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Undergraduate Convexity
Problems and Solutions
Mikkel Slot Nielsen, Victor Ulrich Rohde;;;
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eBook - ePub
Undergraduate Convexity
Problems and Solutions
Mikkel Slot Nielsen, Victor Ulrich Rohde;;;
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This solutions manual thoroughly goes through the exercises found in Undergraduate Convexity: From Fourier and Motzkin to Kuhn and Tucker. Several solutions are accompanied by detailed illustrations and intuitive explanations. This book will pave the way for students to easily grasp the multitude of solution methods and aspects of convex sets and convex functions.
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Topic
MathématiquesSubtopic
Analyse vectorielleChapter 1
Fourier-Motzkin elimination
1.1Introduction
Just like its Gaussian counterpart, Fourier-Motzkin elimination quickly becomes very natural with a bit of practice. The exercises below focus exactly on this practice, and the reader will hopefully feel confident in working with inequalities after carefully going through the material. Some exercises have a more applied flavor and may also serve as a motivation for reducing a system of inequalities.
Being cumbersome, space consuming, and slightly trivial to rewrite the Fourier-Motzkin elimination in detail in every solution, the later solutions become more concise; so, feeling a bit lost may be resolved by simply going back a couple of solutions.
Proposition 1.1. Let α1, …, αr, β1, …βs ∈ ℝ. Then
if and only if αi ≤ βi for every i, j with 1 ≤ i ≤ r and 1 ≤ j ≤ s:
Definition 1.4. The subset
of solutions to a system
of finitely many linear inequalities (here aij and bi are real numbers) is called a polyhedron.
Whilst Proposition 1.1 is very useful for computation, the following result is more of theoretical interest.
Theorem 1.6.Consider the projection π : ℝn → ℝn−1 given by
If P ⊆ ℝn is a polyhedron, then
is a polyhedron.
1.2Exercises and solutions
Exercise 1.1. Sketch the set of solutions to the system
of linear inequalities. Carry out the elimination procedure for (1.1) as illustrated in §1.1.
Solution 1.1. The set of solutions is sketched in Figure 1.1. To carry out the elimination procedure we start by isolating x which gives the new system
This is, according to Proposition 1.1, equivalent to the system
There exists x such that (x, y) is a solution if and only if y satisfies
By another use of Proposition 1.1 we know that y is a solution to (1.2) if and only if y is a solution to
This system is equivalent to
which means that a solution satisfies y ∈ [0, 3]. In conclusion, we have (x, y) ∈ ℝ2 is a solution if and only if y ∈ [0, 3] and
Exercise 1.2. Let
and π : ℝ3 → ℝ2 be given by π(x, y, z) = (y, z).
(i)Compute π(P) as a polyhedron i.e., as the solutions to a set of linear inequalities in y and z.
(ii)Compute η(P), where η : ℝ3 → ℝ is given by η(x, y, z) = x.
(iii)How m...