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Calculus with Convex Sets in a Nutshell

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Calculus with Convex Sets in a Nutshell Calculus with Convex Sets in a Nutshell Christian Schilling April 2, 2019 Preliminaries Basic convex sets Set operations Advanced convex sets Support function Conclusion Overview Preliminaries Basic convex sets Set operations Advanced convex sets Support function Conclusion 1 / 25 Preliminaries Basic convex sets Set operations Advanced convex sets Support function Conclusion Overview Preliminaries Basic convex sets Set operations Advanced convex sets Support function Conclusion 2 / 25 2.0 2.0 1.5 1.5 1.0 1.0 0.5 0.5 0.0 0.0 -1.0 -0.5 0.0 0.5 1.0 0.0 0.5 1.0 1.5 2.0 Convex Not convex Preliminaries Basic convex sets Set operations Advanced convex sets Support function Conclusion Convex sets n • We consider the vector space R Definition (Convex set) X is convex if X = {λ · ~x + (1 − λ) · ~y | ~x,~y ∈ X, λ ∈ [0, 1] ⊆ R} 3 / 25 2.0 2.0 1.5 1.5 1.0 1.0 0.5 0.5 0.0 0.0 -1.0 -0.5 0.0 0.5 1.0 0.0 0.5 1.0 1.5 2.0 Preliminaries Basic convex sets Set operations Advanced convex sets Support function Conclusion Convex sets n • We consider the vector space R Definition (Convex set) X is convex if X = {λ · ~x + (1 − λ) · ~y | ~x,~y ∈ X, λ ∈ [0, 1] ⊆ R} Convex Not convex 3 / 25 Definition (Bounded set) X is bounded if ∃δ ∈ R ∀~x,~y ∈ X : kx − yk ≤ δ Definition (Compact set) A set is compact if it is closed and bounded Preliminaries Basic convex sets Set operations Advanced convex sets Support function Conclusion Compact sets Definition (Closed set) A set is closed if it contains all its boundary points 0 1 2 3 0 1 2 3 1 1 1 ≤ x1 ≤ 2 (⊆ R ) 1 < x1 < 2 (⊆ R ) 4 / 25 Definition (Compact set) A set is compact if it is closed and bounded Preliminaries Basic convex sets Set operations Advanced convex sets Support function Conclusion Compact sets Definition (Closed set) A set is closed if it contains all its boundary points Definition (Bounded set) X is bounded if ∃δ ∈ R ∀~x,~y ∈ X : kx − yk ≤ δ 0 1 2 3 0 1 2 3 1 1 1 ≤ x1 ≤ 2 (⊆ R ) 1 ≤ x1 (⊆ R ) 4 / 25 Preliminaries Basic convex sets Set operations Advanced convex sets Support function Conclusion Compact sets Definition (Closed set) A set is closed if it contains all its boundary points Definition (Bounded set) X is bounded if ∃δ ∈ R ∀~x,~y ∈ X : kx − yk ≤ δ Definition (Compact set) A set is compact if it is closed and bounded 4 / 25 Preliminaries Basic convex sets Set operations Advanced convex sets Support function Conclusion Overview Preliminaries Basic convex sets Set operations Advanced convex sets Support function Conclusion 5 / 25 Preliminaries Basic convex sets Set operations Advanced convex sets Support function Conclusion Simplest examples 2 2 1 1 0 0 0 1 2 0 1 2 Empty set Singleton 6 / 25 Unit ball in ∞-norm Unit ball in 2-norm Unit ball in 1-norm aka hypercube aka hypersphere aka cross-polytope Definition (p-norm) pp p p k~x = (x1,..., xn)kp := |x1| + ··· + |xn| . • Balls in the p-norm are convex for p ≥ 1. Preliminaries Basic convex sets Set operations Advanced convex sets Support function Conclusion More examples 1.0 1.0 1.0 0.5 0.5 0.5 0.0 0.0 0.0 -0.5 -0.5 -0.5 -1.0 -1.0 -1.0 -1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 7 / 25 Preliminaries Basic convex sets Set operations Advanced convex sets Support function Conclusion Unit balls 1.0 1.0 1.0 0.5 0.5 0.5 0.0 0.0 0.0 -0.5 -0.5 -0.5 -1.0 -1.0 -1.0 -1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 Unit ball in ∞-norm Unit ball in 2-norm Unit ball in 1-norm aka hypercube aka hypersphere aka cross-polytope Definition (p-norm) pp p p k~x = (x1,..., xn)kp := |x1| + ··· + |xn| . • Balls in the p-norm are convex for p ≥ 1. 