Optimization methods NOPT048

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1 Optimization methods NOPT048 Jirka Fink fink/ Department of Theoretical Computer Science and Mathematical Logic Faculty of Mathematics and Physics Charles University in Prague Summer semester 2016/17 Last change on May 11, 2017 License: Creative Commons BY-NC-SA 4.0 Jirka Fink Optimization methods 1

2 Jirka Fink: Optimization methods Plan of the lecture Linear and integer optimization Convex sets and Minkowski-Weyl theorem Simplex methods Duality of linear programming Ellipsoid method Unimodularity Minimal weight maximal matching Matroid Cut and bound method Jirka Fink Optimization methods 2

3 Jirka Fink: Optimization methods General information Homepage fink/ Consultations Individual schedule Examination Tutorial conditions Tests Theoretical homeworks Practical homeworks Pass the exam Jirka Fink Optimization methods 3

4 Jirka Fink: Optimization methods Literature A. Schrijver: Theory of linear and integer programming, John Wiley, 1986 W. J.Cook, W. H. Cunningham, W. R. Pulleyblank, A. Schrijver: Combinatorial Optimization, John Wiley, 1997 J. Matoušek, B. Gärtner: Understanding and using linear programming, Springer, J. Matoušek: Introduction to Discrete Geometry. ITI Series , MFF UK, 2003 Jirka Fink Optimization methods 4

5 Outline 1 Linear programming 2 Linear, affine and convex sets 3 Simplex method 4 Duality of linear programming 5 Integer linear programming 6 Vertex Cover 7 Matching Jirka Fink Optimization methods 5

6 Example of linear programming: Optimized diet Express using linear programming the following problem Find the cheapest vegetable salad from carrots, white cabbage and cucumbers containing required amount the vitamins A and C and dietary fiber. Food Carrot White cabbage Cucumber Required per meal Vitamin A [mg/kg] mg Vitamin C [mg/kg] mg Dietary fiber [g/kg] g Price [EUR/kg] Formulation using linear programming Carrot White cabbage Cucumber Minimize 0.75x x x 3 Cost subject to 35x x x Vitamin A 60x x x 3 15 Vitamin C 30x x x 3 4 Dietary fiber x 1, x 2, x 3 0 Jirka Fink Optimization methods 6

7 Matrix notation of Linear programming problem Formulation using linear programming Minimize 0.75x x x 3 subject to 35x x x x x x x x x 3 4 x 1, x 2, x 3 0 Matrix notation Minimize Subject to a x 1, x 2, x T x 1 x 2 x x x x 3 4 Jirka Fink Optimization methods 7

8 Notation: Vector and matrix Matrix A matrix of type m n is a rectangular array of m rows and n columns of real numbers. Matrices are written as A, B, C, etc. Vector A vector is an n-tuple of real numbers. Vectors are written as c, x, y, etc. Usually, vectors are column matrices of type n 1. Scalar A scalar is a real number. Scalars are written as a, b, c, etc. Special vectors 0 and 1 are vectors of zeros and ones, respectively. Transpose The transpose of a matrix A is matrix A T created by reflecting A over its main diagonal. The transpose of a column vector x is the row vector x T. Jirka Fink Optimization methods 8

9 Notation: Matrix product Elements of a vector and a matrix The i-th element of a vector x is denoted by x i. The (i, j)-th element of a matrix A is denoted by A i,j. The i-th row of a matrix A is denoted by A i,. The j-th column of a matrix A is denoted by A,j. Dot product of vectors The dot product (also called inner product or scalar product) of vectors x, y R n is the scalar x T y = n i=1 x i y i. Product of a matrix and a vector The product Ax of a matrix A R m n of type m n and a vector x R n is a vector y R m such that y i = A i, x for all i = 1,..., m. Product of two matrices The product AB of a matrix A R m n and a matrix B R n k a matrix C R m k such that C,j = AB,j for all j = 1,..., k. Jirka Fink Optimization methods 9

10 Notation: System of linear equations and inequalities Equality and inequality of two vectors For vectors x, y R n we denote x = y if x i = y i for every i = 1,..., n and x y if x i y i for every i = 1,..., n. System of linear equations Given a matrix A R m n of type m n and a vector b R m, the formula Ax = b means a system of m linear equations where x is a vector of n real variables. System of linear inequalities Given a matrix A R m n of type and a vector b R m, the formula Ax b means a system of m linear inequalities where x is a vector of n real variables. Example: System of linear inequalities in two different notations ( ) 2x 1 + x 2 + x ( ) x x 2x 1 + 5x 2 + 5x x 3 Jirka Fink Optimization methods 10

11 Optimization Mathematical optimization is the selection of a best element (with regard to some criteria) from some set of available alternatives. Examples Minimize x 2 + y 2 where (x, y) R 2 Maximal matching in a graph Minimal spanning tree Shortest path between given two vertices Optimization problem Given a set of solutions M and an objective function f : M R, optimization problem is finding a solution x M with the maximal (or minimal) objective value f (x) among all solutions of M. Duality between minimization and maximization If min x M f (x) exists, then also max x M f (x) exists and min x M f (x) = max x M f (x). Jirka Fink Optimization methods 11

12 Linear Programming Linear programming problem A linear program is the problem of maximizing (or minimizing) a given linear function over the set of all vectors that satisfy a given system of linear equations and inequalities. Equation form: min c T x subject to Ax = b, x 0 Canonical form: max c T x subject to Ax b, where c R n, b R m, A R m n a x R n. Conversion from the equation form to the canonical form max c T x subject to Ax b, Ax b, x 0 Conversion from the canonical form to the equation form min c T x + c T x subject to Ax Ax + Ix = b, x, x, x 0 Jirka Fink Optimization methods 12

13 Terminology Basic terminology Number of variables: n Number of constrains: m Solution: an arbritrary vector x of R n Objective function: e.g. max c T x Feasible solution: a solution satisfying all constrains, e.g. Ax b Optimal solution: a feasible solution maximizing c T x Infeasible problem: a problem having no feasible solution Unbounded problem: a problem having a feasible solution with arbitrary large value of given objective function Polyhedron: a set of points x R n satisfying Ax b for some A R m n and b R m Polytope: a bounded polyhedron Jirka Fink Optimization methods 13

14 Example of linear programming: Network flow Network flow problem Given a direct graph (V, E) with capacities c R E and a source s V and a sink t V, find the maximal flow from s to t satisfying the flow conservation and capacity constrains. Formulation using linear programming Variables: flow f e for every edge e E Capacity constrains: 0 f c Flow conservation: uv E f uv = vw E f vw for every v V \ {s, t} Objective function: Maximize sw E f sw us E f us Matrix notation Add an auxiliary edge x ts with a sufficiently large capacity c ts Objective function: max x ts Flow conservation: Ax = 0 where A is the incidence matrix Capacity constrains: x c and x 0 Jirka Fink Optimization methods 14

