Pivoting methods for linear programming

Examensarbete Stockholm, 2012 HENRIK SJÖSTRÖM

Pivoting methods for

linear programming

A_K A K A_K

A K

A_ij i j A

xi i x

ei i x ≥ 0

x

xB x B x

x_B x_N

x =!xB

x_N

"

.

x_i≥k x i i ≥ k

ker(A) A v

Av = 0

Bin(p, q) Bin(p, q) = _q!(p−q)!^p!

0 ≤ q ≤ p p q

c^Tx Ax = b x ≥ 0.

A n×m c^Tx c b

m n

b^Ty

A^Ty + s = c s ≥ 00.

C ∀ x, y ∈ C λ ∈ [0, 1]

(1 − λ)x + λy ∈ C

B A |B| = n

A_B A_B i A i ∈ B

B N N = B^C

x

Ax = b B xB ABxB = b

A_B x B x_N

xN = 0

x ≥ 0

C x

C x ∈ C y, z ∈ C λ ∈ (0, 1)

x = (1 − λ)y + λz y, z '= x

x C

x

x xN xB

x =!xB

x_N

"

x = (1 − λ)y + λz z, y '= x λ ∈ (0, 1)

y z y ≥ 0 z ≥ 0

x x =!xB

x_N

"

= (1 − λ)y + λz = (1 − λ)!yB

y_N

"

+ λ!zB

z_N

"

. xN = 0

0 = x_N = (1 − λ)yN+ λz_N. λ > 0 yN ≥ 0 zN ≥ 0

yN = zN = 0.

y, z ∈ C ABx = ABy = ABz = b A⁻¹_B

ABxB = AByB = ABzB= b =⇒ x^B = yB = zB = A⁻¹_B b.

x = y = z y, z '= x x

y z

x x

x x₀ = 0

xD > 0 A A₀ AD

A_D = A_B x AD

A_Dx_D = b.

A_D y

z x = ¹₂y + ¹₂z x = y = z x

A_D p

A_Dp = 0.

"

AD(x ± "p) = A^DxB± "A^Dp = ADxB= b.

x_D > 0 "

x_D+ "p > 0,

x_D− "p > 0.

y = x_D + "p z = x_D − "p x

x AD

AD

x #y^T s^T$T

c^Tx − b^Ty = s^Tx ≥ 0 c^Tx − b^Ty

Ax = b A^Ty + s = c c^Tx − b^Ty

c^Tx − b^Ty = (A^Ty + s)^Tx − (Ax)^Ty = (y^TA + s^T)x − x^TA^Ty.

x^TA^Ty x^TA^Ty = y^TAx (y^TA + s^T)x − x^TA^Ty = y^TAx + s^Tx − y^TAx = s^Tx

c^Tx − b^Ty = s^Tx s^Tx ≥ 0 s ≥ 0 x ≥ 0

Ax = b, A^Ty + s = c, x ≥ 0, s ≥ 0, xjsj = 0 ∀j.

B

x s x_N = 0 s_B= 0

B AB

Ax =#AB AN

$!xB

xN

"

= b.

x_N = 0

A_Bx_B = b =⇒ xB= A⁻¹_B b.

B x

xisi= 0 sB= 0 sB= 0 A^T_By = cB

y = A^−T_B c_B A^T_Ny + s_N = c_N =⇒

A^T_NA^−T_B cB+ sN = cN =⇒ s^N = cN − A^Ny = cN − A^NA^−T_B cB. sN

z w

z_iw_i = 0, w = M z + q,

z ≥ 0, w ≥ 0,

q M

B B^# xB = x_B!

B s s ≥ 0

t t

s

pB = −A⁻¹B ANet

v x

pB xv

v

B

xB = A⁻¹_B b sN = cN − A^TNA^−T_B cB

sN ≥ 0

t = argmin

i

(si|sⁱ< 0).

p_B= −A⁻¹B A_Ne_t p_B ≥ 0 B v = argmin_i%

xi

−pi|i ∈ B pi < 0&

. t ∈ B v ∈ N

x ≥ 0 ABxB = b

x_j =' xB,j j ∈ B 0 j '∈ B

(

sN

s_N ≥ 0 x_B ≥ 0

si ≥ 0 t = argmin_i(si)

1 Ax = b

x

!AB AN

0 I

"

p = et

Ax = c pt= 1

p_j =







pB,j j ∈ B 0 j '∈ B + {t}

1 j = t





 p_B = −A⁻¹B A_Ne_t

c^Tp < 0 p s_t< 0

!A^T_B 0 A^T_N I

" ! y sN

"

