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▼❆■✵✶✸✵ ■♥t❡r✐♦r P♦✐♥t ▼❡t❤♦❞s ❢♦r ▲P ❈❤❛♣t❡r ✼✳✹✲✼✳✽ ❏♦❤❛♥ ❍❛❣❡♥❜❥ör❦

Pr♦❞✉❦t✐♦♥s❡❦♦♥♦♠✐ ✲ ■❊■ ▲✐♥❦ö♣✐♥❣s ✉♥✐✈❡rs✐t❡t

❉❡❝❡♠❜❡r ✶✺✱ ✷✵✶✻

❏♦❤❛♥ ❍❛❣❡♥❜❥ör❦ ❈❤❛♣t❡r ✸

❈♦♥t❡♥ts

✷

❈❡♥tr❛❧ P❛t❤ ❆❧❣♦r✐t❤♠s✳ Pr✐♠❛❧ ❉✉❛❧ ▼❡t❤♦❞s✳ ❙♦❧✈✐♥❣ t❤❡ ▲✐♥❡❛r ❙②st❡♠✳ ❚❤❡ ♠❛t❡r✐❛❧ ✐t ❛ ♠✐① ♦❢ ❬▼✉rt②✱ ✷✵✵✾❪ ❛♥❞ ❬◆♦❝❡❞❛❧ ❛♥❞ ❲r✐❣❤t✱ ✷✵✵✻❪

❏♦❤❛♥ ❍❛❣❡♥❜❥ör❦ ❈❤❛♣t❡r ✸

❆♥❛❧②t✐❝ ❈❡♥t❡r

✸

▲❡t Γ ❜❡ ❛ ❝♦♥✈❡①t ♣♦❧②t♦♣❡ Γ = {x : v = Ax − b ≥ 0} , ✭✶✮ ✇❤❡r❡ v ✐s t❤❡ ✈❡❝t♦r ♦❢ s❧❛❝❦ ✈❛r✐❛❜❧❡s ❛ss♦❝✐❛t❡❞ ✇✐t❤ t❤❡ ✐♥❡q✉❛❧✐t② ❝♦♥str❛✐♥ts✳ ❆♥❛❧②t✐❝ ❈❡♥t❡r ❚❤❡ ❛♥❛❧②t✐❝ ❝❡♥t❡r Γ ✐s ❞❡✜♥❡❞ ❛s t❤❡ ♣♦✐♥t ✐♥ Γ ✇❤✐❝❤ ♠❛①✐♠✐③❡s t❤❡ ♣r♦❞✉❝t ♦❢ t❤❡ s❧❛❝❦ ✈❛r✐❛❜❧❡s ❛ss♦s✐❛t❡❞ ✇✐t❤ t❤❡ ✐♥❡q✉❛❧✐t② ❝♦♥str❛✐♥ts✱ ❛♥❞ ❆ ✐s m × n✳

❏♦❤❛♥ ❍❛❣❡♥❜❥ör❦ ❈❤❛♣t❡r ✸

❆♥❛❧②t✐❝ ❈❡♥t❡r

✹

❚❤✐s ✐s ❡q✉✐✈❛❧❡♥t ♦❢ max

m

• i=1

log vi s✉❜❥❡❝t t♦ v = Ax − b ≥ 0

❏♦❤❛♥ ❍❛❣❡♥❜❥ör❦ ❈❤❛♣t❡r ✸

❈❡♥tr❛❧ P❛t❤

✺

❋✐♥❞✐♥❣ ❛ s❡q✉❡♥❝❡ ♦❢ ✐♥t❡r✐♦r ❢❡❛s✐❜❧❡ s♦❧✉t✐♦♥s ❛❧♦♥❣ ❛ ♣❛t❤ ❝❛❧❧❡❞ t❤❡ ❝❡♥tr❛❧ ♣❛t❤✱ ❝♦♥✈❡r❣✐♥❣ t♦ t❤❡ ❛♥❛❧②t✐❝ ❝❡♥t❡r ♦❢ t❤❡ ♦♣t✐♠✉♠ ❢❛❝❡ ♦❢ t❤❡ ▲P ✐s t❤❡ ❜✉✐❧❞✐♥❣ ❜❧♦❝❦ ♦❢ s♦♠❡ ❛❧❣♦r✐t❤♠s✳ ❈♦♥s✐❞❡r min z(x) = cT x s✉❜❥❡❝t t♦ Ax = b, x ≥ 0 ✭✷✮ ▲❡t µ > 0 ❛♥❞ ❝♦♥s✐❞❡r min z(x) = cT x − µ  

