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Question 1: (20 pts) Consider the following piecewise linear function f(x). Find the minimum of f(x) with a Linear Programming (LP) formulation.

Question 2:

22x + 10

8x + 24 f(x) =

17.5x + 75

10x 35 (40 pts) Consider the following Nonlinear Programming (NLP) problem:

x [0, 1] x [1, 2] x [2, 4] x [4, 5]

(1)

min Z = (x1 3)2 +(x2 3)2 (2) s.t.

4x21 + 9x2 36 (3) x 21 + 3 x 2 = 3 (4) x=(x1,x2)X {x:x1 1} (5)

IE 317 Fall 2021 Homework # 1 2/2 1. (20 pts) Sketch (manually or via MATLAB) the feasible region and the contours of

the objective function. Identify the optimum graphically.

2. (20 pts) Repeat part 1 by replacing minimization with maximization in the problem statement.

Question 3: (40 pts) Consider the following NLP problem:

min f(x) = x + x1

1. (20 pts) Use MATLAB to generate 2D and 3D plots of f(x) in the range [10,10]. Analytically calculate all stationary points and determine whether they are local min/max or inflection point.

2. (20 pts) Use Java (with Eclipse environment) or MATLAB to implement the Bisection Search method. Find the local minimum of f(x) in the region [0.1,10] with precision level = 0.001.

   What is the global optimum solution of NLP? Does the global optimum solution occur at a stationary point?

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