Optimization Lecture 03 10/25/11 Geometry of LPs A (linear) subspace of ő is a subset of points that is closed under vector addition and scalar multiplication. Equivalently, ͍ Ɣ ͬ ∈ ő ͕%ͥ ͬͥ ƍ ⋯ ƍ ͕% ͬ Ɣ0,͞Ɣ1,…,͡ʝ is the set of points satisfying some set of homogeneous linear equations. It has dimension dim ͍ Ɣ ͘Ǝrank ͕$% . Note: dim ʚ͍ʛ is the maximum number of linear independent vectors in ͍. 2 Geometry of LPs An affine subspace ̻ of ő is a linear subspace ͍ translated by a vector ͩ, formally, ̻ Ɣ ͩ ƍ ͬ ͬ ∈ ͍ʝ. We define dim ̻ Ɣ dim ʚ͍ʛ. Equivalently, ̻ Ɣ ͬ ∈ ő ͕%ͥ ͬͥ ƍ ⋯ ƍ ͕% ͬ Ɣ ͖%, ͞ Ɣ 1, … , ͡ʝ Is the set of points satisfying some set of (inhomogenous) linear equations. The dimension of any subset of ő is the smallest dimension of any affine subspace containing it. 3 Geometry of LPs For example, every line segment has dimension 1. Any set of ͟ points, ͟ ƙ ͘ ƍ 1, has dimension at most ͟ Ǝ 1. The feasible region ̀ Ɣʜͬ|̻ͬ Ɣ͖,ͬƚ0ʝ of the linear program max. ͗ͬ s.t. ̻ͬ Ɣ ͖, ̻ an ͡ Ɛ ͘-matrix, ͡ Ɨ ͘ ͬ ƚ 0, has dimension at most ͘ Ǝ ͡. 4 Convex Polytopes An affine subspace of ő of dimension ͘ Ǝ 1 is called a hyperplane . Alternatively, a hyperplane is a set of points ͬ satisfying ͕ͥͬͥ ƍ ⋯ ƍ ͕ͬ Ɣ ͖, with not all ‘s equal to zero. ͕% A hyperplane defines two (closed) halfspaces. These are the sets of points satisfying or ͕ͥͬͥ ƍ ⋯ ƍ ͕ͬ ƚ ͖ , respectively. ͕ͥͬͥ ƍ ⋯ ƍ ͕ͬ ƙ ͖ 5 Convex Polytopes A subset ͍ of ő is called convex , if for any ͬ, ͭ ∈ ͍ and ∈ ʞ0,1ʟ, it holds that ͬ ƍ 1 Ǝ ͭ ∈ ͍. A halfspace is a convex set. Therefore the intersection of halfspaces is convex. The intersection of a finite number of halfspaces, when it is bounded and non-empty, is called a (convex) polytope . We are interested in convex polytopes that are included in the non-negative orthant, i.e., by convention ͘ of the halfspace defining the polytope will always be , . ͬ% ƚ 0 ͞ Ɣ 1, … , ͘ 6 Convex Polytopes Polytope defined by the constraints of LP-1: ͬͥ ƍ ͬͦ ƍ ͬͧ ƙ 4 ͬͥ ƙ 2 ͬͧ ƙ 3 3ͬͦ ƍ ͬͧ ƙ 6 ͬͥ, ͬͦ, ͬͧ ƚ 0 7 Convex Polytopes Let ͊ be a convex polytope of dimension ͘ and ͍͂ a halfspace defined by hyperplane ͂. Let ͚ Ɣ ͊ ∩ ͍͂ . If ͚ ⊆ ͂, i.e., when ͊ and ͍͂ touch just in their exteriors, we call ͚ a face of ͊ and ͂ is the supporting hyperplane defining ͚. There are three distinguished kinds of faces: • A facet is a face of dimension ͘ Ǝ 1. • A vertex is a face of dimension zero (a point). • An edge is a face of dimension one (a line segment). 8 Convex Polytopes Some facets, edges and vertices of the 3-dimensional polytope defined by the constraints of LP-1. 