1. Convex Programming Prove that the following mathematical program is a convex program. You may use without proof any facts claimed about convex functions and sets in the lecture slides. min max(1/1₁, 1/₂) 21.-22 F1, F₂ > 0 x²+x≤1
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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 solutions including original diagram for part a!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!Machine Learning Problem Perform the optimization problem of finding the minimum of J(x) = (2x-3)2 by: (i) defining theta, J(theta), h(theta) as defined in the Stanford Machine Learning videos in Coursera; (ii) plotting J(theta) vs theta by hand then use a program (iii) determining its minimum using gradient descent approach starting from a random initial value of theta = 5. Perform the search for the minimum using the gradient descent approach by hand calculations, i.e., step 1, step 2, etc. showing your work completely
- Prove that the following problem is NP: We're provided S and t. Is there a subset of S whose members add to t? Algorithm and Data Structure Issues4. Reducibility, NP-completeness 4a. Recall that linear ax + b = 0 is reducible to the quadratic ax²+bx+c=0. Show that the quadratic ax2+bx+c=0 is reducible to quartic ax*+bx+cx+dx+e=0 (Lec14.1). Is quartic reducible to quadratic? 4b. Consider the 4-way Partition Problem: Given a set S of positive integers, determine whether S can be partitioned into four subsets S1, S2, S3, and S4 such that the sum of the numbers in S, is equal to S2, S3, and S4. Demonstrate that the problem is NP-complete based on NP-completeness of the 2-way Partition Problem. 4c. Prove that P != NP.Artificial Intelligence - Adversarial Search 1. Consider the following three variants of minimax search: the simple version, alpha-beta search, anddepth-limited search, and consider the games of tic-tac-toe and chess. For the chess game, supposethat the Threefold Repetition Rule and the Fifty-Move Rule, and the similar rules if any, are notconsidered, i.e., the game will not terminate if the same position occurs multiple times. For eachcombination of minimax variant and game, answer the following question: can that minimax variantpossibly never terminate, in computing the best next move? Justify your answer.
- 2 (**) Rubik's Cube Consider the task of solving Rubik's Cube. Please formulate it as a state-space search problem. Specifically, please define the state space and the successor function. For simplicity, let us consider a 2 × 2 × 2 cube.[Introduction to the Design and Analysis of Algorithms, 3rd Edition] Maxima search. A point (xi, yi) in the Cartesian plane is said to be dominated by point (xj , yj ) if xi ≤ xj and yi ≤ yj with at least one of the two inequalities being strict. Given a set of n points, one of them is said to be a maximum of the set if it is not dominated by any other point in the set. For example, in the figure below, all the maximum points of the set are circled. Design an efficient algorithm for finding all the maximum points of a given set of n points in the Cartesian plane. What is the time efficiency class of your algorithm?Suppose you have been given a map of 6 cities connected with each other via different paths. Your job is to visit every city just once covering the minimum distance possible. Solve this problem using Genetic Algorithm. You can start at any point and end at any point. Just make sure that all the cities have been covered. 1. Encode the problem and create an initial population of 3 different chromosomes 2. Choose any one parent from your above solution and identify the following:i. Gene ii. Chromosome 3. Think of an appropriate fitness function to this problem and give proper justification. 4. Use the fitness function to calculate the fitness level of all the chromosomes in your population. Select the fittest 2 chromosomes based on the fitness function. 5. Perform crossover that you have been taught in the class on the selected parents. Now based on the offspring, for this problem do you think that is the best way to perform crossover? If not, explain why. 6. Perform mutation that you…
- This characteristic is met by a problem if it is feasible to develop an optimum solution for it by first creating optimal solutions for its subproblems. a) Subproblems that overlap; b) substructure that's optimum; c) memorization; d) greedyThis 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."If you are given a set S of integers and a number t, prove that this issue falls into the NP class. Is there a subset of S where the total number of items is t? Note: Complexity in Data Structures and Algorithms