n

• j=1

log xj   s✉❜❥❡❝t t♦ Ax = b, (x ≥ 0) ✭✸✮

❏♦❤❛♥ ❍❛❣❡♥❜❥ör❦ ❈❤❛♣t❡r ✸

❈❡♥tr❛❧ P❛t❤

✻

❚❤✐s ♣r♦❜❧❡♠ ✭✸✮ ✐s ❦♥♦✇♥ ❛s t❤❡ ❧♦❣❛r✐t❤♠✐❝ ❜❛rr✐❡r ♣r♦❜❧❡♠✳ ❇② ♣❡♥❛❧✐③✐♥❣ ✈❛r✐❛❜❧❡s ❡♥t❡r✐♥❣ t❤❡ ♥❡❣❛t✐✈❡ r❡❣✐♦♥ ✇❡ ❝❛♥ r❡❧❛① ♥♦♥✲♥❡❣❛t✐✈✐t②✳ ②♦✉t✉✳❜❡✴▼s❣♣❙❧✺❏❘❜■ ■t ❝❛♥ ❜❡ s❤♦✇♥ t❤❛t ✐t ❤❛s ❛ ✉♥✐q✉❡ ♦♣t✐♠✉♠ ❢♦r ❡❛❝❤ µ > 0✳ ▲❡tt✐♥❣ y ❞❡♥♦t❡ t❤❡ ✈❡❝t♦r ♦❢ ❞✉❛❧ ✈❛r✐❛❜❧❡s ♦❢ ▲❛❣r❛♥❣❡ ♠✉❧t✐♣❧✐❡rs ❛ss♦❝✐❛t❡❞ ✇✐t❤ t❤❡ ❡q✉❛❧✐t② ❝♦♥str❛✐♥ts ♦❢ ✭✸✮✱ t❤❡ ♦♣t✐♠❛❧✐t② ❝♦♥❞✐t✐♦♥s ❢♦r ❛ ❢❡❛s✐❜❧❡ s♦❧✉t✐♦♥ x(µ) ✐s t❤❛t t❤❡r❡ ❡①✐sts ❛ y(µ) s❛t✐s❢②✐♥❣ t❤❡ ❑❑❚ ❝♦♥❞✐t✐♦♥s ❢♦r ✭✸✮✳ Ax = b −AT y − s = −cT Xs = µe x, s > 0 ✭✹✮ ✇❤❡r❡ X = diag(x1, ..., xn)✳

❏♦❤❛♥ ❍❛❣❡♥❜❥ör❦ ❈❤❛♣t❡r ✸

❈❡♥tr❛❧ P❛t❤

✼

■t ❝❛♥ ❜❡ s❤♦✇♥ t❤❛t t❤❡ tr❛❥❡❝t♦r② tr❛❝❡❞ ❜② (x(µ), y(µ), s(µ)) ❡①✐sts ❛♥❞ ✐s ✉♥✐q✉❡ ❢♦r ❛❧❧ µ ✐❢ t❤❡ ♦r✐❣✐♥❛❧ ♣r♦❜❧❡♠ ✭✷✮ ❛♥❞ ✐ts ❞✉❛❧ ❤❛✈❡ ✐♥t❡r✐♦r ❢❡❛s✐❜❧❡ s♦❧✉t✐♦♥s✳ ❚❤✐s tr❛❥❡❝t♦r② ✐s ❝❛❧❧❡❞ t❤❡ ♣r✐♠❛❧✲❞✉❛❧ ❝❡♥tr❛❧ ♣❛t❤✳ ❊✈❡r② ♣♦✐♥t x(µ) ♦❢ t❤❡ ❝❡♥tr❛❧ ♣❛t❤ ✐s t❤❡ ❛♥❛❧②t✐❝ ❝❡♥t❡r ♦❢ t❤❡ ✐♥t❡rs❡❝t✐♦♥ ♦❢ K ✇✐t❤ t❤❡ ♦❜❥❡❝t✐✈❡ ♣❧❛♥❡ t❤r♦✉❣❤ t❤❛t ♣♦✐♥t✳

