Undergraduate convexity. From Fourier and Motzkin to Kuhn by Niels Lauritzen

By Niels Lauritzen

In response to undergraduate educating to scholars in laptop technological know-how, economics and arithmetic at Aarhus college, this can be an effortless creation to convex units and convex capabilities with emphasis on concrete computations and examples.

ranging from linear inequalities and Fourier-Motzkin removal, the speculation is built by means of introducing polyhedra, the double description approach and the simplex set of rules, closed convex subsets, convex capabilities of 1 and a number of other variables finishing with a bankruptcy on convex optimization with the Karush-Kuhn-Tucker stipulations, duality and an inside element set of rules.

Readership: Undergraduates targeting convexity and optimization.

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Can T have 3 extreme points? 2? (vi) Let Li = [ui , vi ] for i = 1, . . , m, where ui , vi ∈ Rn . 15) can have. Show that Z is the image of the unit cube [0, 1]m ⊆ Rm under a suitable affine map. 15)) is called a zonotope. 16. Give an example of a non-convex cone. 17. Prove in detail that C = {(x, y, z) ∈ R3 | z ≥ 0, x2 + y 2 ≤ z 2 } is a convex cone. Is C finitely generated? 18. 3) is a convex cone, where C is a convex subset. Perhaps the identity x + (n + λ)d = (1 − λ)(x + nd) + λ(x + (n + 1)d) might come in handy.

The key point is that a finitely generated cone is the union of finitely many simplicial subcones. A convex cone is called simplicial if it is generated by finitely many linearly independent vectors. 5 5 Constantin Carathéodory (1873–1950). Greek mathematician. 14 (Carathéodory). Let V = {v1 , . . , vm } ⊆ Rd . If v ∈ cone(V ), then v belongs to the cone generated by a linearly independent subset of V . Proof. Suppose that v = λ1 v1 + · · · + λm vm with λ1 , . . , λm > 0 and v1 , . . , vm linearly dependent.

Prove in detail that C = {(x, y, z) ∈ R3 | z ≥ 0, x2 + y 2 ≤ z 2 } is a convex cone. Is C finitely generated? 18. 3) is a convex cone, where C is a convex subset. Perhaps the identity x + (n + λ)d = (1 − λ)(x + nd) + λ(x + (n + 1)d) might come in handy. 19. What is the recession cone of a bounded convex subset? 20. Can you give an example of an unbounded convex subset C ⊆ R2 with rec(C) = {0}? 21. Let 2 1 C = cone 1 , 2 . (i) Show that 1 −2 C ◦ = cone , −2 1 . (ii) Suppose that a c C = cone b , d , where ab cd is an invertible matrix.

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