15 Graphical method: Set of feasible solutions Example Draw the set of all feasible solutions (x 1, x 2 ) satisfying the following conditions. x 1 + 6x x 1 x 2 10 x 1 + x 2 1 x 1, x 2 0 Solution x 1 0 x 2 x 1 1 x 1 + 6x 2 15 x 2 0 (0, 0) 4x 1 x 2 10 Jirka Fink Optimization methods 15

16 Graphical method: Optimal solution Example Find the optimal solution of the following problem. Solution Maximize x 1 + x 2 x 1 + 6x x 1 x 2 10 x 1 + x 2 1 x 1, x 2 0 c T x = 2 c T x = 5 c T x = 0 c T x = 1 (3, 2) (1, 1) (0, 0) Jirka Fink Optimization methods 16

17 Graphical method: Multiple optimal solutions Example Find all optimal solutions of the following problem. Maximize x 2 x 1 + 6x x 1 x 2 10 x 1 + x 2 1 x 1, x 2 0 Solution c T x = 10 3 ( 1 6, 1) (0, 0) Jirka Fink Optimization methods 17

18 Graphical method: Unbounded problem Example Show that the following problem is unbounded. Maximize x 1 + x 2 x 1 + x 2 1 x 1, x 2 0 Solution (1, 1) (0, 0) Jirka Fink Optimization methods 18

19 Graphical method: Infeasible problem Example Show that the following problem has no feasible solution. Maximize x 1 + x 2 x 1 + x 2 2 x 1, x 2 0 Solution x 1 0 x 2 0 (0, 0) x 1 + x 2 2 Jirka Fink Optimization methods 19

20 Related problems Integer linear programming Integer linear programming problem is an optimization problem to find x Z n which maximizes c T x and satisfies Ax b where A R m n and b R m. Mix integer linear programming Some variables are integer and others are real. Binary linear programming Every variable is either 0 or 1. Complexity A linear programming problem is efficiently solvable, both in theory and in practice. The classical algorithm for linear programming is the Simplex method which is fast in practice but it is not known whether it always run in polynomial time. Polynomial time algorithms the ellipsoid and the interior point methods. No strongly polynomial-time algorithms for linear programming is known. Integer linear programming is NP-hard. Jirka Fink Optimization methods 20

21 Example of integer linear programming: Vertex cover Vertex cover problem Given an undirected graph (V, E), find the smallest set of vertices U V covering every edge of E; that is, U e for every e E. Formulation using integer linear programming Variables: cover x v {0, 1} for every vertex v V Covering: x u + x v 1 for every edge uv E Objective function: Minimize v V x v Matrix notation Variables: cover x {0, 1} V (i.e. 0 x 1 and x Z V ) Covering: A T x 1 where A is the incidence matrix Objective function: Minimize 1 T x Jirka Fink Optimization methods 21

22 Relation between optimal integer and relaxed solution Non-empty polyhedron may not contain an integer solution Integer feasible solution may not be obtained by rounding of a relaxed solution Integral optimum c Relaxed optimum Jirka Fink Optimization methods 22

23 Example: Ice cream production planning Problem description An ice cream manufacturer needs to plan production of ice cream for next year The estimated demand of ice cream for month i {1,..., n} is d i (in tons) Storage facilities for 1 ton of ice cream cost a per month Changing the production by 1 ton from month i 1 to month i cost b Produced ice cream cannot be stored longer than one month The total cost has to be minimized Jirka Fink Optimization methods 23

24 Example: Ice cream production planning Solution Variable x i determines the amount of produced ice cream in month i {0,..., n} Variable s i determines the amount of stored ice cream from month i 1 month i The stored quantity is computed by s i = s i 1 + x i d i for every i {1,..., n} Durability is ensured by s i d i for all i {1,..., n} Non-negativity of the production and the storage x, s 0 Objective function min b n i=1 x i x i 1 + a n i=1 s i is non-linear Let y i 0 and z i 0 be the increment and the decrement of production, reps., and x i x i 1 = y i z i Linear programming problem formulation Minimize b n i=1 (y i + z i ) + a n i=1 s i subject to s i 1 s i + x i = d i for i {1,..., n} s i d i for i {1,..., n} x, s, y, z 0 We can bound the initial and final amount of ice cream s 0 a s n and also bound the production x 0 Jirka Fink Optimization methods 24

25 Finding shortest paths from a vertex s in an oriented graph Linear programming problem Maximize u V x u subject to x v x u c uv for every edge uv x s = 0 Proof (the optimal solution x u gives the distance from s to u u V ) 1 Let y u be the length of the shortest path from s to u 2 It holds that y x Let P be edges on the shortest path from s to z y z = uv P c uv uv x v x u = x z y s = x z 3 It holds that y = x For the sake of contradiction assume that y x So y x and u V y u > u V x u But y is a feasible solution and x is an optimal solution Jirka Fink Optimization methods 25

26 Outline 1 Linear programming 2 Linear, affine and convex sets 3 Simplex method 4 Duality of linear programming 5 Integer linear programming 6 Vertex Cover 7 Matching Jirka Fink Optimization methods 26

27 Linear space Definition: Linear (vector) space A set (V, +, ) is called a linear (vector) space over a field T if + : V V V i.e. V is closed under addition + : T V V i.e. V is closed under multiplication by T (V, +) is an Abelian group For every x V it holds that 1 x = x where 1 T For every a, b T and every x V it holds that (ab) x = a (b x) For every a, b T and every x V it holds that (a + b) x = a x + b x For every a T and every x, y V it holds that a (x + y) = a x + a y Observation If V is a linear space and L V, then L is a linear space if and only if 0 L, x + y L for every x, y L and αx L for every x L and α T. Jirka Fink Optimization methods 27

28 Linear and affine spaces in R n Observation A non-empty set V R n is a linear space if and only if αx + βy V for all α, β R, x, y V. Definition If V R n is a linear space and a R n is a vector, then V + a is called an affine space where V + a = {x + a; x V }. Basic observations If L R n is an affine space, then L + x is an affine space for every x R n. If L R n is an affine space, then L x is a linear space for every x L. 1 If L R n is an affine space, then L x = L y for every x, y L. 2 An affine space L R n is linear if and only if L contains the origin 0. 3 System of linear equations The set of all solutions of Ax = 0 is a linear space and every linear space is the set of all solutions of Ax = 0 for some A. 4 The set of all solutions of Ax = b is an affine space and every affine space is the set of all solutions of Ax = b for some A and b, assuming Ax = b is consistent. 5 Jirka Fink Optimization methods 28

