HW3_solutions_v2.pdf - Homework 3 Introduction to...

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Homework 3Introduction to OptimizationMS&E 111 / MS&E 211 / ENGR 62, Stanford UniversityInstructor: Ashish GoelDue: May 2nd, 1:30 pm on Gradescope.See the syllabus for submission instructions, in particular those pertaining to Excel work.Problem 1State whether each of the following statements is true or false, and explain your answer briefly. Any answer,even correct, without an explanation will be considered incorrect and no credit will be given.a) Consider a linear program with a bounded feasible set. Ifxis an optimal solution, then it must be anoptimal basic feasible solution.b) Any linear program with a non-empty bounded feasible set has an optimal solution.c) The union of a finite number of convex sets is convex.d) The feasible region of any linear program is convex.e) A linear program may have no basic feasible solutions.f) Any linear program has at most two optimal basic feasible solutionsProblem 1 Solution
e) True. Consider the following linear programmaximizex1+x2stx10.It has no basic solutions.1
Problem 2A policy maker in Sacramento county needs to allocate some water to a group ofnfarmers in Californiafrommreservoirs. Because the media will highlight any farmer who is very unhappy, the policy maker wantsto maximize the minimum amount that any of the farmers receives. Reservoirihas capacityui. If farmerjis served from reservoiri, then a fractionfijof the water gets lost in evaporation while being channeled tothe farmer. Imagine that for every farmer who receives a non-zero amount of water from a given reservoir,a separate measuring meter is installed in that reservoir.

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