=!cB

cN

"

c^Tp =!!A^T_B 0 A^T_N I

" ! y s_N

""T

p

=#y^T s^T_N$!AB A_N

0 I

" !AB A_N

0 I

"−1

et

=#y^T s^T_N$ et= st.

s_t< 0 c^Tp < 0.

xB+kpB k

k kpB = −x^B =⇒ k = −^−xpB^B

pi > 0

k k = min_i%

xi

−pi|i ∈ B p_i < 0&

i pi < 0 i ∈ B

k

(t, v) B

pB

k

st< 0

#−2 −1 0 0$ x,

!1 1 1 0 1 0 0 1

"

x =!2 1

"

, x ≥ 0.

B = 3, 4 AB = I

AN =!1 1 1 0

"

xB sN

xB = I⁻¹!2 1

"

=!2 1

"

,

sN = cN− A^TNA^−T_B cB =!−2

−1

"

. x1

p_B=

! −1 −1

−1 0

" !1 0

"

=!−1

−1

"

. min(−xi/p_i) = min(2, 1) x₄

(1, 3)

η c

t

B

xB = A⁻¹_B b η =!−A⁻¹B AN

I

"

IN

η_i η ^c_|η^Tηi

i| ≥ 0 ∀i η

t = argmin

i

! c^Tηi

|ηi|

"

.

s = argmin

i

! −xi

ηi,t |ηi,t < 0 i ∈ B

"

. i ηi,t < 0

(t, s) s t

B

ηi i

ηi

|ηi| c

pi= c^Tηi

|ηⁱ|.

pi≥ 0 ∀i

t p_t

−c

pB = ηt ηt η

n ≥ 1 n

0 ≤ x1≤ 1,

βx_j−1 ≤ x^j ≤ δ^j− βxj−1 = 2, ..., n.

β = 0 δi= 1

x₁≤ 1,

βx_j−1− x^j ≤ 0, j > 1, βx_j−1+ xj ≤ δ^j, j > 1,

x ≥ 0

θ > 2 β ≥ 2 δj = (θβ)^j−1 ci = −βⁱ. θ > 2 β ≥ 1

xi

0 0.2 0.4 0.6 0.8 1 0

1 2 3 4 5 6

y1

y2

β = 2 θ = 3

βx_i−1< δ_i− βxi−1 θ

δi = (θβ)ⁱ⁻¹

βx_i−1 ≤ xⁱ≤ δⁱ− βxi−1 =⇒ 2βxi−1 < δi, x_i−1< δi

2β = (θβ)ⁱ⁻¹ 2β =! θ

2

"

(θβ)ⁱ⁻². x₁≤ 1 i = 2

x₁<! θ 2

"

(θβ)⁰ =! θ 2

"i

. βx_i−1 < δ_i− βxi−1 x₁ ≤ 1 <#_θ

2

$i

∀i > 1 θ > 2

a n a_i = 1 x_i

ai = 0 xi

a B

B x

xi =' βxi−1 a_i = 0, δi− βxi−1 ai = 1.

(

xi

B B^#

a

a^# k x

x^#

x^#_i− xi =







0 if i < k,

(δ_k− 2βxk−1)(1 − 2ak) i = k, β^i−k(δk− 2βxk−1)/i

j=k(1 − 2a^j) i > k.

x_i x_j<i i < k

x^#_i = xi x^#_i− xⁱ = 0 ai = a^#_i

x^#_k− xk =

' −(δ^k− 2βxk−1) ak= 1, (δ_k− 2βxk−1) a_k= 0.

ak = 1 1 − 2a^k = −1 ak = 0 1 − 2a^k = 1 x^#_k− xk= (δ_k− 2βxk−1)(1 − 2ak).

i > k a_i= a^#_i x^#_i− xi =

' (δ_i− βx^#i−1) − (δi− βxi−1) = −β(x^#i−1− xi−1) a_i = 1, βx^#_i−1− βxⁱ⁻¹= β(x^#_i−1− xⁱ⁻¹) ai = 0.

x^#_k− xk

x^#_i− xⁱ = β(x^#_i−1− xi−1)(1 − 2aⁱ).

x^#_i− xⁱ= β^i−k(x^#_k− x^k)

i

0

j=k+1

(1 − 2a^j).

x^#_k− xk x^#_i − xi

x_k−1

x^#_i− xⁱ= β^i−k(δk− 2βxk−1)(1 − 2a^k)

i

0

j=k+1

(1 − 2a^j)

= β^i−k(δ_k− 2βxk−1)

i

0

j=k

(1 − 2aj).

x^#− x

|x^#− x|.

x^!_i−xi

|x^!−x| = 0 i < k x^#_i− xⁱ

|x^#− x| = β^i−k(δ_k− 2βxk−1)/i

j=k(1 − 2aj) 1

2n

j=kβ^2(j−k)(δk− 2βxk−1)²/j

w=k(1 − 2a^w)², if i ≥ k.