29 1 By definition, L = V + a for some linear space V and some vector a R n. Observe that L x = V + (a x) and we prove that V + (a x) = V which implies that L x is a linear space. There exists y V such that x = y + a. Hence, a x = a y a = y V. Since V is closed under addition, it follows that V + (a x) V. Similarly, V (a x) V which implies that V V + (a x). Hence, V = V + (a x) and the statement follows. 2 We proved that L = V + a for some linear space V R n and some vector a R n and L x = V + (a x) = V for every x L. So, L x = V = L y. 3 Every linear space must contain the origin by definition. For the opposite implication, we set x = 0 and apply the previous statement. 4 If V is a linear space, then we can obtain rows of A from the basis of the orthogonal space of V. 5 If L is an affine space, then L = V + a for some vector space V and some vector a and there exists a matrix A such that V = {x; Ax = 0}. Hence, V + a = {x + a; Ax = 0} = {y; Ay Aa = 0} = {y; Ay = b} where we substitute x + a = y and set b = Aa. If L = {x; Ax = b} is non-empty, then let y be an arbitrary vertex of L. Furthermore, L y = {x y; Ax = b} = {z; Ay + Az = b} = {z; Az = 0} is a linear space since Ay = b. Jirka Fink Optimization methods 28

30 Convex set Observation A set S R n is an affine space if and only if S contains whole line given every two points of S. Definition A set S R n is convex if S contains whole segment between every two points of S. Example a u b v Jirka Fink Optimization methods 29

31 Linear, affine and convex hulls Observation The intersection of linear spaces is also a linear space. 1 The non-empty intersection of affine spaces is an affine space. 2 The intersection of convex sets is also a convex set. 3 Definition Let S R n be an non-empty set. The linear hull span(s) of S is the intersection of all linear sets containing S. The affine hull aff(s) of S is the intersection of all affine sets containing S. The convex hull conv(s) of S is the intersection of all convex sets containing S. Observation Let S R n be an non-empty set. A set S is linear if and only if S = span(s). 4 A set S is affine if and only if S = aff(s). 5 A set S is convex if and only if S = conv(s). 6 span(s) = aff(s {0}) Jirka Fink Optimization methods 30

32 1 Use definition and logic. 2 Let L i be affine space for i in an index set I and L = i I L i and a L. We proved that L a = i I (L i a) is a linear space which implies that L is an affine space. 3 Use definition and logic. 4 Similar as the convex version. 5 Similar as the convex version. 6 We proved that conv(s) is convex, so if S = conv(s), then S is convex. In order to prove that S = conv(s) if S is convex, we observe that conv(s) S since conv(s) = M S,M convex and S is included in this intersection. Similarly, conv(s) S since every M in the intersection contains S. Jirka Fink Optimization methods 30

33 Linear, affine and convex combinations Definition Let v 1,..., v k be vectors of R n where k is a positive integer. The sum k i=1 α iv i is called a linear combination if α 1,..., α k R. The sum k i=1 α iv i is called an affine combination if α 1,..., α k R k i=1 α i = 1. The sum k i=1 α iv i is called a convex combination if α 1,..., α k 0 and k i=1 α i = 1. Lemma Let S R n be a non-empty set. The set of all linear combinations of S is a linear space. 1 The set of all affine combinations of S is an affine space. 2 The set of all convex combinations of S is a convex set. 3 Lemma A linear space S contains all linear combinations of S. 4 An affine space S contains all affine combinations of S. 5 A convex set S contains all convex combinations of S. 6 Jirka Fink Optimization methods 31

34 1 We have to verify that the set of all linear combinations has closure under addition and multiplication by scalars. In order to verify the closure under multiplication, let k i=1 α iv i be a linear combination of S and c R be a scalar. Then, c k i=1 α iv i = k i=1 (cα i)v i is a linear combination of of S. Similarly, the set of all linear combinations has closure under addition and it contains the origin. 2 Similar as the convex version: Show that S contains whole line defined by arbitrary pair of points of S. 3 Let k i=1 α iu i and l j=1 β jv j be two convex combinations of S. In order to prove that the set of all convex combinations of S contains the line segment between k i=1 α iu i and l j=1 β jv j, let us consider γ 1, γ 2 0 such that γ 1 + γ 2 = 1. Then, γ k 1 i=1 α iu i + γ l 2 j=1 β jv j = k i=1 (γ 1α i )u i + l j=1 (γ 2β j )v j is a convex combination of S since (γ 1 α i ), (γ 2 β j ) 0 and k i=1 (γ 1α i ) + l j=1 (γ 2β j ) = 1. 4 Similar as the convex version. 5 Let k i=1 α iv i be an affine combination of S. Since S v k is a linear space, the linear combination k i=1 α i(v i v k ) of S v k belongs into S v k. Hence, v k + k i=1 α i(v i v k ) = k i=1 α iv i belongs to S. 6 We prove by induction on k that S contains every convex combination k i=1 α iv i of S. The statement holds for k 2 by the definition of a convex set. Let k i=1 α iv i be a convex combination of k vectors of S and we assume that α k < 1, otherwise α 1 = = α k 1 = 0 so k i=1 α iv i = v k S. Hence, k i=1 α iv i = (1 α k ) k i=1 α i 1 α k v i + α k v k = (1 α k )y + α k v k where we observe Jirka Fink Optimization methods 31

35 that y := k i=1 α i 1 α k v i is a convex combination of k 1 vectors of S which by induction belongs to S. Furthermore, (1 α k )y + α k v k is a convex combination of S which by induction also belongs to S. Jirka Fink Optimization methods 31

36 Linear, affine and convex combinations Theorem Let S R n be a non-empty set. The linear hull of a set S is the set of all linear combinations of S. 1 The affine hull of a set S is the set of all affine combinations of S. 2 The convex hull of a set S is the set of all convex combinations of S. 3 Jirka Fink Optimization methods 32

37 1 Similar as the convex version. 2 Similar as the convex version. 3 Let T be the set of all convex combinations of S. First, we prove that conv(s) T. The definition states that conv(s) = M S,M convex M and we proved that T is a convex set containing S, so T is included in this intersection which implies that conv(s) is a subset of T. In order to prove conv(s) T, we again consider the intersection conv(s) = M S,M convex M. We proved that a convex set M contains all convex combinations of M which implies that if M S then M also contains all convex combinations of S. So, in this intersection every M contains T which implies that conv(s) T. Jirka Fink Optimization methods 32

