Questions and Exercises
1. In the design space, plot the objective function contours for the following unconstrained problem and identify the Pareto optimal set:
Minimize: f1(x) = (x1 − 2)2 + (x2 − 3)2
f2(x) = (x1 − 5)2 + (x2 − 8)2
Draw the gradients of each function at any point on the Pareto optimal set in the design space. Comment on the relationship between the two gradients.
2. Sketch the Pareto optimal set for Problem 1 in the criterion space. Find the utopia point. Is the utopia point attainable?
3. A constrained optimization of two variables and two objectives is given below:
Minimize: f1(x) = (x1 − 2)2 + (x2 − 3)2
f2(x) = (x1 − 5)2 + (x2 − 8)2
Subject to: g1 = −x1 − x2 + 10 ≤ 0
g2 = −10 − 2x1 + 3x2 ≤ 0
a. Sketch the ...
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