/i

j=k(1 − 2a^j)² = 1 aj

(δk− 2βxk−1)²β^−2k x^#_i− xi

|x^#− x| = β^i−k(δk− 2βxk−1)/i

j=k(1 − 2a^j) β^−k(δ_k− 2βxk−1)1

2n j=kβ^2j

= βⁱ/_i

j=k(1 − 2aj) 12n

j=kβ^2j .

c·(x^!−x)

|x^!−x| ci = −βⁱ

ci(x^#_i− xⁱ)

|x^#i− xⁱ| = −βⁱβⁱ/_i

j=k(1 − 2aj) 12n

j=kβ^2j

= −β²ⁱ/_i

j=k(1 − 2aj) 12n

j=kβ^2j .

c · (x^#− x)

|x^#− x| = − 2_n

i=kβ²ⁱ/_i

j=k(1 − 2aj) 12n

j=kβ^2j

.

2n

i=kβ²ⁱ/i

j=k(1 − 2aj) /n

j=k(1 − 2aj)

wp =/p

j=k(1 − 2a^j) wi = ±1 wn= 1

n

3

i=k

β²ⁱ⁻¹

i

0

j=k

(1 − 2aj) > 0

β⁻¹

n

3

i=k

β²ⁱw_i= β²ⁿ+

n−1

3

i=k

β²ⁱ⁻¹w_i.

wi<n= −1 k = 1

β²ⁿ+

n−1

3

i=k

β²ⁱwi ≥ β²ⁿ−

n−1

3

i=1

β²ⁱ 2n−1

i=1 y²ⁱ= −^y_(y^2−y2−1)²ⁿ

β²ⁿ+

n−1

3

i=k

β²ⁱwi ≥ β²ⁿ+β²− β²ⁿ (β²− 1)

β²ⁿ+ ^β_(β^2−β₂₋₁₎²ⁿ > 0 β

β²ⁿ(β²− 1) + β²− β²ⁿ > 0, β²− 1 + β²

β²ⁿ − 1 > 0, β²+ β²

β²ⁿ > 2, β²+ 1

β²⁽ⁿ⁻¹⁾ > 2 =⇒ β² > 2.

β²> 2 2n

i=kβ²ⁱ⁻¹wi> 0 wn= 1 wn= −1

n

3

i=k

β²ⁱ

i

0

j=k

(1 − 2aj) < 0.

wi<n= 1 k = 1

−β²ⁿ+

n−1

3

i=k

β²ⁱwi ≤ −β²ⁿ+

n−1

3

i=1

β²ⁱ

β²ⁿ − 2_n−1

i=k β²ⁱ > 0

−β²ⁿ⁻¹+

n−1

3

i=1

β²ⁱ⁻¹< 0,

−

β ≥ 2 m /n

j=m(1 −

2aj) = 1 k > m /n

j=k(1 − 2a^j) = 1

c·(x^m−x)

|x^m−x| < ^c·(x_|xk^k−x|^−x) x^m x m

β

− 2n

i=mβ²ⁱ/i

j=m(1 − 2a^j) 12n

j=mβ^2j < − 2n

i=kβ²ⁱ/i

j=k(1 − 2a^j) 12n

j=kβ^2j

.

Φ =

k−1

3

i=m

β²ⁱ∀i

i

0

j=m

(1 − 2a^j) = 1,

Ψ =

k−1

3

i=m

β²ⁱ∀i

i

0

j=m

(1 − 2a^j) = −1,

Ω =

n

3

i=k

β²ⁱ∀i

i

0

j=m

(1 − 2aj) = 1,

Λ =

n

3

i=k

β²ⁱ∀i

i

0

j=m

(1 − 2aj) = −1.

Φ, Ω > 0 Ψ, Λ ≥ 0

−√Φ − Ψ +Ω − Λ

Φ + Ψ + Ω + Λ < −√Ω − Λ Ω +Λ.

(Φ − Ψ +Ω − Λ)²(Ω + Λ) − (Ω − Λ)²(Φ + Ψ + Ω + Λ) > 0 (Φ − Ψ + Ω − Λ)²(Ω + Λ) − (Ω − Λ)²(Φ + Ψ + Ω + Λ)

= (Φ − Ψ)²(Ω + Λ) + 2(Φ − Ψ)(Ω²− Λ²) − (Ω − Λ)²(Φ + Ψ)

= (Φ − Ψ)²(Ω + Λ) + (Ω − Λ)[Ω(Φ − 3Ψ) + Λ(3Φ − Ψ)].