38 Independence and base Definition A set of vectors S R n is linearly independent if no vector of S is a linear combination of other vectors of S. A set of vectors S R n is affinely independent if no vector of S is an affine combination of other vectors of S. Observation (Homework) Vectors v 1,..., v k R n are linearly dependent if and only if there exists a non-trivial combination α 1,..., α k R such that k i=1 α iv i = 0. Vectors v 1,..., v k R n are affinely dependent if and only if there exists a non-trivial combination α 1,..., α k R such that k i=1 α iv i = 0 a k i=1 α i = 0. Observation Vectors v 0,..., v k R n are affinely independent if and only if vectors v 1 v 0,..., v k v 0 are linearly independent. 1 Vectors v 1,..., v k R n are linearly independent if and only if vectors 0, v 1,..., v k are affinely independent. 2 Jirka Fink Optimization methods 33

39 1 If vectors v 1 v 0,..., v k v 0 are linearly dependent, then there exists a non-trivial combination α 1,..., α k R such that k i=1 α i(v i v 0 ) = 0. In this case, 0 = k i=1 α i(v i v 0 ) = k i=1 α iv i v k 0 i=1 α i = k i=0 α iv i is a non-trivial affine combination with k i=0 α i = 0 where α 0 = k i=1 α i. f v 0,..., v k R n are affinely dependent, then there exists a non-trivial combination α 0,..., α k R such that k i=0 α iv i = 0 a k i=0 α i = 0. In this case, 0 = k i=0 α iv i = α 0 v 0 + k i=1 α iv i = k i=1 α i(v i v 0 ) is a non-trivial linear combination of vectors v 1 v 0,..., v k v 0. 2 Use the previous observation with v 0 = 0. Jirka Fink Optimization methods 33

40 Basis Definition Let B R n and S R n. B is a base of a linear space S if B are linearly independent and span(b) = S. B is an base of an affine space S if B are affinely independent and aff(b) = S. Observation All linear bases of a linear space have the same cardinality. All affine bases of an affine space have the same cardinality. 1 Observation Let S be a linear space and B S \ {0}. Then, B is a linear base of S if and only if B {0} is an affine base of S. Definition The dimension of a linear space is the cardinality of its linear base. The dimension of an affine space is the cardinality of its affine base minus one. The dimension dim(s) of a set S R n is the dimension of affine hull of S. Jirka Fink Optimization methods 34

41 1 For the sake of contradiction, let a 1,..., a k and b 1,..., b l be two basis of an affine space L = V + x where V a linear space and l > k. Then, a 1 x,..., a k x and b 1 x,..., b l x are two linearly independent sets of vectors of V. Hence, there exists i such that a 1 x,..., a k x, b i x are linearly independent, so a 1,..., a k, b i are affinely independent. Therefore, b i cannot be obtained by an affine combination of a 1,..., a k and b i / aff(a 1,..., a k ) which contradicts the assumption that a 1,..., a k is a basis of L. Jirka Fink Optimization methods 34

42 Carathéodory Theorem (Carathéodory) Let S R n. Every point of conv(s) is a convex combinations of affinely independent points of S. 1 Corollary Let S R n be a set of dimension d. Then, every point of conv(s) is a convex combinations of at most d + 1 points of S. Jirka Fink Optimization methods 35

43 1 Let x conv(s). Let x = k i=1 α ix i be a convex combination of points of S with the smallest k. If x 1,..., x k are affinely dependent, then there exists a combination 0 = β i x i such that β i = 0 and β 0. Since this combination is non-trivial, there exists j such that β j > 0 and α j β j x = i j γ i x i i j γ i = 1 γ i 0 for all i j which contradicts the minimality of k. is minimal. Let γ i = α i α j β i. Observe that β j Jirka Fink Optimization methods 35

44 Outline 1 Linear programming 2 Linear, affine and convex sets 3 Simplex method 4 Duality of linear programming 5 Integer linear programming 6 Vertex Cover 7 Matching Jirka Fink Optimization methods 36

45 Notation Notation used in the Simplex method Equation form: Maximize c T x such that Ax = b and x 0 where A R m n and b R m. We assume that rows of A are linearly independent. For a subset B {1,..., n}, let A B be the matrix consisting of columns of A whose indices belong to B. Similarly for vectors, x B denotes the coordinates of x whose indices belong to B. The set N = {1,..., n} \ B denotes the remaining columns. Example Consider B = {2, 4}. Then, N = {1, 3, 5} and ( ) ( ) A = A B = 4 9 A N = ( 1 5 ) Note that Ax = A B x B + A N x N. x T = (3, 4, 6, 2, 7) x T B = (4, 2) x T N = (3, 6, 7) Jirka Fink Optimization methods 37

46 Geometrical interpretation of basis solutions 1 For a system Ax = b with n variables and n linearly independent conditions, there exists the inverse matrix A 1 and the only feasible solution of Ax = b is x = A 1 b. 2 Consider a system Ax b with n = rank(a) variables and m n conditions and select n linearly independent rows A x b. Then, the system A x = b has a solution x = A 1 b. Moreover, if Ax b, then x is a vertex of the polyhedron Ax b. 1 3 Consider the equation form Ax = b and x 0 and let N be n m rows of x 0. If rows of the system Ax = 0 and x N = 0 are linearly independent, then b = Ax = A B x B + A N x N = A B x B, so x = (x B, x N) = (A 1 B b, 0) where B = {1,..., n} \ N. Moreover, if x B 0, then x is a vertex of Ax = b and x 0. 4 Consider the equation form again. If we choose m linearly independent columns B of A, then conditions Ax = b and x N = 0 are linearly independent. Jirka Fink Optimization methods 38

47 1 The solution x = A 1 b will be called a basis solution. Vertices of a polyhedron will be formally defined later, so we use a geometrical intuition now. Jirka Fink Optimization methods 38

48 Basic feasible solutions Definitions Consider the equation form Ax = b and x 0 with n variables and rank(a) = m rows. A set B {1,..., n} of linearly independent columns of A is called a basis. 1 The basic solution x corresponding to a basis B is x N = 0 and x B = A 1 B b. A basic solution satisfying x 0 is called a basic feasible solution. x B are called basis variables and x N are called non-basis variables. 2 Observation A feasible solution x of a Ax = b and x 0 is basis if and only if columns of A K are linearly independent where K = {j {1,..., n} ; x j > 0}. 3 4 Observation Linear program in the equation form has at most ( ) n m basis solutions. 5 Jirka Fink Optimization methods 39