❏♦❤❛♥ ❍❛❣❡♥❜❥ör❦ ❈❤❛♣t❡r ✸

❚❤❡ ❆✣♥❡ ❙❝❛❧✐♥❣ ▼❡t❤♦❞

✽

❈♦♥s✐❞❡r t❤❡ st❛♥❞❛r❞ ♣r♦❜❧❡♠ ✭✷✮ ❛♥❞ ❧❡t xr ❜❡ ❛♥ ✐♥t❡r✐♦r ❢❡❛s✐❜❧❡ s♦❧✉t✐♦♥✳ ❚❤❡ ♠❡t❤♦❞ ❝r❡❛t❡s ❛♥ ❡❧❧✐♣s♦✐❞ ✐♥ Rn ❛r♦✉♥❞ xr ❜② r❡♣❧❛❝✐♥❣ x ≥ 0 ✐♥t♦ x ∈ Er =   x : n

• i=1

xi − xr

i

x2

i

2 ≤ 1    ✭✺✮ ❲❡ t❤❡♥ ♦❜t❛✐♥ t❤❡ ♣r♦❜❧❡♠ min z(x) = cT x s✉❜❥❡❝t t♦ Ax = b n

• i=1

xi − xr

i

x2

i

2 ≤ 1 ✭✻✮

❏♦❤❛♥ ❍❛❣❡♥❜❥ör❦ ❈❤❛♣t❡r ✸

❚❤❡ ❆✣♥❡ ❙❝❛❧✐♥❣ ▼❡t❤♦❞

✾

❚❤❡ ✐♥t❡rs❡❝t✐♦♥ ♦❢ t❤❡ ❡❧❧✐♣s♦✐❞ ✇✐t❤ t❤❡ ❝♦♥str❛✐♥ts Ax = b ✐s ❛♥ ❡❧❧✐♣s♦✐❞ ¯ Er ✇✐t❤ ❝❡♥t❡r xr✳ ❚❤❡ ♦♣t✐♠✉♠ ♦❢ t❤✐s ♣r♦❜❧❡♠ ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ❛♥❛❧②t✐❝❛❧❧②✳ ❆♥ ❡❧❧✐♣s♦✐❞ ✐♥ Rn ✐s t❤❡ s❡t ♦❢ E =

• x : (x − x0)T D(x − x0) ≤ ρ2

, ✭✼✮ ✇❤❡r❡ D ✐s ❛ ♣♦s✐t✐✈❡ ✜♥✐t❡ ♠❛tr✐①✳

❏♦❤❛♥ ❍❛❣❡♥❜❥ör❦ ❈❤❛♣t❡r ✸

❚❤❡ ❆✣♥❡ ❙❝❛❧✐♥❣ ▼❡t❤♦❞

✶✵

❈♦♥s✐❞❡r t❤❡ ♣r♦❜❧❡♠ min z(x) = cT x s✉❜❥❡❝t t♦ Ax = b (x − x0)T D(x − x0) ≤ ρ2 ✭✽✮ ❋♦r t❤❡ s♣❡❝✐❛❧ ❝❛s❡ ✇❤❡r❡ t❤❡ ❡❧❧✐♣s♦✐❞ ✐s ❛ s♣❤❡r❡✱ D = I✱ ✇❡ ❣❡t t❤❡ s♦❧✉t✐♦♥ x⋆ = x0 + ρ(−cT )/c ✭✾✮