(Φ − Ψ)²> 0 (Ω + Λ) > 0 (Ω − Λ)[Ω(Φ − 3Ψ) + Λ(3Φ − Ψ)]

(Φ−Ψ)²(Ω+Λ)+(Ω−Λ)[Ω(Φ−3Ψ)+Λ(3Φ−Ψ)] > (Ω−Λ)[Ω(Φ−3Ψ)+Λ(3Φ−Ψ)].

Ω > Λ (Ω −

Λ)[Ω(Φ − 3Ψ) + Λ(3Φ − Ψ)] Ω(Φ − 3Ψ) + Λ(3Φ − Ψ) (Φ − 3Ψ) ≥ 0

(3Φ − Ψ) > 0

n

0

j=k

(1 − 2a^j) = 1

1 =

n

0

j=m

(1 − 2a^j) =

k−1

0

j=m

(1 − 2a^j)

n

0

j=k

(1 − 2a^j) =⇒

k−1

0

j=m

(1 − 2a^j) = 1.

/_k−1

j=m(1−2a^j) = 1 /j≤k−2

j=m (1−2a^j) = −1

Ψ Φ

(Φ − 3Ψ) ≥ β^2k− 32_k−2

i=mβ²ⁱ

= β^2k−2(β²− 32_k−2

i=mβ^{−2(k−i−1)})

≥ β^2k−2(β²−_β^3β2−1² ).

β²−β^3β2−1² ≥ 0 β ≥ 2

2ⁿ− 1

m

n

0

j=m

(1 − 2a^j) = 1

am

n = 2 ai = 0 ∀i a⁰ = [0, 0]

/n

j=i(1 − 2a^j) = 1∀i a¹ = [1, 0]

a m

0 [0, 0] 1

1 [1, 0] 2

2 [1, 1] 1

3 [0, 1] −

n − 1 2ⁿ⁻¹− 1 n

1 − 2aⁿ= 1|^an=0

n − 1

aⁿ⁻¹ = [0...0, 1_n−1, 0]

i /n

j=i(1 − 2a^j) = 1 n aⁿ= [0...0, 1_n−1, 1n]

1 − 2aⁿ= −1|^an=1 n−1

0

j=i

(1 − 2a^j)|^an=1= −

n−1

0

j=i

(1 − 2a^j)|^an=0. n − 1

n − 1

2 ∗ (2²⁻¹− 1) an= 1

2 ∗ (2²⁻¹− 1) + 1 = 2ⁿ− 1.

r

c·(x^!−x)+cslack·(r^!−r))

|x^!−x+r^!−r| c_slack = 0

c·(x^!−x)

|x^!−x+r^!−r|

d c^Tx,

Ax +^I_dr_d= b, x, r ≥ 0.

B

x r d r = dr_d

c · (x^#− x)

|x^#− x + r^#− r| = c · (x^#− x)

|x^#− x + d(r^#changed− rchanged)|.

d 0

d→0lim

c · (x^#− x)

|x^#− x + d(r^#d− rd)| = c · (x^#− x)

|x^#− x| .

d

c^Tx

dAx + r_d= db, x, r ≥ 0,

c b

ci =

' βⁱ i ≤ n 0 i > n

( ,

bi=' δi i ≤ n 0 i > n

( .

n A n −1

A =

































1 0 0 ... 0

β 1 ... 0

0 β 1 ... 0

0 ... 0 β 1

β −1 ... 0

0 β −1 ... 0

0 0 ... β −1































 .

β ≥ 2 θ > 2 d > 0 d

2ⁿ− 1 x_i>1 r_{n≤i≤2n−1}

0 0.2 0.4 0.6 0.8 1 0

1 2 3 4 5 6

y1

y2

1

2 3 4

β = 2 θ = 3

c^Tηiki

s

x x^# = x + k_tη_t B

xB = A⁻¹_B b η =!−A⁻¹B AN

I

"

IN

ηi η ^c_|η^Tηi

i| ≥ 0 ∀i

η_i k_i

k_i = min

j

! xj∈B

ηij |ηij < 0

"

.

t = argmin_i(c^Tηiki) s s = argmin

j

! xj∈B

ηtj |η^tj< 0

"

.

