.15, S0 = 120 and interest rate, r = 0.02: (a) by implementing, using Python, R, C++ or Matlab, a binomial model where n = 100, 1000, or 10000. For each value of n compute the error (difference between option value obtained using the B-S model and the one from the binomial model). Compute ∆0.
- Suppose we use the following KB (where x, y, z are variables and r1, r2, r3, goal are constants) to determine whether a particular robot can score. (a) Open(x) ∧ HasBall(x) → CanScore(x)(b) Open(x) ∧ CanAssist(y, x) ∧ HasBall(y) → CanScore(x) (c) PathClear(x,y) → CanAsist(x,y)(d) PathClear(x,z) ∧ CanAssist(z,y) → CanAssist(x,y) (e) PathClear(x,goal) → Open(x)(f) PathClear(y,x) → PathClear(x,y) (g) HasBall(r3)(h) PathClear(r1,goal) (i) PathClear(r2,r1) (j) PathClear(r3,r2) (k) PathClear(r3,goal)Suppose we use the following KB (where x, y, z are variables and r1, r2, r3, goal are constants) to determine whether a particular robot can score. (a) Open(x) ∧ HasBall(x) → CanScore(x)(b) Open(x) ∧ CanAssist(y, x) ∧ HasBall(y) → CanScore(x) (c) PathClear(x,y) → CanAsist(x,y)(d) PathClear(x,z) ∧ CanAssist(z,y) → CanAssist(x,y) (e) PathClear(x,goal) → Open(x)(f) PathClear(y,x) → PathClear(x,y) (g) HasBall(r3)(h) PathClear(r1,goal) (i) PathClear(r2,r1) (j) PathClear(r3,r2) (k) PathClear(r3,goal) Intuitively, CanScore(x) means x can score on goal. CanAssist(x, y) means there exists some series of passes that can get the ball from x to y. Open(x) means x can shoot on goal directly. And P athClear(x, y) means the path between x and y is clear. Provide a SLD-derivation for the query CanScore(x) in which the answer provided is r1. Provide a SLD-derivation for the query CanScore(x) in which the answer provided is r3. How many “distinct” derivations (i.e., involving different…We examine a problem in which we are handed a collection of coins and are tasked with forming a sum of money n out of the coins. The currency numbers are coins = c1, c2,..., ck, and each coin can be used as many times as we want. What is the bare amount of money required?If the coins are the euro coins (in euros) 1,2,5,10,20,50,100,200 and n = 520, we need at least four coins. The best option is to choose coins with sums of 200+200+100+20.