49 1 Observe that B {1,..., n} is a basis if and only if A B is a regular matrix. 2 Remember that non-basis variables are always equal to zero. 3 If x is a basic feasible solution and B is the corresponding basis, then x N = 0 and so K B which implies that columns of A K are also linearly independent. If columns of A K are linearly independent, then we can extend K into B by adding columns of A so that columns of A B are linearly independent which implies that B is a basis of x. 4 Note that basis variables can also be zero. In this case, the basis B corresponding to a basis solution x may not be unique since there may be many ways to extend K into a basis B. This is called degeneracy. 5 There are ( n m) subsets B {1,..., n} and for some of these subsets AB may not be regular. Jirka Fink Optimization methods 39

50 Optimal basis feasible solutions Theorem If the linear program max c T x subject to Ax = b and x 0 has a feasible solution and the objective function is bounded from above of the set of all feasible solutions, then there exists an optimal solution. Moreover, if an optimal solution exists then there is a basis feasible solution which is optimal. 1 Lemma If the objective function of a linear program in the equation form is bounded above, then for every feasible solution x there exists a basis feasible solution x with the same or larger value of the objective function, i.e. c T x c T x. 2 Jirka Fink Optimization methods 40

51 1 If the problem is bounded, one may try to find the optimal solution by finding all basis feasible solutions. However, this is not an efficient algorithm since the number of basis grows exponentially. 2 Let x{ be a feasible solution with } c T x c T x and the smallest possible size of the set K = j {1,..., n} ; x j > 0. Let N = {1,..., n} \ K. If columns of A K are linearly independent, then x is a basis solution. There exists a non-zero vector v K such that A K v K = 0. Let v N = 0. WLOG: c T v 0 since we can replace v by v. Consider the line x(t) = x + tv for t R. For every t R: Ax(t) = b and (x(t)) N = 0. For every t 0: c T x(t) c T x. If c T v > 0 and v 0, then points x(t) are feasible for every t 0 and the objective function c T x(t) = c T x + tc T v converges to infinity which contradicts assuptions. x j If v j < 0 for some j K, then consider j K with v j < 0 and minimal. Let v j t = x j v j. Since x( t) 0 and (x( t)) j = 0, the solution x( t) is feasible with smaller number of positive components than x which is a contradiction. The remaining case is c T v = 0 and v j 0. Since v K is a non-trivial combination, there exists j K with v j > 0. Replace v by v and apply the previous case. Jirka Fink Optimization methods 40

52 Convex polyhedrons Definition A hyperplane is a set { x R n ; a T x = b } where a R n \ {0} and b R. A half-space is a set { x R n ; a T x b } where a R n \ {0} and b R. A polyhedron is an intersection of finitely many half-spaces. A polytope is a bounded polyhedron. Observation For every a R n and b R, the set of all x R n satisfying a T x b is convex. Corollary Every polyhedron Ax b is convex. Examples n-dimensional hypercube: {x R n ; 0 x 1} n-dimensional crosspolytope: { x R n ; n i=1 x i 1 } 1 n-dimensional simplex: { x R n+1 ; x 0, 1x = 1 } 2 Jirka Fink Optimization methods 41

53 1 Formally, n i=1 x i 1 is not a linear inequality. However, it can be replaced by 2 n linear inequalities dx 1 for all d { 1, 1} n. 2 n-dimensional simplex is a convex hull of n + 1 affinely independent points. Jirka Fink Optimization methods 41

54 Faces of a polyhedron Definition Let P be a polyhedron. A half-space α T x β is called a supporting hyperplane of P if the inequality α T x β holds for every x P and the hyperplane α T x = β has a non-empty intersection with P. Definition If α T x β is a supporting hyperplane of a polyhedron P, then P { x; α T x = β } is called a face of P. By convention, the empty set and P are also called faces, and the other faces are proper faces. 1 Definition Let P be a d-dimensional polyhedron. A 0-dimensional face of P is called a vertex of P. A 1-dimensional face is of P called an edge of P. A (d 1)-dimensional face of P is called an facet of P. Jirka Fink Optimization methods 42

55 1 Observe, that every face of a polyhedron is also a polyhedron. Jirka Fink Optimization methods 42

56 Vertices Observations The set of all optimal solutions of a linear program max c T x over a polyhedron P is a face of P. 1 Every proper face of P is a set of all optimal solutions of a linear program max cx over a polyhedron P for some c R n. 2 3 Vertices are unique solutions of linear programs max c T x over P for some c. Theorem Let P be the set of all solutions of a linear program in the equation form and v P. Then the following statements are equivalent. 1 v is a vertex of a polyhedron P. 2 v is a basis feasible solution of the linear program. 4 Theorem If the linear program max c T x subject to Ax = b and x 0 has a feasible solution and the objective function is bounded from above of the set of all feasible solutions, then there exists an optimal solution. Moreover, if an optimal solution exists then there is a basis feasible solution which is optimal. Jirka Fink Optimization methods 43

57 1 Let F be the set of all optimal solutions. If F = or F = P, then F is a face of P by definition. Otherwise, d = max { c T x; x P } exists. Since c T x = d is a supporting hyperplane of P and F = P { x; c T x = d }, it follows that F is a face of P. 2 A proper face F of P is defined as the intersection of P and a supporting hyperplane c T x = d, so F is the set of all optimal solutions of the linear program max c T x over P. 3 Note that P is also the set of all optimal solutions of a linear program for c = 0. On the other hand, if P is non-empty and bounded, then the empty set cannot be express as a set of all optimal solutions for any c. 4 Follows from the following theorem. Let B be the basis defining v and let c B = 0 and c N = 1. Then c T v = c T B v B + c T N v N = 0 and for every feasible x it holds holds that x 0, so c T x 0. Hence, v is a optimal solution of the linear program with the objective function max c T x. Furthermore, v is the only optimal solution since every optimal solution x must satisfy x N = 0. In this case, x B = A 1 b is unique. B Jirka Fink Optimization methods 43

58 Example: Initial simplex tableau Canonical form Equation form Simplex tableau Maximize x 1 + x 2 subject to x 1 + x 2 1 x 1 3 x 2 2 x 1, x 2 0 Maximize x 1 + x 2 subject to x 1 + x 2 + x 3 = 1 x 1 + x 4 = 3 x 2 + x 5 = 2 x 1, x 2, x 3, x 4, x 5 0 x 3 = 1 + x 1 x 2 x 4 = 3 x 1 x 5 = 2 x 2 z = x 1 + x 2 Jirka Fink Optimization methods 44