❏♦❤❛♥ ❍❛❣❡♥❜❥ör❦ ❈❤❛♣t❡r ✸

❚❤❡ ❆✣♥❡ ❙❝❛❧✐♥❣ ▼❡t❤♦❞

✶✶

❏♦❤❛♥ ❍❛❣❡♥❜❥ör❦ ❈❤❛♣t❡r ✸

❚❤❡ ❆✣♥❡ ❙❝❛❧✐♥❣ ▼❡t❤♦❞

✶✷

▲❡t x0 ❜❡ ❛ ❣✐✈❡♥ ♣♦✐♥t ❛♥❞ ❝♦♥s✐❞❡r t❤❡ ♣r♦❜❧❡♠ min z(x) = cT x s✉❜❥❡❝t t♦ Ax = b (x − x0)T (X0)−2(x − x0) ≤ ρ2 ✭✶✵✮ ✇❤✐❝❤ ❝❛♥ ❜❡ tr❛♥s❢♦r♠❡❞ ✐♥t♦ t❤❡ s♣❤❡r❡ ♣r♦❜❧❡♠ ✭✽✮ ❜② yT = eT + (x − x0)T (X0)−1 ✭✶✶✮ ✇❤✐❝❤ ❣✐✈❡s ✉s t❤❡ ❡①♣r❡ss✐♦♥ ❢♦r t❤❡ ♣♦✐♥t x ❛s xT = (y − e)T X0(x0)T = yT X0 ✭✶✷✮

❏♦❤❛♥ ❍❛❣❡♥❜❥ör❦ ❈❤❛♣t❡r ✸

❚❤❡ ❆✣♥❡ ❙❝❛❧✐♥❣ ▼❡t❤♦❞

✶✸

❚❤✐s ❣✐✈❡s ✉s min z(x) = cT x s✉❜❥❡❝t t♦ (y − e)T (y − e) ≤ ρ2 ✭✶✸✮ ✇✐t❤ t❤❡ ♦♣t✐♠❛❧ s♦❧✉t✐♦♥ x⋆ = x0 − ρ(X0)2cT X0cT

❏♦❤❛♥ ❍❛❣❡♥❜❥ör❦ ❈❤❛♣t❡r ✸

❚❤❡ ❆✣♥❡ ❙❝❛❧✐♥❣ ▼❡t❤♦❞

✶✹

◆♦✇ ❝♦♥s✐❞❡r t❤❡ ♣r♦❜❧❡♠ min z(x) = cT x s✉❜❥❡❝t t♦ Ax = b (x − x0)T (x − x0) ≤ ρ2 ✭✶✹✮ ✇❤❡r❡ A ✐s ❛ ♠❛tr✐① ♦❢ ♦r❞❡r m × n ❛♥❞ ❢✉❧❧ r♦✇ r❛♥❦ m✳ ❆♥❞ x0 ✐s ❛ ♣♦✐♥t ✐♥ t❤❡ s♣❛❝❡ H = {x : Ax = b}✳ ❉❡♥♦t✐♥❣ t❤❡ ❡❧❧✐♣s♦✐❞ B ✇❡ ❦♥♦✇ t❤❛t t❤❡ ❝❡♥t❡r x0 ♦❢ B ✐s ✐♥ H ❛♥❞ t❤❛t G ∩ B ✐s ❛♥♦t❤❡r ❜❛❧❧ ✇❤✐❝❤ ❤❛s ❝❡♥t❡r x0 ❛♥❞ r❛❞✐✉s ρ ❛♥❞ ✐s t♦t❛❧❧② ❝♦♥t❛✐♥❡❞ ✐♥ H✳

❏♦❤❛♥ ❍❛❣❡♥❜❥ör❦ ❈❤❛♣t❡r ✸

❚❤❡ ❆✣♥❡ ❙❝❛❧✐♥❣ ▼❡t❤♦❞

✶✺

❙✐♥❝❡ A ✐s ♦❢ ❢✉❧❧ r❛♥❦✱ t❤❡ ♦rt❤♦❣♦♥❛❧ ♣r♦❥❡❝t✐♦♥ ♦❢ cT ✐♥t♦ t❤❡ s✉❜s♣❛❝❡ {x : Ax = 0} ✐s PcT ✇❤❡r❡ P = I − AT (AAT )−1A ✐s t❤❡ ♣r♦❥❡❝t✐♦♥ ♠❛tr✐①✳