9 Convex Polytopes The hyperplane defining a facet corresponds to a defining halfspace of the polytope. The converse is not always true. Vertices are the polytope‘s corners. An edge is a line segment joining two vertices. Not every pair of vertices defines an edge, though! Theorem 2 (a) Every convex polytope is the convex hull of its vertices. (b) Conversely, if ͐ is a finite set of points, then the convex hull of ͐ is a convex polytope ͊. The set of vertices of ͊ is a subset of ͐. What‘s the formal relation between standard-form LPs and polytopes? 10 Standard-Form LPs and Polytopes Given an LP max. ͗ͬ s.t. ̻ͬ Ɣ ͖ ͬ ƚ 0, in standard form, we assume w.l.o.g. that the equations ̻ͬ Ɣ ͖ are of the form )ͯ( ͬ$ Ɣ ͖$ Ǝ ȕ ͕$% ͬ% , ͝Ɣ͢Ǝ͡ƍ1,…,͢. %Ͱͥ We can always achieve this by finding a basis Č of ̻, multi- plying ̻ͬ Ɣ ͖ by ̼ͯͥ and rearranging ̻ͬ Ɣ ͖ so Č corres- ponds to the last ͡ columns of ̻. 11 Standard-Form LPs and Polytopes So ̻ͬ Ɣ ͖, ͬ ƚ 0, is in fact equivalent to the inequalities )ͯ( ͖$ Ǝ ȕ ͕$% ͬ% ƚ0, ͞Ɣ͢Ǝ͡ƍ1,…,͢ %Ͱͥ . ͬ% ƚ0, ͞ Ɣ1,…,͢Ǝ͡ This, however, is an intersection of halfspaces (and bounded by Assumption 3) and, thus, defines a convex polytope ͊ ⊂ ő)ͯ(. Thus, we can view the feasible region ̀ ⊂ ő) of a standard- form LP ̻ͬ Ɣ ͖, ͬ ƚ 0, as a convex polytope ͊ ⊂ ő)ͯ(. 12 Standard-Form LPs and Polytopes Conversely, let ͊ be a polytope in the non-negative orthant of ő)ͯ(. The ͢ halfspaces defining ͊ can be expressed as ͜$ͥ ͬͥ ƍ ⋯ ƍ ͜$ )ͯ( ͬ)ͯ( ƍ ͛$ ƙ0, ͝Ɣ1,…,͢, where w.l.o.g. the first ͢ Ǝ ͡ inequalities are of the form ͬ$ ƚ0, ͝Ɣ1,…,͢Ǝ͡. Let ͂ be the matrix of coefficients of the remaining inequali- ties. We introduce slack variables to obtain ͡ ͬ)ͯ(ͮͥ, … , ͬ) the equivalent formulation ̻ͬ Ɣ ͖, ͬ ƚ 0, with matrix , , and ͡ Ɛ ͢ ̻ Ɣ ʞ͂|̓ʟ ͖ Ɣ Ǝ ͛)ͯ(ͮͥ, … , ͛) ). ͬ ∈ ő 13 Standard-Form LPs and Polytopes Thus, we can alternatively view any polytope ͊ ⊂ ő)ͯ( as the feasible region ̀ ⊂ ő) of a standard-form LP ̻ͬ Ɣ ͖, ͬ ƚ 0. The transformation of solutions between these two eqivalent views is as follows: Given a point , we ͬȤ Ɣ ͬͥ, … , ͬ)ͯ( ∈ ͊ obtain by defining ͬ Ɣ ͬͥ, … , ͬ) ∈ ̀ )ͯ( ͬ$ Ɣ Ǝ͛$ Ǝ ȕ ͜$% ͬ% , ͝Ɣ͢Ǝ͡ƍ1,…,͢. %Ͱͥ Conversely, can be transformed into ͬ Ɣ ͬͥ, … , ͬ) ∈ ̀ by truncating the last coordinates ͬȤ Ɣ ͬͥ, … , ͬ)ͯ( ∈ ͊ ͡ of . ͬ 14 Standard-Form LPs and Polytopes Theorem 3 Let ͊ ∈ ő)ͯ( be a convex polytope, ̀ Ɣ ͬ ̻ͬ Ɣ ͖, ͬ ƚ 0 ⊂ ő) the corresponding feasible region of a standard-form LP, and . Then the following are ͬȤ Ɣ ͬͥ, … , ͬ)ͯ( ∈ ͊ equivalent: (a) The point ͬȤ is a vertex of ͊. (b) If ͬȤ Ɣ ͬȤɑ ƍ 1 Ǝ ͬȤ′′ , with ͬȤɑ, ͬȤɑɑ ∈ ͊ and 0 Ɨ Ɨ 1, then ͬȤɑ Ɣ ͬȤɑɑ Ɣ ͬȤ. (c) The corresponding vector ͬ ∈ ̀ with )ͯ( ͬ$ Ɣ Ǝ͛$ Ǝ ∑%Ͱͥ ͜$% ͬ% , ͝Ɣ͢Ǝ͡ƍ1,…,͢ is a basic feasible solution of ̀. 15.