More on Polyhedra and Farkas Lemma Marco Chiarandini Department of - PowerPoint PPT Presentation

DM545 Linear and Integer Programming Lecture 8 More on Polyhedra and Farkas Lemma Marco Chiarandini Department of Mathematics & Computer Science University of Southern Denmark

Farkas Lemma Outline Beyond the Simplex 1. Farkas Lemma 2. Beyond the Simplex 2

Farkas Lemma Outline Beyond the Simplex 1. Farkas Lemma 2. Beyond the Simplex 3

Farkas Lemma Beyond the Simplex We now look at Farkas Lemma with two objectives: • (giving another proof of strong duality) • understanding a certificate of infeasibility 5

Farkas Lemma Farkas Lemma Beyond the Simplex Lemma (Farkas) Let A ∈ R m × n and b ∈ R m . Then, ∃ x ∈ R n : A x = b and x ≥ 0 either I . ∃ y ∈ R m : y T A ≥ 0 T and y T b < 0 or II . Easy to see that both I and II cannot occur together: y T A x = y T b ( 0 ≤ ) ( < 0 ) 6

Farkas Lemma Geometric interpretation of Farkas L. Beyond the Simplex Linear combination of a i with nonnegative terms generates a convex cone: { λ 1 a 1 + . . . + λ n a n , | λ 1 , . . . , λ n ≥ 0 } Polyhedral cone: C = { x | A x ≤ 0 } , intersection of many ax ≤ 0 Convex hull of rays p i = { λ i a i , λ i ≥ 0 } Either point b lies in convex cone C ∃ hyperplane h passing through point 0 h = { x ∈ R m : y T x = 0 } for or y ∈ R m such that all vectors a 1 , . . . , a n (and thus C ) lie on one side and b lies (strictly) on the other side (ie, y T a i ≥ 0 , ∀ i = 1 . . . n and y T b < 0). 7

Farkas Lemma Variants of Farkas Lemma Beyond the Simplex Corollary ⇒ ∀ y ∈ R m with y T A ≥ 0 T , y T b ≥ 0 (i) A x = b has sol x ≥ 0 ⇐ ⇒ ∀ y ≥ 0 with y T A ≥ 0 T , y T b ≥ 0 (ii) A x ≤ b has sol x ≥ 0 ⇐ (iii) A x ≤ 0 has sol x ∈ R n ⇐ ⇒ ∀ y ≥ 0 with y T A = 0 T , y T b ≥ 0 i) = ⇒ ii): ¯ A = [ A | I m ] ⇒ ¯ A x ≤ b has sol x ≥ 0 ⇐ A ¯ x = b has sol ¯ x ≥ 0 By (i): ∀ y ∈ R m relation with Fourier & y T b ≥ 0 , y T ¯ y T A ≥ 0 A ≥ 0 Moutzkin method y ≥ 0 8

Farkas Lemma Certificate of Infeasibility Beyond the Simplex Farkas Lemma provides a way to certificate infeasibility. Theorem Given a certificate y ∗ it is easy to check the conditions (by linear algebra): A T y ∗ ≥ 0 by ∗ < 0 Why would y ∗ be a certificate of infeasibility? Proof (by contradiction) Assume, A T y ∗ ≥ 0 and by ∗ < 0. Moreover assume ∃ x ∗ : A x ∗ = b , x ∗ ≥ 0 ,then: ( y ∗ ) T A x ∗ = ( y ∗ ) T b ( ≥ 0 ) ( < 0 ) Contradiction 12

Farkas Lemma Beyond the Simplex General form: infeasible ⇔ ∃ y ∗ max c T x A 1 x = b 1 b T 1 y 1 + b T 2 y 2 + b T 3 y 3 > 0 A 2 x ≤ b 2 A T 1 y 1 + A T 2 y 2 + A T 3 y 3 ≤ 0 A 3 x ≥ b 3 y 2 ≤ 0 x ≥ 0 y 3 ≥ 0 Example b T 1 y 1 + b T 2 y 2 > 0 y 1 + 2 y 2 > 0 max c T x A T 1 y 1 + A T 2 y 2 ≤ 0 y 1 + y 2 ≤ 0 x 1 ≤ 1 y 1 ≤ 0 y 1 ≤ 0 x 1 ≥ 2 y 2 ≥ 0 y 2 ≥ 0 y 1 = − 1 , y 2 = 1 is a valid certificate. 13

Farkas Lemma Beyond the Simplex • Observe that it is not unique! • It can be reported in place of the dual solution because same dimension. • To repair infeasibility we should change the primal at least so much as that the certificate of infeasibility is no longer valid. • Only constraints with y i � = 0 in the certificate of infeasibility cause infeasibility 14

Farkas Lemma Duality: Summary Beyond the Simplex • Derivation: 1. bounding 2. multipliers 3. recipe 4. Lagrangian • Theory: • Symmetry • Weak duality theorem • Strong duality theorem • Complementary slackness theorem • Farkas Lemma: Strong duality + Infeasibility certificate • Dual Simplex • Economic interpretation • Geometric Interpretation • Sensitivity analysis 15

Farkas Lemma Resume Beyond the Simplex Advantages of considering the dual formulation: • proving optimality (although the simplex tableau can already do that) • gives a way to check the correctness of results easily • alternative solution method (ie, primal simplex on dual) • sensitivity analysis • solving P or D we solve the other for free • certificate of infeasibility 16

Farkas Lemma Outline Beyond the Simplex 1. Farkas Lemma 2. Beyond the Simplex 17

Farkas Lemma Interior Point Algorithms Beyond the Simplex • Ellipsoid method: cannot compete in practice but weakly polynomial time (Khachyian, 1979) • Interior point algorithm(s) (Karmarkar, 1984) competitive with simplex and polynomial in some versions • affine scaling algorithm (Dikin) • logarithmic barrier algorithm (Fiacco and McCormick) ≡ Karmakar’s projective method 1. Start at an interior point of the feasible region 2. Move in a direction that improves the objective function value at the fastest possible rate while ensuring that the boundary is not reached 3. Transform the feasible region to place the current point at the center of it 20

Farkas Lemma Beyond the Simplex • because of patents reasons, now mostly known as barrier algorithms • one single iteration is computationally more intensive than the simplex (matrix calculations, sizes depend on number of variables) • particularly competitive in presence of many constraints (eg, for m = 10 , 000 may need less than 100 iterations) • bad for post-optimality analysis � crossover algorithm to convert a solution of barrier method into a basic feasible solution for the simplex 21