Given a sequence X[1, . . . , n], find an increasing subsequence with the maximum sum. Precisely define the subproblem. Provide the recurrence equation. Describe the algorithm in pseudocode to compute the optimal value. Describe the algorithm in pseudocode to print out an optimal solution.

Please write it on paper

There are n people who want to carpool during m days. On day i, some subset si ofpeople want to carpool, and the driver di must be selected from si . Each person j hasa limited number of days fj they are willing to drive. Give an algorithm to find a driverassignment di ∈ si each day i such that no person j has to drive more than their limit fj. (The algorithm should output “no” if there is no such assignment.) Hint: Use networkflow.For example, for the following input with n = 3 and m = 3, the algorithm could assignTom to Day 1 and Day 2, and Mark to Day 3. Person Day 1 Day 2 Day 3 Limit 1 (Tom) x x x 2 2 (Mark) x   x 1 3 (Fred)   x x 0

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Let x0 = (2, 1). Following the general optimization algorithm find solution x2 if for iteration #1 step size α0 = 2 and search direction p0 = (0, 1), and for iteration #2 α1 = 3 and p1 = (2, 0). Show solutions x0, x1, x2 using cartesian coordinates

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