Applied Integer Programming

Transcription

1 Applied Integer Programming D.S. Chen; R.G. Batson; Y. Dang Fahimeh April 21, 2015

2 Context 8.2. Convex sets 8.3. Describing a bounded polyhedron 8.4. Describing unbounded polyhedron 8.5. Faces, facets and dimensions of a polyhedron 8.6. Describing a polyhedron by facets 8.7. Correspondence between algebraic and geometric terms

3 8.2. Convex sets Convex sets and polyhedra Directions of unbounded convex sets Convex and polyhedral cones Convex and concave functions

4 Convex sets and polyhedra Definition of a convex set: The solution of every linear equation or inequality forms a convex set. The intersection of two or more convex sets forms a convex set. A set containing nonlinear constraints may or may not be convex

5 Convex sets and polyhedra

6 Convex sets and polyhedra Definition of a hyperplane: X = x: p T x = k p E n and nonzero, k is a constant Hyperplane is the solution set to a linear equation A hyperplane separates E n into two halfspaces H 1 = x: p T x k and H 2 = x: p T x k

7 Example Convex sets and polyhedra

8 Convex sets and polyhedra Definition of an extreme point and an edge:

9 Convex sets and polyhedra Definition of ray: Definition of convex hull Conv(P): If it is finite it is called convex polytope

10 8.2. Convex sets Convex sets and polyhedra Directions of unbounded convex sets Convex and polyhedral cones Convex and concave functions

11 Directions of unbounded convex sets Direction: Distinct directions: d 1 kd 2 Extreme direction: It cannot be expressed as a positive combination of two distinct directions. Any other direction can be expressed as a positive combination of extreme directions. Extreme ray: a ray whose direction is an extreme direction

12 8.2. Convex sets Convex sets and polyhedra Directions of unbounded convex sets Convex and polyhedral cones Convex and concave functions

13 Convex and polyhedral cones Convex cone Polyhedral cone

14 Convex and polyhedral cones

15 8.2. Convex sets Convex sets and polyhedra Directions of unbounded convex sets Convex and polyhedral cones Convex and concave functions

16 Convex and concave functions Convex function:

17 Convex and concave functions Concave function:

18 Convex and concave functions Some properties: f is concave(convex) if and only if g=-f is convex(concave) f is linear if and only if f is both convex and concave. The definitions can be reduced to a specific subset of XϵE n, some functions are concave or convex for specific intervals.

19 Convex and concave functions Relations between convex and concave function in E n and E n+1 Epigraph and hypergraph of a function: A function f(x), xϵ E n is convex if and only if its epigraph is a convex set in in E n+1 A function f(x), xϵ E n is concave if and only if its hypergraph is a convex set in in E n+1

20 Context 8.2. Convex sets 8.3. Describing a bounded polyhedron 8.4. Describing unbounded polyhedron 8.5. Faces, facets and dimensions of a polyhedron 8.6. Describing a polyhedron by facets 8.7. Correspondence between algebraic and geometric terms

21 8.3. Describing a bounded polyhedron Representation by extreme points Given a nonempty bounded polyhedron P = x: Ax b, x 0 with extreme points x 1, x 2, x n, any point x P can be represented as a convex combination of extreme points, that is x = x j for some p j=1 particular values of α j 0 where α j = 1 α j p j=1

22 8.3. Describing a bounded polyhedron Example:

23 Context 8.2. Convex sets 8.3. Describing a bounded polyhedron 8.4. Describing unbounded polyhedron 8.5. Faces, facets and dimensions of a polyhedron 8.6. Describing a polyhedron by facets 8.7. Correspondence between algebraic and geometric terms

24 8.4. Describing unbounded polyhedron By the set of all extreme points and the set of all extreme directions Finding extreme directions algebraically Representing by extreme points and extreme directions (representation theorem)

25 Finding extreme directions algebraically Direction of an unbounded polyhedron: The set of recession directions

26 Finding extreme directions algebraically

27 Finding extreme directions algebraically Example:

28 Representing by extreme points and extreme directions Representation theorem:

29 Context 8.2. Convex sets 8.3. Describing a bounded polyhedron 8.4. Describing unbounded polyhedron 8.5. Faces, facets and dimensions of a polyhedron 8.6. Describing a polyhedron by facets 8.7. Correspondence between algebraic and geometric terms

30 8.5. Faces, facets and dimensions of a polyhedron X = x: Ax b, x 0 A: m n, x: n 1, b: (m 1) Number of inequality constraints (defining half-spaces) is m+n In a unique solution of x, n defining hyperplanes pass through an extreme point x Degenerate extreme point, degenerate polyhedron

31 8.5. Faces, facets and dimensions of a polyhedron

32 8.5. Faces, facets and dimensions of a polyhedron Binding (active) constraint A proper face F of X is a nonempty set of points in X formed by the intersection of some set of binding defining hyperplanes of X. dim(f)=n-rank(f), 0 dim F n 1 rank(f)=maximum number of linearly independent defining hyperplanes binding at all points of F. 1 rank F n

33 8.5. Faces, facets and dimensions of a polyhedron

34 Context 8.2. Convex sets 8.3. Describing a bounded polyhedron 8.4. Describing unbounded polyhedron 8.5. Faces, facets and dimensions of a polyhedron 8.6. Describing a polyhedron by facets 8.7. Correspondence between algebraic and geometric terms

35 8.6. Describing a polyhedron by facets Interesting for cutting plane and branch-and-cut methods A full dimensional polyhedron contains n linearly independent directions. At any interior point in polyhedron P, there exists a set of directions d and ε 0 >0 such that x 0 + εd ϵp, 0<ε<ε 0, An important property is that there is no hypeplane H such that P is a subset of H. A full dimensional polyhedron can be uniquely represented as a set of inequalities that each inequaliy is unique and defines a facet.

36 8.6. Describing a polyhedron by facets Valid inequality

37 Context 8.2. Convex sets 8.3. Describing a bounded polyhedron 8.4. Describing unbounded polyhedron 8.5. Faces, facets and dimensions of a polyhedron 8.6. Describing a polyhedron by facets 8.7. Correspondence between algebraic and geometric terms

38 P = {x: n j=1 a ij x j b i, i = 1,, m; x j 0, j = 1,, n}