Math 407A: Linear Optimization Lecture 12: The Geometry of Linear Programming Math Dept, University of Washington Lecture 12: The Geometry of Linear Programming (Math Dept, UniversityMath 407A: of Washington)Linear Optimization 1 / 49 1 The Geometry of Linear Programming Hyperplanes Convex Polyhedra Vertices 2 The Geometry of Degeneracy 3 The Geometry of Duality Lecture 12: The Geometry of Linear Programming (Math Dept, UniversityMath 407A: of Washington)Linear Optimization 2 / 49 Fact: H ⊂ Rn is a hyperplane if and only if the set H − x0 = fx − x0 : x 2 Hg n where x0 2 H is a subspace of R of dimension (n − 1). The Geometry of Linear Programming Hyperplanes Definition: A hyperplane in Rn is any set of the form H(a; β) = fx : aT x = βg where a 2 Rn n f0g and β 2 R. Lecture 12: The Geometry of Linear Programming (Math Dept, UniversityMath 407A: of Washington)Linear Optimization 3 / 49 The Geometry of Linear Programming Hyperplanes Definition: A hyperplane in Rn is any set of the form H(a; β) = fx : aT x = βg where a 2 Rn n f0g and β 2 R. Fact: H ⊂ Rn is a hyperplane if and only if the set H − x0 = fx − x0 : x 2 Hg n where x0 2 H is a subspace of R of dimension (n − 1). Lecture 12: The Geometry of Linear Programming (Math Dept, UniversityMath 407A: of Washington)Linear Optimization 3 / 49 Points What are the hyperplanes in R2? Lines What are the hyperplanes in R3? Planes What are the hyperplanes in Rn? Translates of (n − 1) dimensional subspaces. Hyperplanes What are the hyperplanes in R? Lecture 12: The Geometry of Linear Programming (Math Dept, UniversityMath 407A: of Washington)Linear Optimization 4 / 49 What are the hyperplanes in R2? Lines What are the hyperplanes in R3? Planes What are the hyperplanes in Rn? Translates of (n − 1) dimensional subspaces. Hyperplanes What are the hyperplanes in R? Points Lecture 12: The Geometry of Linear Programming (Math Dept, UniversityMath 407A: of Washington)Linear Optimization 4 / 49 Lines What are the hyperplanes in R3? Planes What are the hyperplanes in Rn? Translates of (n − 1) dimensional subspaces. Hyperplanes What are the hyperplanes in R? Points What are the hyperplanes in R2? Lecture 12: The Geometry of Linear Programming (Math Dept, UniversityMath 407A: of Washington)Linear Optimization 4 / 49 What are the hyperplanes in R3? Planes What are the hyperplanes in Rn? Translates of (n − 1) dimensional subspaces. Hyperplanes What are the hyperplanes in R? Points What are the hyperplanes in R2? Lines Lecture 12: The Geometry of Linear Programming (Math Dept, UniversityMath 407A: of Washington)Linear Optimization 4 / 49 Planes What are the hyperplanes in Rn? Translates of (n − 1) dimensional subspaces. Hyperplanes What are the hyperplanes in R? Points What are the hyperplanes in R2? Lines What are the hyperplanes in R3? Lecture 12: The Geometry of Linear Programming (Math Dept, UniversityMath 407A: of Washington)Linear Optimization 4 / 49 What are the hyperplanes in Rn? Translates of (n − 1) dimensional subspaces. Hyperplanes What are the hyperplanes in R? Points What are the hyperplanes in R2? Lines What are the hyperplanes in R3? Planes Lecture 12: The Geometry of Linear Programming (Math Dept, UniversityMath 407A: of Washington)Linear Optimization 4 / 49 Translates of (n − 1) dimensional subspaces. Hyperplanes What are the hyperplanes in R? Points What are the hyperplanes in R2? Lines What are the hyperplanes in R3? Planes What are the hyperplanes in Rn? Lecture 12: The Geometry of Linear Programming (Math Dept, UniversityMath 407A: of Washington)Linear Optimization 4 / 49 Hyperplanes What are the hyperplanes in R? Points What are the hyperplanes in R2? Lines What are the hyperplanes in R3? Planes What are the hyperplanes in Rn? Translates of (n − 1) dimensional subspaces. Lecture 12: The Geometry of Linear Programming (Math Dept, UniversityMath 407A: of Washington)Linear Optimization 4 / 49 This division defines two closed half-spaces. The two closed half-spaces associated with the hyperplane are n T H+(a; β) = fx 2 R : a x ≥ βg and n T H−(a; β) = fx 2 R : a x ≤ βg: Hyperplanes Every hyperplane divides the space in half. H(a; β) = fx : aT x = βg Lecture 12: The Geometry of Linear Programming (Math Dept, UniversityMath 407A: of Washington)Linear Optimization 5 / 49 The two closed half-spaces associated with the hyperplane are n T H+(a; β) = fx 2 R : a x ≥ βg and n T H−(a; β) = fx 2 R : a x ≤ βg: Hyperplanes Every hyperplane divides the space in half. This division defines two closed half-spaces. H(a; β) = fx : aT x = βg Lecture 12: The Geometry