(t, s) s t B

ηi i

x_j∈B− ηijk_i = 0 j xj∈B− η^ijki≥ 0 j

k_i = min

j

! xj

η_ij|ηij < 0, j ∈ B

"

.

c^Tη_ik_i η_i k_i

ηi

p c^Tx

ψ(p) p ψ(p) ∈ [1, 0]

y

0 0.2 0.4 0.6 0.8 1

0 0.2 0.4 0.6 0.8 1

x1

y1

p1

y1

y₁

y1 y1

v w

pi ∈ p [(1 − ψ(pⁱ))v + ψ(pi)w, pi]∀i [v₁, p₁] v₁

[v₁, p₁] p [w₁, pmax]

[w₂, p_max] w [w₃, p_max]

p w v pmax

p₁ p v

v w v

w

p

v w

p

0 2 4 6 8 10 12 14

−0.5 0 0.5 1 1.5 2 2.5 3 3.5

x₂

y2

v1 w1

v w

y₂

B

i x_i < 0 s_i < 0 t

t ∈ B

η_N v η_v < 0

t ∈ N v

pB v

pv< 0 t v B

N

B B

x_B = A⁻¹_B b s_N = c_N − A^TNA^−T_B c_B xB≥ 0 sN ≥ 0

t = min

i ({i|xi < 0ors_i < 0}).

t ∈ B η_N = A^T_NA^−T_B e_t η_N ≥ 0 B v = mini(i|ηⁱ < 0)

t ∈ N pB = −A⁻¹B ANet pB ≥ 0 B

v = mini(i|pⁱ < 0 i ∈ B)

(t, v) B

t

x s x s

t v

t ∈ B xB

s_N

t x_i < 0 s_i< 0

t ∈ B

t ∈ N

(t, v) t ∈ B t ∈ N v ∈ B t ∈ N t ∈ B v ∈ N

t ∈ N

t ∈ B

x^# x^## s^∗ s^∗∗ (s^∗− s^∗∗)^T(x^#− x^##) = 0

K = ker(A) AK = 0

x Ax = b Ax^#− Ax^##= b − b = 0 A(x^#− x^##) = 0 x^#− x^##∈ K

s A^Ty + s = c

A^Ty^∗+ s^∗− A^Ty^∗∗+ s^∗∗= c − c = 0.

AK = 0 K^TA^T = 0 K^T

K^TA^T(y^∗− y^∗∗) + K^T(s^∗− s^∗∗) = K^T(s^∗− s^∗∗) = 0.

(K^T(s^∗− s^∗∗))^T = (s^∗ − s^∗∗)^TK = 0 (s^∗− s^∗∗)^T K

x^# − x^## ∈ K (s^∗ − s^∗∗)^T K (s^∗− s^∗∗)^T(x^#− x^##) = 0

t

B t B^# t

k t t t t

(s¹− s²)^T(x³− x⁴) = 0 s¹ s²

x³ x⁴ x³ x B^#

x⁴

x³ x³− x⁴ = lp^# l l

p^# p

p^#_k= 1

p^#_j =







−A⁻¹B^!ANej j ∈ B^#

1 j = k

0 j ∈ N^#\k





 i sixi = 0

s s^# B B^# s s^#

t

sj =' ≥ 0 j < t

< 0 j = t (

,

s^#_j =







f ree j > t

0 j = t

< 0 j = k





 .

0 = (s − s^#)^Tp^# = s_kp^#_k+ s_tp_t^# − s^#kp^#_k− s^#tp^#_t+ 3

i∈N ∩B^!\t,k

s_ip^#_i− s^#ip^#_i.

p^#_i∈N!\k= 0 s^#_i∈B! = 0 s^#_i)=kp^#_i)=k= 0 p^#_k= 1

0 = (s − s^#)^Tp^#= s_k+ s_tp^#_t− s^#k+ 3

i∈N ∩B^!\t,k

s_ip^#_i.

sk≥ 0 s^t< 0 p^#_t< 0 s^#_k< 0 s_{i∈N ∩B}!\t,k ≥ 0 p^#_{i∈N ∩B}!\t,k ≥ 0

0 = (s − s^#)^Tp^#= s_k+ s_tp^#_t− s^#k+ 3

i∈N ∩B^!\t,k

s_ip^#_i> 0.

t

p^# η

s^# s x^# x

0 = (s − s^#)^T(x − x^#) s^Tx = s^#Tx^# = 0

0 = (s − s^#)^T(x − x^#) = −s^Tx^#− s^#Tx.

sj =' ≥ 0 j < t

< 0 j = t (

,

xj =

' ≥ 0 j < t

0 j = t

( ,

s^#_j =' ≥ 0 j < t

0 j = t

( ,

x^#_j =

' ≥ 0 j < t

< 0 j = t (

.