❏♦❤❛♥ ❍❛❣❡♥❜❥ör❦ ❈❤❛♣t❡r ✸

❚❤❡ ❆✣♥❡ ❙❝❛❧✐♥❣ ▼❡t❤♦❞

✶✻

◆♦✇ ❝♦♥s✐❞❡r t❤❡ ♣r♦❜❧❡♠ min z(x) = cT x s✉❜❥❡❝t t♦ Ax = b (x − x0)T (X0)−2(x − x0) ≤ ρ2 ✭✶✺✮ ✇❤❡r❡ A ✐s ❛ ♠❛tr✐① ♦❢ ♦r❞❡r m × n ❛♥❞ ❢✉❧❧ r♦✇ r❛♥❦ m✳ ❆♥❞ x0 ✐s ❛ ♣♦✐♥t ✐♥ t❤❡ s♣❛❝❡ H = {x : Ax = b}✳ ❙♦❧✈✐♥❣ ✭✶✺✮ ✐s ❡q✉✐✈❛❧❡♥t t♦ ♠✐♥✐♠✐③✐♥❣ cPx ♦♥ H ∩ B ❛♥❞ ✉s✐♥❣ t❤❡ s♦❧✉t✐♦♥ t♦ t❤❡ s♣❤❡r❡ ♣r♦❜❧❡♠ ✭✾✮ ✇❡ ❣❡t x0 − ρ PcT PcT ✭✶✻✮

❏♦❤❛♥ ❍❛❣❡♥❜❥ör❦ ❈❤❛♣t❡r ✸

❚❤❡ ❆✣♥❡ ❙❝❛❧✐♥❣ ▼❡t❤♦❞

✶✼

❯s✐♥❣ t❤❡ tr❛♥s❢♦r♠❛t✐♦♥ ♦❢ ✈❛r✐❛❜❧❡s ♣r❡✈✐♦✉s❧② ❞❡✜♥❡❞ ✐♥✭✶✶✮✱ t❤❡ ♣r♦❜❧❡♠ ❜❡❝♦♠❡s min z(x) = cX0y + ❝♦♥st❛♥t s✉❜❥❡❝t t♦ AX0y = b (y − e)T (y − e) ≤ ρ2 ✭✶✼✮ ❲❡ ❦♥♦✇ t❤❛t t❤❡ ♦♣t✐♠❛❧ ② ✐s ❣✐✈❡♥ ❜② ¯ y = e − ρ P0X0cT P0X0cT ✇❤❡r❡ P0 ✐s t❤❡ ♣r♦❥❡❝t✐♦♥ ♠❛tr✐① P0 = I − X0AT (A(X0)2AT )−1AX0 ✐s t❤❡ ♣r♦❥❡❝t✐♦♥ ♠❛tr✐①✳

