Problem 1 To keep healthy, a cat needs to eat n ingredients over a week. The unit cost for each ingredients is 03-. Assume each ingredient j has nutritional content at; for nutrient 2'. Also, for each nutrient 2', the cat needs a minimum level b, over a week. Formulate a linear programming problem whose objective is to minimize the total cost of the cat food while keep the cat healthy over a week. (Hint: assume the quantities of n ingredients you buy over a week is 933', j = 1, - - - ,n.) Problem 2 Consider a school district with I neighborhoods, J schools, and G grades at each school. Each school j has a capacity of ng for grade 9. In each neighborhood 2', the student population of grade 2' is Sig. Finally, the distance of school 3' from neighborhood 2' is dij. Formulate a linear programming problem whose objective is to assign all students to schools, while minimizing the total distance traveled by all students. (Ignore the fact that numbers of students must be integer.) (Hint: assume the decision variable $ng represent the number of students from neighborhood 2' that go to school j, in grade 9.) Problem 3 Consider the problem minimize f (1:) = :81 subject to (9:1 1)2 + m3 = 1 ($1 + 1)2+a: = 1. Graph the feasible set. Are there local minimizer? Are there global minimizer? Problem 4 Consider a feasible region 3 dened by a set of linear constraints 3 = {1: : Am 5 b} Prove S is convex