59 Example: Initial simplex tableau Simplex tableau Initial basic feasible solution Pivot B = {3, 4, 5}, N = {1, 2} x = (0, 0, 1, 3, 2) Two edges from the vertex (0, 0, 1, 3, 2): x 3 = 1 + x 1 x 2 x 4 = 3 x 1 x 5 = 2 x 2 z = x 1 + x 2 1 (t, 0, 1 + t, 3 t, 2) when x 1 is increased by t 2 (0, r, 1 r, 3, 2 r) when x 2 is increased by r These edges give feasible solutions for: 1 t 3 since x 3 = 1 + t 0 and x 4 = 3 t 0 and x 5 = r 1 since x 3 = 1 r 0 and x 4 = 3 0 and x 5 = 2 r 0 In both cases, the objective function is increasing. We choose x 2 as a pivot. Jirka Fink Optimization methods 45

60 Example: Pivot step Simplex tableau Basis Original basis B = {3, 4, 5} x 2 enters the basis (by our choice). x 3 = 1 + x 1 x 2 x 4 = 3 x 1 x 5 = 2 x 2 z = x 1 + x 2 (0, r, 1 r, 3, 2 r) is feasible for r 1 since x 3 = 1 r 0. Therefore, x 3 leaves the basis. New base B = {2, 4, 5} New simplex tableau x 2 = 1 + x 1 x 3 x 4 = 3 x 1 x 5 = 1 x 1 + x 3 z = 1 + 2x 1 x 3 Jirka Fink Optimization methods 46

61 Example: Next step Simplex tableau Next pivot x 2 = 1 + x 1 x 3 x 4 = 3 x 1 x 5 = 1 x 1 + x 3 z = 1 + 2x 1 x 3 Basis B = {2, 4, 5} with a basis feasible solution (0, 1, 0, 3, 1). This vertex has two incident edges but only one increases the objective function. The edge with increasing objective function is (t, 1 + t, 0, 3 t, 1 t). Feasible solutions for x 2 = 1 + t 0 and x 4 = 3 t 0 and x 5 = 1 t 0. Therefore, x 1 enters the basis and x 5 leaves the basis. New simplex tableau x 1 = 1 + x 3 x 5 x 2 = 2 x 5 x 4 = 2 x 3 + x 5 z = 3 + x 3 2x 5 Jirka Fink Optimization methods 47

62 Example: Last step Simplex tableau Next pivot x 1 = 1 + x 3 x 5 x 2 = 2 x 5 x 4 = 2 x 3 + x 5 z = 3 + x 3 2x 5 Basis B = {1, 2, 4} with a basis feasible solution (1, 2, 0, 2, 0). This vertex has two incident edges but only one increases the objective function. The edge with increasing objective function is (1 + t, 2, t, 2 t, 0). Feasible solutions for x 1 = 1 + t 0 and x 2 = 2 0 and x 4 = 2 t 0. Therefore, x 3 enters the basis and x 4 leaves the basis. New simplex tableau x 1 = 3 x 4 x 2 = 2 x 5 x 3 = 2 x 4 + x 5 z = 5 x 4 x 5 Jirka Fink Optimization methods 48

63 Example: Optimal solution Simplex tableau x 1 = 3 x 4 x 2 = 2 x 5 x 3 = 2 x 4 + x 5 z = 5 x 4 x 5 No other pivot Basis B = {1, 2, 3} with a basis feasible solution (3, 2, 2, 0, 0). This vertex has two incident edges but no one increases the objective function. We have an optimal solution. Why this is an optimal solution? Consider an arbitrary feasible solution ỹy. The value of objective function is z = 5 ỹy 4 ỹy 5. Since ỹy 4, ỹy 5 0, the objective value is z = 5 ỹy 4 ỹy 5 5 = z. Jirka Fink Optimization methods 49

64 Example: Unboundedness Canonical form Maximize x 1 subject to x 1 x 2 1 x 1 + x 2 2 x 1, x 2 0 Equation form Maximize x 1 subject to x 1 x 2 + x 3 = 1 x 1 + x 2 + x 4 = 2 x 1, x 2, x 3, x 4 0 Initial simplex tableau x 3 = 1 x 1 + x 2 x 4 = 2 + x 1 x 2 z = x 1 Jirka Fink Optimization methods 50

65 Example: Unboundedness Simplex tableau x 3 = 1 x 1 + x 2 x 4 = 2 + x 1 x 2 z = x 1 First pivot Basis B = {3, 4} with a basis feasible solution (0, 0, 1, 2). This vertex has two incident edges but only one increases the objective function. The edge with increasing objective function is (t, 0, 1 t, 2 + t). Feasible solutions for x 3 = 1 t 0 and x 4 = 2 + t 0. Therefore, x 1 enters the basis and x 3 leaves the basis. Simplex tableau x 1 = 1 + x 2 x 3 x 4 = 3 x 3 z = 1 + x 2 x 3 Jirka Fink Optimization methods 51

66 Example: Unboundedness Simplex tableau x 1 = 1 + x 2 x 3 x 4 = 3 x 3 z = 1 + x 2 x 3 Unboundedness Basis B = {1, 4} with a basis feasible solution (1, 0, 0, 3). This vertex has two incident edges but only one increases the objective function. The edge with increasing objective function is (1 + t, t, 0, 3). Every point (1 + t, t, 0, 3) for t 0 is feasible. The value of the objective function is 1 + t. Therefore, this problem is unbounded. Jirka Fink Optimization methods 52

67 Example: Degeneracy Canonical form Maximize x 2 subject to x 1 + x 2 0 x 1 2 x 1, x 2 0 Convert to the equation form by adding slack variables x 3 and x 4. Same solution with different basis x 2 = x 1 x 3 x 4 = 2 x 1 z = x 1 x 3 Basis feasible solution (0, 0, 0, 2) with the basis {2, 4}. Initial simplex tableau x 3 = x 1 x 2 x 4 = 2 x 1 z = x 2 Basis feasible solution (0, 0, 0, 2) with the basis {3, 4}. Optimal simplex tableau x 1 = 2 x 4 x 2 = 2 x 3 x 4 z = 2 x 3 x 4 Optimal solution (2, 2, 0, 0) with the basis {1, 2}. Jirka Fink Optimization methods 53

68 Simplex tableau in general Definition A simplex tableau determined by a feasible basis B is a system of m + 1 linear equations in variables x 1,..., x n, and z that has the same set of solutions as the system Ax = b, z = c T x, and in matrix notation looks as follows: x B = p + Qx N z = z 0 + r T x N where x B is the vector of the basis variables, x N is the vector on non-basis variables, p R m, r R n m, Q is an m (n m) matrix, and z 0 R. Example x 3 = 5 + x 1 x 2 x 4 = 2 x 1 z = 3 + x 1 + 2x 2 ( ) 1 1 Q = 1 0 ( ) ( ) x x B = 3 x, x N = 1, p = x 4 z 0 = 3, r T = (1, 2) x 2 ( ) 5 2 Jirka Fink Optimization methods 54