7 / 25 Preliminaries Basic convex sets Set operations Advanced convex sets Support function Conclusion Unit balls 1.0 1.0 1.0 0.5 0.5 0.5 0.0 0.0 0.0 -0.5 -0.5 -0.5 -1.0 -1.0 -1.0 -1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 Unit ball in 3-norm Unit ball in 42-norm Unit ball in (π − 2)-norm Definition (p-norm) pp p p k~x = (x1,..., xn)kp := |x1| + ··· + |xn| . • Balls in the p-norm are convex for p ≥ 1. 7 / 25 Preliminaries Basic convex sets Set operations Advanced convex sets Support function Conclusion Unit balls https://commons.wikimedia. org/wiki/File:Astroid.svg Unit ball in 2/3-norm (not convex!) Definition (p-norm) pp p p k~x = (x1,..., xn)kp := |x1| + ··· + |xn| . • Balls in the p-norm are convex for p ≥ 1. 7 / 25 Preliminaries Basic convex sets Set operations Advanced convex sets Support function Conclusion Unbounded sets 1.5 1.5 1.0 1.0 0.5 0.5 0.0 0.0 -0.5 -0.5 -0.5 0.0 0.5 1.0 1.5 -0.5 0.0 0.5 1.0 1.5 Hyperplane h~d,~xi = c Half-space h~d,~xi ≤ c 8 / 25 Preliminaries Basic convex sets Set operations Advanced convex sets Support function Conclusion Overview Preliminaries Basic convex sets Set operations Advanced convex sets Support function Conclusion 9 / 25 Preliminaries Basic convex sets Set operations Advanced convex sets Support function Conclusion Minkowski sum Definition X ⊕ Y := {~x + ~y | ~x ∈ X,~y ∈ Y } 10 / 25 Preliminaries Basic convex sets Set operations Advanced convex sets Support function Conclusion Minkowski sum Definition X ⊕ Y := {~x + ~y | ~x ∈ X,~y ∈ Y } 2.5 2.0 1.5 1.0 0.5 0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 Translation 10 / 25 Preliminaries Basic convex sets Set operations Advanced convex sets Support function Conclusion Minkowski sum Definition X ⊕ Y := {~x + ~y | ~x ∈ X,~y ∈ Y } 1.00 0.75 0.50 0.25 0.00 0.0 0.5 1.0 1.5 Square ⊕ circle centered in the origin 10 / 25 Preliminaries Basic convex sets Set operations Advanced convex sets Support function Conclusion Linear map Definition M · X := {M · ~x | ~x ∈ X} 11 / 25 Preliminaries Basic convex sets Set operations Advanced convex sets Support function Conclusion Linear map Definition M · X := {M · ~x | ~x ∈ X} 3.5 3.0 2 0.5 M = 2.5 0 1.5 2.0 1.5 1 2 3 4 Invertible map 11 / 25 Preliminaries Basic convex sets Set operations Advanced convex sets Support function Conclusion Linear map Definition M · X := {M · ~x | ~x ∈ X} 2.4 2.2 0 0 M = 2.0 0 1 1.8 1.6 0.0 0.5 1.0 1.5 Projection 11 / 25 Preliminaries Basic convex sets Set operations Advanced convex sets Support function Conclusion Convex hull Definition CH(X) := smallest set Y s.t. Y = X ∪ {λ · ~x + (1 − λ) · ~y | ~x,~y ∈ Y , λ ∈ [0, 1] ⊆ R} 12 / 25 Preliminaries Basic convex sets Set operations Advanced convex sets Support function Conclusion Convex hull Definition CH(X) := smallest set Y s.t. Y = X ∪ {λ · ~x + (1 − λ) · ~y | ~x,~y ∈ Y , λ ∈ [0, 1] ⊆ R} 3.5 3.0 2.5 