. P. 4.3 Basic feasible solutions and vertices of polyhedra. x 1. x 2

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1 4. Basic feasible solutios ad vertices of polyhedra Due to the fudametal theorem of Liear Programmig, to solve ay LP it suffices to cosider the vertices (fiitely may) of the polyhedro P of the feasible solutios. x.. P x 2 Sice the geometrical defiitio of vertex caot be exploited algorithmically, we eed a algebraic characterizatio. E. Amaldi -- Foudatios of Operatios Research -- Politecico di Milao

2 Which are the vertices of P { x R : Ax b, x } with oly iequalities? Example: mi - x - x 2 s.t. x + x 2 6 (I) 2x + x 2 8 (II) x, x 2 E. Amaldi -- Foudatios of Operatios Research -- Politecico di Milao 2 x 2 P c (II) Vertex correspods to the itersectio of the hyperplaes associated to iequalities. Example: 2 Vertex (6) is the itersectio of hyperplaes of (I) ad (II), i.e., solutio of equatios x + x 2 6 ad 2x + x ) ) ) 6) 5) 4) (I) x

3 What about the vertices of polyhedra expressed i stadard form? P {x R : Ax b, x } We wat to solve LPs i stadard form Easier to describe if we start from a polyhedro P {x R : Ax b, x }, trasform it ito stadard form P {x R : Ax + s b, x, s } ad reame: A: [ A I ], x : [ x s ]. E. Amaldi -- Foudatios of Operatios Research -- Politecico di Milao

4 Example: (P) x + x 2 6 (I) 2x + x 2 8 (II) x, x 2 P 2) (P ) x + x 2 + s 6 2x + x 2 + s 2 8 x, x 2, s, s 2 ) x 2 ) 6) 5) 4) (II) (I) x Takig the itersectio of the lies associated to (I) ad (II) i P, amouts i P to let s s 2. Observatio: Every costrait i P correspods to a variable i P, whe the variable is set to the costrait is satisfied with. Example: s x + x 2 6, x x Vertex of P is the itersectio of iequalities i P settig the correspodig variables i P to. E. Amaldi -- Foudatios of Operatios Research -- Politecico di Milao 4

5 Example (cotiued): Compute all the itersectios x 2 x + x 2 + s 6 (I) 2x + x 2 + s 2 8 (II) x, x 2, s, s 2 ) x, x 2 s 6, s 2 8 2) x, s x 2 6, s 2 2 ) x, s 2 x 2 8, s -2 4) x 2, s x 6, s 2-4 5) x 2, s 2 x 4, s 2 6) s, s 2 x 2, x 2 4 P 2) ) ) 6) 5) 4) (II) (I) x The itersectios where some x j or s i are < yield ifeasible solutios. E. Amaldi -- Foudatios of Operatios Research -- Politecico di Milao 5

6 Which are the vertices of a polyhedro i stadard form? Visualize the example P{x R : x +x 2 +x, x } Property: For ay polyhedro P { x R : Ax b, x } the facets (edges i R 2 ) are obtaied by settig oe variable to, the vertices are obtaied by settig -m variables to. I the example: -2 variables set to for vertices. E. Amaldi -- Foudatios of Operatios Research -- Politecico di Milao 6

7 Algebraic characterizatio of the vertices Cosider ay P {x R : Ax b, x } i stadard form. Assumptio: A R m with m of rak m there are o redudat costraits. Example: 2 x + x 2 + x 2 (I) Sice (I) (II) + (III), x + x 2 (II) the (I) ca be dropped. x + x (III) x, x 2, x If m, uique solutio of Ax b. ( x A - b ) If m <, solutio of Ax b: the system has -m degrees of freedom (-m variables ca be fixed arbitrarily). If we fix them to, we get a vertex. E. Amaldi -- Foudatios of Operatios R.search -- Politecico di Milao 7

8 P {x R : Ax b, x } variables, m costraits, A i R m Defiitio: A basis of such a matrix A is a subset of m colums of A that are liearly idepedet ad form a m m o sigular matrix B. A B N m -m First permute the colums of A, the partitio A ito [B N] E. Amaldi -- Foudatios of Operatios Research -- Politecico di Milao 8

9 Let x x B x N m compoets -m compoets Ay system Ax b ca be writte as B x B + N x N b B is osigular For ay set of values for x N, we have Defiitios: x B B - b B - N x N A basic solutio is a solutio obtaied by settig x N ad, cosequetly, lettig x B B - b. A basic solutio with x B is a basic feasible solutio. The variables i x B are the basic variables ad those i x N are the o basic variables. E. Amaldi -- Foudatios of Operatios Research -- Politecico di Milao 9

10 Note: x B satisfies Ax b by costructio. Theorem: x R is a basic feasible solutio if ad oly if x is a vertex of P { x R : Ax b, x }. E. Amaldi -- Foudatios of Operatios Research -- Politecico di Milao

11 mi z 2x + x 2 + 5x s.t. x + x 2 + x + x 4 4 x + x 5 2 x + x 6 x 2 + x + x 7 6 x i i,, 7 E. Amaldi -- Foudatios of Operatios Research -- Politecico di Milao Example: b A Choosig colums 4, 5, 6, 7, we have: B B - B x B B - b b basic feasible solutio

12 2 mi z 2x + x 2 + 5x s.t. x + x 2 + x + x 4 4 x + x 5 2 x + x 6 x 2 + x + x 7 6 x i i,, 7 E. Amaldi -- Foudatios of Operatios Research -- Politecico di Milao Example (cotiued): Choosig colums 2, 5, 6, 7, we have: B b A B x B feasible 2 4-6

13 E. Amaldi -- Foudatios of Operatios Research -- Politecico di Milao # basic feasible solutios m m m m ))! ( ( )! (! Oe for each choice of -m variables out of (obasic variables) Number of basic feasible solutios