69 Simplex tableau in general Definition A simplex tableau determined by a feasible basis B is a system of m + 1 linear equations in variables x 1,..., x n, and z that has the same set of solutions as the system Ax = b, z = c T x, and in matrix notation looks as follows: x B = p + Qx N z = z 0 + r T x N where x B is the vector of the basis variables, x N is the vector on non-basis variables, p R m, r R n m, Q is an m (n m) matrix, and z 0 R. Observation For each basis B there exists exactly one simplex tableau, and it is given by Q = A 1 B A N p = A 1 B b z 0 = c T BA 1 B b r = c N (c T BA 1 B A N) T 1 Jirka Fink Optimization methods 55

70 1 Since A B x B + A N x N = b and A B is a regular matrix, it follows that x B = A 1 B b A 1 B A Nx N where A 1 b = p and A 1 B B A N = Q. The objective function is c T B x B + c T N x N = c T B A 1 B b (ct B A 1 B A N + c T N )x N, where c T B A 1 B b = z 0 and c T B A 1 B A N + c T N = r T. Jirka Fink Optimization methods 55

71 Properties of a simplex tableau Simplex tableau in general Observation Basis B is feasible if and only if p 0. Observation x B = p + Qx N z = z 0 + r T x N If r 0, then the solution corresponding to a basis B is optimal. Idea of the pivot step Choose v N. Which is the last feasible point of the half-line x(t) for t 0 where x v(t) = t x N\{v} (t) = 0 x B (t) = p + Q,vt? Observation If there exists a non-basis variable x v such that r v > 0 and Q,v 0, then the problem is unbounded. Jirka Fink Optimization methods 56

72 Pivot step Simplex tableau in general x B = p + Qx N z = z 0 + r T x N Find a pivot If r 0, then we have an optimal solution. Otherwise, choose an arbitrary entering variable x v such that r v > 0. If Q,v 0, then the problem is also unbounded. Otherwise, find a leaving variable x u which limits the increment of the entering variable most strictly, i.e. Q u,v < 0 and p u Q u,v is minimal. Jirka Fink Optimization methods 57

73 Pivot rules Pivot rules Largest coefficient Choose an improving variable with the largest coefficient. Largest increase Choose an improving variable that leads to the largest absolute improvement in z, e.i. c T (x new x old ) is maximal. Steepest edge Choose an improving variable whose entering into the basis moves the current basic feasible solution in a direction closest to the direction of the vector c, i.e. c T (x new x old ) x new x old Bland s rule Choose an improving variable with the smallest index, and if there are several possibilities of the leaving variable, also take the one with the smallest index. Random edge Select the entering variable uniformly at random among all improving variables. Jirka Fink Optimization methods 58

74 Pivot step Simplex tableau in general Gaussian elimination x B = p + Qx N z = z 0 + r T x N New basis variables are (B \ {u}) {v} and new non-basis variables are (N \ {v}) {u} Row x u = p u + Q u,vx v + j N\{v} Q u,jx j is replaced by row x v = p u Q u,v + 1 Q u,v x u + Q u,j j N\{v} Q u,v x j. Rows x i = p i + Q i,v x v + j N\{v} Q i,jx j for i B \ {u} are replaced by rows x i = (p i + Q i,v Q u,v p u ) + Q i,v Q u,v x u + j N\{v} (Q i,j + Q u,j Q i,v Q u,v )x j. Objective function z = z 0 + r vx v + j N\{v} r j x j is replaced by objective function z = (z 0 + Observation p u Q u,v ) + r v Q u,v x u + j N\{v} (r j + r v Q i,v Q u,v )x j. Pivot step does not change the set of all feasible solutions. Jirka Fink Optimization methods 59

75 Bland s rule Simplex tableau in general x B = p + Qx N z = z 0 + r T x N Observation Let B is a basis with the corresponding solution x and let B a new basis with the corresponding solution x after a single pivot step. Then, x = x or c T x < c T x. 1 Observation If the simplex method loops endlessly, then basis occuring in the loop correspond to the same vertex. 2 Theorem The simplex method with Bland s pivot rule is always finite. 3 Jirka Fink Optimization methods 60

76 1 Consider the half-line x(t) providing the pivot step and let t = max {t 0; x(t) 0}. Clearly, c T x = x( t). If t = 0, then x = x(0) = x. If t > 0, then c T x = c T x( t) = z 0 + r v t > z 0 = c T x since r t > 0. 2 Consider that the simplex method iteraters over basis B ( 1),..., B (k), B (k+1) = B (1) with the corresponding solutions x (1),..., x (k), x (k+1) = x (1). By the previous observation holds that c T x (1) c T x (2) c T x (k) c T x (k+1) = c T x (1). Hence, c T x (1) = = c T x (k+1) and the previous observation implies that x (1) = = x (k+1). 3 For the sake of contradiction, we assume that the simplex method with Bland s pivot rule loops endlessly. Consider all basis in the loop. Let F be the set of all entering variables and let x v F be the variable with largest index. Let B be a basis in the loop just before x v enters. Note that variables of B \ F and N \ B are always basis and non-basis variables during the loop, respectively. Consider the following auxiliary problem. Maximize c T x subject to Ax = b x F \{v} 0 x v 0 x N\F = 0 x B\F R B\F ( ) We prove that ( ) has an optimal solution and it is also unbounded which is a contradiction. Jirka Fink Optimization methods 60

77 r v > 0 since x v is the entering variable r i 0 for every i (F N) \ {v} since x v is the improving variable with the smallest index (Bland s rule) For every solution x satisfying ( ) holds that c T x = z 0 + r T x N = z 0 + r T v x v + r T (F N)\{v} x (F N)\{v} + r T N\F x N\F z 0. Hence, the solution corresponding to the basis B is an optimal solution to ( ). Now, we prove that ( ) is unbounded. Let B be a basis in the loop just before x v leaves and let Q, p and r be the parameter of the simplex tableau corresponding to B. Let x u be the entering variable. Hence, r u > 0. Q v,u < 0 since v is the leaving variable. From Bland s rule it follows that Q i,u 0 for every i (F B ) \ {v} p F B = 0 since degenerated basis variables are zero Consider the half-line x(t) for t 0 where x u(t) = t and x N \{v}(t) = 0 and x B (t) = p + Q,v t. x (F N )\{u}(t) = 0 since non-basis variables remains zero x i (t) = p i + Q i,u t 0 for every i (F B ) \ {v} Hence, x F \{v} (t) 0 x v (t) = p v + Q v,u t 0 x N \F (t) = 0 since non-basis variables remains zero Hence, x(t) satisfies ( ) for every t 0 r u > 0 since x u is the entering variable c T x(t) = z 0 + r ut for t Hence, ( ) is unbounded. Jirka Fink Optimization methods 60