of Linear Programming (Math Dept, UniversityMath 407A: of Washington)Linear Optimization 5 / 49 are n T H+(a; β) = fx 2 R : a x ≥ βg and n T H−(a; β) = fx 2 R : a x ≤ βg: Hyperplanes Every hyperplane divides the space in half. This division defines two closed half-spaces. The two closed half-spaces associated with the hyperplane H(a; β) = fx : aT x = βg Lecture 12: The Geometry of Linear Programming (Math Dept, UniversityMath 407A: of Washington)Linear Optimization 5 / 49 and n T H−(a; β) = fx 2 R : a x ≤ βg: Hyperplanes Every hyperplane divides the space in half. This division defines two closed half-spaces. The two closed half-spaces associated with the hyperplane H(a; β) = fx : aT x = βg are n T H+(a; β) = fx 2 R : a x ≥ βg Lecture 12: The Geometry of Linear Programming (Math Dept, UniversityMath 407A: of Washington)Linear Optimization 5 / 49 Hyperplanes Every hyperplane divides the space in half. This division defines two closed half-spaces. The two closed half-spaces associated with the hyperplane H(a; β) = fx : aT x = βg are n T H+(a; β) = fx 2 R : a x ≥ βg and n T H−(a; β) = fx 2 R : a x ≤ βg: Lecture 12: The Geometry of Linear Programming (Math Dept, UniversityMath 407A: of Washington)Linear Optimization 5 / 49 Define the half-spaces T Hj = fx : ej x ≥ 0g for j = 1;:::; n and T Hn+i = fx : ai· x ≤ bi g for i = 1;:::; m; where ai· is the ith row of A. Then n+m \ Ω = Hk : k=1 That is, the constraint region of an LP is the intersection of finitely many closed half-spaces. Intersections of Closed Half-Spaces Consider the constraint region to an LP Ω = fx : Ax ≤ b; 0 ≤ xg: Lecture 12: The Geometry of Linear Programming (Math Dept, UniversityMath 407A: of Washington)Linear Optimization 6 / 49 Then n+m \ Ω = Hk : k=1 That is, the constraint region of an LP is the intersection of finitely many closed half-spaces. Intersections of Closed Half-Spaces Consider the constraint region to an LP Ω = fx : Ax ≤ b; 0 ≤ xg: Define the half-spaces T Hj = fx : ej x ≥ 0g for j = 1;:::; n and T Hn+i = fx : ai· x ≤ bi g for i = 1;:::; m; where ai· is the ith row of A. Lecture 12: The Geometry of Linear Programming (Math Dept, UniversityMath 407A: of Washington)Linear Optimization 6 / 49 That is, the constraint region of an LP is the intersection of finitely many closed half-spaces. Intersections of Closed Half-Spaces Consider the constraint region to an LP Ω = fx : Ax ≤ b; 0 ≤ xg: Define the half-spaces T Hj = fx : ej x ≥ 0g for j = 1;:::; n and T Hn+i = fx : ai· x ≤ bi g for i = 1;:::; m; where ai· is the ith row of A. Then n+m \ Ω = Hk : k=1 Lecture 12: The Geometry of Linear Programming (Math Dept, UniversityMath 407A: of Washington)Linear Optimization 6 / 49 Intersections of Closed Half-Spaces Consider the constraint region to an LP Ω = fx : Ax ≤ b; 0 ≤ xg: Define the half-spaces T Hj = fx : ej x ≥ 0g for j = 1;:::; n and T Hn+i = fx : ai· x ≤ bi g for i = 1;:::; m; where ai· is the ith row of A. Then n+m \ Ω = Hk : k=1 That is, the constraint region of an LP is the intersection of finitely many closed half-spaces. Lecture 12: The Geometry of Linear Programming (Math Dept, UniversityMath 407A: of Washington)Linear Optimization 6 / 49 If a convex polyhedron in Rn is contained within a set of the form fx j ` ≤ x ≤ u g ; where `; u 2 Rn with ` ≤ u, then it is called a convex polytope. A linear program is simply the problem of either maximizing or minimizing a linear function over a convex polyhedron. We now develop the geometry of convex polyhedra. Convex Polyhedra Definition: Any subset of Rn that can be represented as the intersection of finitely many closed half spaces is called a convex polyhedron. Lecture 12: The Geometry of Linear Programming (Math Dept, UniversityMath 407A: of Washington)Linear Optimization 7 / 49 A linear program is simply the problem of either maximizing or minimizing a linear function over a convex polyhedron. We now develop the geometry of convex polyhedra. Convex Polyhedra Definition: Any subset of Rn that can be represented as the intersection of finitely many closed half spaces is called a convex polyhedron. If a convex polyhedron in Rn is contained within a set of the form fx j ` ≤ x ≤ u g ; where `; u 2 Rn with ` ≤ u, then it is called a convex polytope. Lecture 12: The Geometry of Linear Programming (Math Dept, UniversityMath 407A: of Washington)Linear Optimization 7 / 49 We now develop the geometry of convex polyhedra. Convex Polyhedra Definition: Any subset of Rn that can be represented as the intersection of finitely many closed half spaces is called a convex polyhedron.