0 = −s^Tx^#− s^#Tx = −s^i<tx^#_i<t− s^tx^#_t− s^#i<txi<t− s^#txt. s^#_t= 0 xt= 0

0 = −s^Tx^#− s^#Tx = −s^i<tx^#_i<t− s^tx^#_t− s^#i<txi<t. si<tx^#_i<t ≥ 0 s^tx^#_t > 0 s^#_i<txi<t ≥ 0

0 = −s^Tx^#− s^#Tx = −s^i<tx^#_i<t− s^tx^#_t− s^#i<txi<t< 0.

t

n ≥ 1 n

0 ≤ y1 ≤ 1,

εy_j−1≤ yj ≤ 1 − εyj−1 2 ≤ j ≤ n.

ε = 0 ε#0,¹₂$

0 0.2 0.4 0.6 0.8 1 0

0.2 0.4 0.6 0.8 1

y1

y2

The klee−minty polytope in 2 dimensions with ε=0.2

ε = 0.2

−y¹ ≤ 0, y₁ ≤ 1, εy_j−1− y^j ≤ 0, εyj−1+ yj ≤ 1.

yi

yi

A^Ty =























−1 0 0 ... 0

1 0 0 ... 0

ε −1 0 ... 0

ε 1 0 ... 0

0 ... 0 ε −1

0 ... 0 ε 1











































 y₁ y2

y₃ y₄ y_n−1

y_n























≤





















 0 1 0 1 0 1





















 .

yn

yn,

−y1≤ 0, y₁≤ 1,

εy_j−1− yj ≤ 0 2 ≤ j ≤ n, εy_j−1+ yj ≤ 1 2 ≤ j ≤ n.

2n i=1x^u_i,

Ax =











−1 1 ε ε 0 0 0 ... 0

0 0 −1 1 ε ε 0 ... 0

0 0 0 0

0 0 0 0 0 0 0 −1 1































 x^l₁ x^u₁ x^l₂ x^u₂ x^l_n x^u_n























=





















 0 0 0 0 0 1





















 ,

x ≥ 0.

x x^u x^l x^u

y x^l

y1 = 1 x^u₁

x^l x^u i x^l_i x^u_i

x^u_i yi

εy_j−1+ yj = 1 x^l_i εy_j−1− y^j = 0

y_i−1 ∈ [0, 1] y_i y_i = 1 − εyi−1

εy_j−1 ≤ y^j

εy_j−1 ≤ 1 − εyi−1 2εy_j−1 ≤ 1

ε ∈ (0,¹₂)

y_i−1 ∈ [0, 1] yi yi = εy_i−1

y_i ≤ 1 − εyi−1 εy_i−1 ≤ 1 − εyi−1 =⇒ 2εyj−1 ≤ 1 ε ∈ (0,¹₂)

0 ≤ y1 ≤ 1 y_i−1 ∈ [0, 1] i yi ∈ [εyi−1, 1 − εyi−1] ⊆ [0, 1]

i x^l_i x^u_i

x^l_i x^u_i i

x^l_i x^u_i

A_B A_N A⁻¹_B

AB =

















±1 ε 0 . . . 0

0 ±1 ε 0

0 0 0

0 0 0 ±1 ε

0 0 0 0 ±1















 .

A_B,i,i = 1 x^l_i ∈ B A_B,i,i = −1 x^u_i ∈ B

A⁻¹_B diag(AB) = diag(A⁻¹_B ) AB

±1 A^−T_B

diag(A^−T_B ) = diag(A_B)

AN,i,j = AB,i,j ∀i '= j A_N,i,i = −AB,i,i diag(A_N) = − diag(AB)

A_N =

















∓1 ε 0 . . . 0

0 ∓1 ε 0

0 0 0

0 0 0 ∓1 ε

0 0 0 0 ∓1















 .

A^T_N

A^T_NA^−T_B

diag(A^T_NA^−T_B ) = diag(A_N^T) diag(A^−T_B ) = − diag(AB) diag(A_B) = −1.

A^T_NA^−T_B diag(A^T_NA^−T_B ) < 0

η_N = A^T_NA^−T_B e_i=













 01

0_i−1

−1ⁱ η_N,j>i













 .

η_j < 0 i : th

N x^l_i x^u_i

x^u_i x^l_i ∈ B x^u_i x^l_i x^u_i < 0 x^l_i < 0

xB,i < 0 |{j|x^uj ∈ B ∀j ≥ i}|

x^u_j≥i xB,i < 0

AB=















±1 ε 0 ... 0

0 ±1 ε 0 ...