78 Initial feasible basis Linear programming problem in the equation form Maximize c T x subject to Ax = b and x 0. Assume that b 0 1 Auxiliary problem We add auxiliary variables y R m to obtain the auxiliary problem maximize y 1 y m subject to Ax + Iy = b a x, y 0. Observation Initial feasible basis for the auxiliary problem is B = {y 1,..., y m } with the initial tableau Observation The following statements are equivalent y = b Ax z = 1 T b + (1 T A)x 1 The original problem max { c T x; Ax = b, x 0 } has a feasible solution. 2 Optimal value of the objective function of the auxiliary problem is 0. 3 Auxiliary problem has a feasible solution satisfying y = 0. Jirka Fink Optimization methods 61

79 1 We multiply every equation with negative right hand side by 1. Jirka Fink Optimization methods 61

80 Outline 1 Linear programming 2 Linear, affine and convex sets 3 Simplex method 4 Duality of linear programming 5 Integer linear programming 6 Vertex Cover 7 Matching Jirka Fink Optimization methods 62

81 Duality of linear programming: Example Find an upper bound for the following problem Maximize 2x 1 + 3x 2 subject to 4x 1 + 8x x 1 + x 2 3 3x 1 + 2x 2 4 x 1, x 2 0 Simple estimates 2x 1 + 3x 2 4x 1 + 8x x 1 + 3x 2 1 (4x x 2 ) 6 2 2x 1 + 3x 2 = 1 (4x x 2 + 2x 1 + x 2 ) 5 3 What is the best combination of conditions? Every non-negative linear combination of inequalities which gives an inequality d 1 x 1 + d 2 x 2 h with d 1 2 and d 2 3 provides the upper bound 2x 1 + 3x 2 d 1 x 1 + d 2 x 2 h. Jirka Fink Optimization methods 63

82 1 The first condition 2 A half of the first condition 3 A third of the sum of the first and the second conditions Jirka Fink Optimization methods 63

83 Duality of linear programming: Example Consider a non-negative combination y of inequalities Observations Maximize 2x 1 + 3x 2 subject to 4x 1 + 8x 2 12 / y 1 2x 1 + x 2 3 / y 2 3x 1 + 2x 2 4 / y 3 x 1, x 2 0 Every feasible solution x and non-negative combination y satisfies (4y 1 + 2y 2 + 3y 3 )x 1 + (8y 1 + y 2 + 2y 3 )x 2 12y 1 + 3y 2 + 4y 3. If 4y 1 + 2y 2 + 3y 3 2 and 8y 1 + y 2 + 2y 3 3, then 12y 1 + 2y 2 + 4y 3 is an upper for the objective function. Dual program 1 Minimize 12y 1 + 2y 2 + 4y 3 subject to 4y 1 + 2y 2 + 3y 3 2 8y 1 + y 2 + 2y 3 3 y 1, y 2, y 3 0 Jirka Fink Optimization methods 64

84 1 The primal optimal solution is x T = ( 1 2, 5 4 ) and the dual solution is y T = ( 5 16, 0, 1 4 ), both with the same objective value Jirka Fink Optimization methods 64

85 Duality of linear programming: General Primal linear program Maximize c T x subject to Ax b and x 0 Dual linear program Minimize b T y subject to A T y c and y 0 Weak duality theorem For every primal feasible solution x and dual feasible solution y hold c T x b T y. Corollary If one program is unbounded, then the other one is infeasible. Duality theorem Exactly one of the following possibilities occurs 1 Neither primal nor dual has a feasible solution 2 Primal is unbounded and dual is infeasible 3 Primal is infeasible and dual is unbounded 4 There are feasible solutions x and y such that c T x = b T y Jirka Fink Optimization methods 65

86 Dualization Every linear programming problem has its dual, e.g. Maximize c T x subject to Ax b and x 0 Primal program Maximize c T x subject to Ax b and x 0 Equivalent formulation Minimize b T y subject to A T y c and y 0 Dual program Minimize b T y subject to A T y c and y 0 Simplified formulation A dual of a dual problem is the (original) primal problem Minimize b T y subject to A T y c and y 0 Dual program -Maximize b T y subject to A T y c and y 0 Equivalent formulation -Minimize c T x subject to Ax b and x 0 Dual of the dual program -Minimize c T x subject to Ax b and x 0 Simplified formulation Maximize c T x subject to Ax b and x 0 The original primal program Jirka Fink Optimization methods 66

87 Dualization: General rules Primal linear program Dual linear program Variables x 1,..., x n y 1,..., y m Matrix A A T Right-hand side b c Objective function max c T x min b T y Constraints i-the constraint has y i 0 i-the constraint has y i 0 i-the constraint has = y i R x j 0 j-th constraint has x j 0 j-th constraint has x j R j-th constraint has = Jirka Fink Optimization methods 67

88 Linear programming: Feasibility versus optimality Feasibility versus optimality Finding a feasible solution of a linear program is computationally as difficult as finding an optimal solution. Using duality The optimal solutions of linear programs Primal: Maximize c T x subject to Ax b and x 0 Dual: Minimize b T y subject to A T y c and y 0 are exactly feasible solutions satisfying Ax b A T y c c T x b T y x, y 0 Jirka Fink Optimization methods 68

89 Complementary slackness Theorem Feasible solutions x and y of linear programs Primal: Maximize c T x subject to Ax b and x 0 Dual: Minimize b T y subject to A T y c and y 0 are optimal if and only if x i = 0 or A T i, y = c i for every i = 1,..., n and y j = 0 or A j, x = b j for every j = 1,..., m. Proof c T x = n c i x i i=1 n (y T A,i )x i = y T Ax = i=1 m y j (A j, x) j=1 m y j b j = b T y j=1 Jirka Fink Optimization methods 69

90 Proof of duality using simplex method with Bland s rule Notation Primal: Maximize c T x subject to Ax b and x 0 Primal with slack variables: Maximize c T x subject to Ā x = b and x 0 1 Dual: Minimize b T y subject to A T y c and y 0 Simplex tableau x B = p + Q x N z = z 0 + r T x N Simplex tableau is unique for every basis B Q = Ā 1 B ĀN p = Ā 1 B b z 0 = c T BĀ 1b Lemma B r = c N ( c T BĀ 1 B ĀN) T If B is a basis with an optimal solution x of the primal problem, then y = ( an optimal solution of the dual problem and c T x = b T y. 2 c TĀ 1 B )T is Jirka Fink Optimization methods 70