1. Model this problem as a constrained nonlinear "maxi-min" problem and then convert the program to a "mini-max" problem. 2. For the mini-max version, characterise the set of points for which the objective func- tion is NOT differentiable when X = R2. Illustrate the characterisation on the following example: y¹ = (0,0), y² = (1,0), y³ = (0,2), y = (4,4). Include a figure showing all non-differentiable points. 3. Let X = {(x1, x2) R²: x₂ ≥ 0, x² + x² ≤ 1}. For this X, rewrite the mini-max problem as a nonlinear program with a smooth objective.
1. Model this problem as a constrained nonlinear "maxi-min" problem and then convert the program to a "mini-max" problem. 2. For the mini-max version, characterise the set of points for which the objective func- tion is NOT differentiable when X = R2. Illustrate the characterisation on the following example: y¹ = (0,0), y² = (1,0), y³ = (0,2), y = (4,4). Include a figure showing all non-differentiable points. 3. Let X = {(x1, x2) R²: x₂ ≥ 0, x² + x² ≤ 1}. For this X, rewrite the mini-max problem as a nonlinear program with a smooth objective.
Glencoe Algebra 1, Student Edition, 9780079039897, 0079039898, 2018
18th Edition
ISBN:9780079039897
Author:Carter
Publisher:Carter
Chapter9: Quadratic Functions And Equations
Section: Chapter Questions
Problem 4PT
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Transcribed Image Text:Project Description
We are given N points y¹,..., y on the plane and a convex and closed set XC R². Our
objective is to find a point x EX for which the distance to the nearest point y is the
largest possible.
Tasks:
1. Model this problem as a constrained nonlinear "maxi-min" problem and then convert
the program to a "mini-max" problem.
=
For the mini-max version, characterise the set of points for which the objective func-
tion is NOT differentiable when X = R2. Illustrate the characterisation on the
following example: y¹ = (0,0), y² =
showing all non-differentiable points.
(1,0), y³ = (0,2), y = (4,4). Include a figure
3. Let X = {(x₁, x₂) R²: x₂ ≥ 0, x² + x² ≤ 1}. For this X, rewrite the mini-max
problem as a nonlinear program with a smooth objective.
4. Write out all the KKT conditions for the program in (3). Explain why we can
essentially ignore points for which the constraints are not smooth.
5. Bonus question: Provide a geometric interpretation of the points that satisfy the
KKT conditions. (Hint: try X = R² if that simplifies things).
6. Implement an algorithm for solving your model in (3). Do a computational study
(different initial solutions, parameter settings, etc.) for instances of different values of
N and various random sets of N points. Note, X stay fixed as defined. Compare your
algorithm with other algorithms or pre-existing optimisation functions in Matlab.
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