0 0 0

0 0 0 ±1 ε

0 0 0 0 ±1













 ,

AB,i,i= −1 x^l_i ∈ B AB,i,i= 1 x^u_i ∈ B b = e_n ABxB= b = en.

AB xB,i= x^l_i xB,i = εxB,i+1

xB,n = x^l_n xB,n= −1

x_B,i = x^u_i x_B,i = −εxB,i+1 x_B,n = x^u_n x_B,n = 1 xB,i = −εⁿ⁻ⁱ/n

j=i|x^u_j∈B−1 xB,i < 0 x^u_j ∈ B ∀j ≥ i

2ⁿ− 1

x_B,i< 0 {j|x^uj ∈ B ∀j ≥ i}| = 0 x^l_i x^u_i

x_B = x^l

xB < 0 |{j|x^uj ∈ B ∀j ≥ i}| = 0 2ⁿ− 1

x^u_i

x^l_i i u

l u x_B,i= x^u_i l x_B,i = x^l_i (l, u, u, l, l, u) (x^l₁, x^u₂, x^u₃, x^l₄, x^l₅, x^u₆)

n = 2

(l, l)

x^u₁ 1 u^#s

(u, l), (u, u), (l, u).

n = 2 n

2ⁿ− 1

(−, ..., −, un).

l

|{j|x^uj ∈ B ∀j ≥ i}|

n + 1 l

(l, ..., l_n−1, u_n, l_n+1).

n + 1 u

|{j|x^uj ∈ B ∀j ≥ i}|

(l, ..., l_n−1, un, u_n+1).

n

2ⁿ− 1 (l, ..., l, u_n+1)

(2ⁿ− 1) + 1 + (2ⁿ− 1) = 2ⁿ⁺¹− 1

0 0.2 0.4 0.6 0.8 1 0

0.2 0.4 0.6 0.8 1

y1

y2

The klee−minty polytope in 2 dimensions with ε=0.2

1

2 3 4

ε = 0.2

ziwi = 0 ∀i

w = M z + q,

w ≥ 0,

z ≥ 0,

w_iz_i = 0.

M n × n w z q

n

c^Tx, Ax ≥ b, x ≥ 0, b^Ty,

−A^Ty ≥ −c, y ≥ 0,

a^Tx = c a^Tx ≥ c −a^Tx ≥ −c

c^Tx

Ax − r = b x, r ≥ 0, b^Ty

A^Ty + s = c y, s ≥ 0.

r = Ax − b s = −A^Ty + c

x ≥ 0

y ≥ 0

s ≥ 0

r ≥ 0

riyi = 0 s_ix_i = 0

!s r

"

= ! 0 −A^T

A 0

" !x y

"

+! c

−b

"

,

!x y

"

≥ 0,

!s r

"

≥ 0,

#x y$!s r

"

= 0.

A n × m n '= m det! 0 −A^T

A 0

"

= 0.

j z_jw_j '= 0 z_iw_i= 0

∀i '= j

z_iw_i = 0 z₀w₀ = 0,

! w w₀

"

=!M 0

"

z +!q 0

"

+!1 1

"

z₀,

! z z₀

"

≥ 0,

! w w₀

"

≥ 0,

q M

w₀ = z₀ w₀z₀ = 0 w₀= z₀

z₀ = 0 w₀

z₀ = 0 z₀ = 0

ziwi = 0,

P =#I −M −1$ −1

−1 P



 w

z z₀



= q,

! z z₀

"

≥ 0, w ≥ 0, z₀ = 0,

q M

w

z0 = −min(q) w = z0+ q w

0

z₀ ∈ B

p z₀

w P PB PN

q ≥ 0 w = q z₀

z₀ = −min(q) w w = z₀+ q wi

t w_t = 0 z_t = 0



 w

z z0





B

= P_B⁻¹q.

et t

p p = −PB⁻¹PNet.

s s = argmin

j∈B (−xj

p_j|p^j < 0).

j ∈ B −^x_pj^j < 0 p_j < 0

(s, t) z₀ = 0

z₀ > 0 wt= 0 zt= 0

z₀

z₀

w_i

z0

i

wi zi zi

w_i

z0 = 0 z₀

(s, t)

#−1 −1 −1$ x

#−10 −10 −1$ x ≥ #−10$

x ≥ 0

z_iw_i = 0,

w =









0 0 0 10

0 0 0 10

0 0 0 1

−10 −10 −1 0







 z +









−1

−1

−1 10







 ,

z ≥ 0, w ≥ 0.