2.0 1.5 1.0 0.0 0.5 1.0 1.5 2.0 2.5 Union (not convex) 12 / 25 Preliminaries Basic convex sets Set operations Advanced convex sets Support function Conclusion Convex hull Definition CH(X) := smallest set Y s.t. Y = X ∪ {λ · ~x + (1 − λ) · ~y | ~x,~y ∈ Y , λ ∈ [0, 1] ⊆ R} 3.5 3.0 2.5 2.0 1.5 1.0 0.0 0.5 1.0 1.5 2.0 2.5 Convex hull of the union 12 / 25 Preliminaries Basic convex sets Set operations Advanced convex sets Support function Conclusion Intersection Definition X ∩ Y := {~x | ~x ∈ X ∧ ~x ∈ Y } 13 / 25 Preliminaries Basic convex sets Set operations Advanced convex sets Support function Conclusion Intersection Definition X ∩ Y := {~x | ~x ∈ X ∧ ~x ∈ Y } 4 3 2 1 0 -1 -1 0 1 2 3 4 Disjoint 13 / 25 Preliminaries Basic convex sets Set operations Advanced convex sets Support function Conclusion Intersection Definition X ∩ Y := {~x | ~x ∈ X ∧ ~x ∈ Y } 3 2 1 0 -1 -1 0 1 2 3 Intersecting 13 / 25 Preliminaries Basic convex sets Set operations Advanced convex sets Support function Conclusion Overview Preliminaries Basic convex sets Set operations Advanced convex sets Support function Conclusion 14 / 25 Preliminaries Basic convex sets Set operations Advanced convex sets Support function Conclusion Generalizations of balls 1.5 1.5 1.0 1.0 0.5 0.5 0.0 0.0 -0.5 -0.5 -1.0 -1.0 -1.5 -1.5 -1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 Hyperrectangle (Hyper-)Ellipsoid 15 / 25 Preliminaries Basic convex sets Set operations Advanced convex sets Support function Conclusion Polytopes Definition (Vertex representation) A polytope is the convex hull of the union of finitely many singletons 2.5 2.0 1.5 1.0 0.5 0.0 -0.5 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 16 / 25 Preliminaries Basic convex sets Set operations Advanced convex sets Support function Conclusion Polytopes Definition (Vertex representation) A polytope is the convex hull of the union of finitely many singletons Definition (Constraint (or half-space) representation) A polytope is the bounded intersection of finitely many half-spaces 2.5 2.0 1.5 1.0 0.5 0.0 -0.5 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 16 / 25 Preliminaries Basic convex sets Set operations Advanced convex sets Support function Conclusion Polytopes Definition (Vertex representation) A polytope is the convex hull of the union of finitely many singletons Definition (Constraint (or half-space) representation) A polytope is the bounded intersection of finitely many half-spaces 2.5 2.0 1.5 1.0 0.5 0.0 -0.5 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 Polyhedron (unbounded) 16 / 25 1.0 0.5 0.0 -0.5 -1.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 • Centrally symmetric polytope Preliminaries Basic convex sets Set operations Advanced convex sets Support function Conclusion Zonotopes • Minkowski sum of line segments ( p ) X ~c + ξi · ~gi ξi ∈ [−1, 1] i=1 17 / 25 1.0 0.5 0.0 -0.5 -1.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 Preliminaries Basic convex sets Set operations Advanced convex sets Support function Conclusion Zonotopes • Minkowski sum of line segments ( p ) X ~c + ξi · ~gi ξi ∈ [−1, 1] i=1 • Centrally symmetric polytope 17 / 25 Preliminaries Basic convex sets Set operations Advanced convex sets Support function Conclusion Closure properties X ⊕ Y
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