Modified Branch and bound for finding k-best tours for the traveling salesman problem Input: (G, d), the set of tours H and an integer so that 1 ≤ k ≤|H| Output: A set H(k) of best tours in the (G, d).
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- Generate the graph of f(xk) vs k where k is the iteration number and xk is the current estimate of x at iteration k. This graph should convey the decreasing nature of function values.Let f(x) = x¹ Hx-x¹b, where H and b are constant, independent of x, and H is symmetric positive definite. Given vectors x0) and p0), find the value of the scalar a that minimizes f(x0) + ap0)). This is the formula for the stepsize ak in the linear conjugate gradient algorithm.Minimize the function using Karnaugh map method. F(A, B, C, D) = E(1, 2, 3, 8, 9, 10, 11, 14) + Ed(7, 15).
- Determine P(A x B) – (A x B) where A = {a} and B = {1, 2}.Suppose a salesperson is planning a sales trip that includes n cities. Each city is connected to some of the other cities by a road. To minimize travel time, the salesperson wants to determine the shortest route that starts at the salesperson’s home city, visits each of the cities once, and ends up at the home city. This problem of finding the shortest route is called the Travelling Salesperson Problem (TSP) and is a well-known problem that can be solved using Dynamic Programming. Research about the TSP problem and find an algorithm based on Dynamic Programming for that. Explain the approach, the algorithm, and its time complexity. Use an example to explain the way the algorithm operatesDetermine the recurrence relation for the running time, the worst case time complexity and worst case space complexity of the function bar(m, lx, ly, n).Provide explanations on how the answers were achieved.
- It was claimed that:(a, b) ≤ (c, d) ⇔ (a < c) ∨ (a = c ∧ b ≤ d) defines a well-ordering on N x N. Show that this is actually the case.Give a proof by cases that x ≤ |x| for all real numbers x.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
- Given functions f:R → R and g: R → R. Suppose that go f turns out as go f(x) = (x2 + 7]. Identify the g(x) and f(x) that produce the go f(x) as stated. Also, plot g(x) and f(x).Incorrect solution will be reported. Let A and B be the given two sets: A={ 1,2,3} and B={ 1,4} Let set Z = (A - B) x B. Where x represents the cartesian product The cardinality of Z isGIVEN: n red points and n blue points in the plane in general position (i.e., no 3 points are on the same line) PROVE: there exists a matching (i.e., 1-1 correspondence) between red and blue points such that the segments connecting the corresponding points do not intersect. EXTRA/HINT: describe an algorithm for finding such matching