M =









0 0 0 10

0 0 0 10

0 0 0 1

−10 −10 −1 0







 ,

q =









−1

−1

−1 10







 .

P



 w

z z₀



=









1 0 0 0 0 0 0 −10 −1 0 1 0 0 0 0 0 −10 −1

0 0 1 0 0 0 0 −1 −1

0 0 0 1 10 10 1 0 −1











 w

z z₀



=









−1

−1

−1 10







 .

z₀ = −min









−1

−1

−1 10









= 1 w = q + z₀ =







 0 0 0 11







 w₁ = w₂ = w₃ = 0

w₁ (w₂, w₃, w₄, z₀)

w₁ z₀ 0

z₁ w₁ w₁ z₁

PB=









0 0 0 −1 1 0 0 −1 0 1 0 −1 0 0 1 −1







 ,

PNe₁ =







 0 0 0 10







 , .

p = −PB⁻¹PNez₁ =







 0 0

−10 0







 ,



 w

z z₀





B

=







 w₂ w₃ w₄ z₀









=







 0 0 11

1







 .

j pj < 0 j = 3 w4

(w₂, w₃, z₁, z₀) 1

w4 z4

P_B=









0 0 0 −1 1 0 0 −1 0 1 0 −1 0 0 10 −1







 ,

PNez₄ =









−10

−10

−1 0







 ,

p = −PB⁻¹PNez₄ =







 0

−9

−1

−10







 ,



 w

z z₀





B

=







 w2

w₃ z₁ z0









=







 0 0

11 10

1







 .

0

−9 w₃ z₀

M

M x = λx t x M =! 0 −A^T

A 0

"

x =!x₁ x₂

"

M x =! 0 −A^T

A 0

" !x1

x₂

"

=!−A^Tx₂ Ax₁

"

.

λx^T = x^TM =#x1 x₂$! 0 −A^T

A 0

"

=#x2A −x1A^T$ . λx = λ(x^T)^T

!−A^Tx₂ Ax₁

"

=! A^Tx₂

−Ax1

"

.

B κ > 0

Ki Ki κ

B K_i

B Ki

f (Ki, A, b, c) Ki f (Ki, A, b, c)

B f (K_i, A, b, c)

f (B, A, b, c) Ki f (Ki, A, b, c) < f (B, A, b, c) B

f (K_i, A, b, c) f (K_i, A, b, c)

B

κ > 0

K_i 0 < |B ∩ Ki| ≤ |B| − κ |Ki| = |B|

Ki f (Ki, A, b, c)

v_i

B S_t t = argmin_i(v_i)

vt vt

B Ki

K₁ A |A ∩ S| = |A| − 1

Kκ κ A |A ∩ K| = |A| − κ

κ B

B

f (K, A, b, c)

κ

p q

κ

2

t=1

Bin(p, t) Bin(q, t)

B |B| = p N

|N| = q

1...κ 1...κ

t ant_out(t) =

Bin(p, t) antin(t) = Bin(q, t)

t B Bin(p, t) Bin(q, t)

0 < t ≤ κ

κ

3

t=1

Bin(p, t) Bin(q, t).

κ (κ = 2)

p = 50 q = 50

≈ 1.5 ∗ 10⁶ κ = 3

385 ∗ 10⁶ κ = 5

4.5 ∗10¹² κ

f (K, A, b, c) B

f (Ki, A, b, c) =

' ∞ A⁻¹_K

ib '≥ 0 c^T_K

iA⁻¹_K

ib A⁻¹_K

ib ≥ 0 (

.

κ A⁻¹_K

ib = xB f (Ki, A, b, c) = ∞

c^T_K

iA⁻¹_K

ib κ = |B|

Ki

c^Tx_nonslack

#A I$!xnonslack

x_slack

"

= b

!xnonslack

xslack

"

≥ 0.

A, b 1 : 100 A p p

2p c

−100 : −1 x_slack = b

|{i|xⁱ < 0 si< 0}|

−2

i(x_i+ s_i)|xi < 0 s_i < 0

−2

i(x²_i + s²_i)|xⁱ< 0 si < 0

−min(x√! ⁱ+ si)|xⁱ < 0 si< 0

ix²_i

|x| +

√!

is²_i

|s| |xⁱ< 0 si< 0

√!

ix²_i+s²_i

√|x|²+|s|² |xⁱ < 0 si < 0

f (Ki, A, b, c)

κ ≤ 3 κ

B A b

A c^T

A b c

Examensarbete E361 i Optimeringslära och systemteori Juni 2012

www.math.kth.se/optsyst/