❏♦❤❛♥ ❍❛❣❡♥❜❥ör❦ ❈❤❛♣t❡r ✸

❚❤❡ ❆✣♥❡ ❙❝❛❧✐♥❣ ▼❡t❤♦❞

✶✽

❯s✐♥❣ ✭✶✷✮ ✇❡ ❣❡t t❤❡ ♦♣t✐♠❛❧ ♣♦✐♥t x ¯ x = X0¯ y = X0

• e − ρ P0X0cT

P0X0cT

• =

x0 − ρ

• e − ρ P0X0cT

P0X0cT

• ❏♦❤❛♥ ❍❛❣❡♥❜❥ör❦

❈❤❛♣t❡r ✸

❖♣t✐♠❛❧✐t② ❈♦♥❞✐t✐♦♥s

✶✾

■❢ ¯ xr

j = 0 ❢♦r ❛t❧❡❛st ♦♥❡ j = 1, ..., n✱ t❤❡♥ ¯

xr

j ✐s ❛♥ ♦♣t✐♠✉♠

s♦❧✉t✐♦♥ ♦❢ ✭✷✮✳ ■❢ t❤❡ t❡♥t❛t✐✈❡ ❞✉❛❧ s❧❛❝❦ ✈❡❝t♦r sr = cT − AT yr ≤ 0 t❤❡ ♦❜❥❡❝t✐✈❡ ❢✉♥❝t✐♦♥ ✐s ✉♥❜♦✉♥❞❡❞ ✐♥ t❤❡ ♣r♦❜❧❡♠ ✭✷✮✳ ■❢ ❡✐t❤❡r ♦❢ t❤❡ ❛❜♦✈❡ ✐s s❛t✐s✜❡❞ t❤❡ ❛❧❣♦r✐t❤♠ t❡r♠✐♥❛t❡s✳ ❊❧s❡ ❛ st❡♣ ✐s t❛❦❡♥ ✐♥ t❤❡ ❞✐r❡❝t✐♦♥ ❣✐✈❡♥ ❜② dr = ¯ xr − xr✳

❏♦❤❛♥ ❍❛❣❡♥❜❥ör❦ ❈❤❛♣t❡r ✸

❙t❡♣ ▲❡♥❣t❤

✷✵

❚❤❡ ♠❛①✐♠✉♠ st❡♣ ❧❡♥❣t❤ θr ✐s t❤❡ ♠❛①✐♠✉♠ ✈❛❧✉❡ t❤❛t ❦❡❡♣s xr

j + θdr j ≥ 0,

∀j. ■t ❝❛♥ ❜❡ ✈❡r✐✜❡❞ t❤❛t t❤❛t ✐s θr = min

• j, Xrsr

xr

jsr j

: sr

j > 0

• xr+1 = xr + αθrdr, 0 < α < 1.

❚②♣✐❝❛❧❧② α = 0.95

❏♦❤❛♥ ❍❛❣❡♥❜❥ör❦ ❈❤❛♣t❡r ✸

◆❡✇t♦♥✬s ▼❡t❤♦❞

✷✶

◆❡✇t♦♥✬s ♠❡t❤♦❞ ✐s ❛ ♠❡t❤♦❞ ❢♦r s♦❧✈✐♥❣ ♥♦♥✲❧✐♥❡❛r ❡q✉❛t✐♦♥s✳ xr+1 = xr − (∇f(xr))−1f(xr) ■ t❤❡ ❏❛❝♦❜✐❛♥ ✐s ♥♦♥✲s✐♥❣✉❧❛r✱ t❤❡ ♠❡t❤♦❞ t❛❦❡s ❛ st❡♣ ♦❢ ❧❡♥❣t❤ 1 ✐♥ t❤✐s ❞✐r❡❝t✐♦♥✳

❏♦❤❛♥ ❍❛❣❡♥❜❥ör❦ ❈❤❛♣t❡r ✸

Pr✐♠❛❧✲❉✉❛❧ P❛t❤ ❋♦❧❧♦✇✐♥❣ ▼❡t❤♦❞s

✷✷

❚❤❡ ❝❡♥tr❛❧ ♣❛t❤✲❢♦❧❧♦✇✐♥❣ ♣r✐♠❛❧✲❞✉❛❧ ♠❡t❤♦❞s ❛r❡ s♦♠❡ ♦❢ t❤❡ ♠♦st ♣♦♣✉❧❛r ♠❡t❤♦❞s✳ min cT x s✉❜❥❡❝t t♦ Ax = b x ≥ 0 ✭✶✽✮ ❆♥❞ ✐ts ❞✉❛❧ max bT y s✉❜❥❡❝t t♦ AT y + s = c x, s ≥ 0 ✭✶✾✮ ❚❤❡ ❝♦♠♣❧❡♠❡♥t❛r② s❧❛❝❦♥❡ss ❝♦♥❞✐t✐♦♥s ❛r❡ xjsj = 0✳

❏♦❤❛♥ ❍❛❣❡♥❜❥ör❦ ❈❤❛♣t❡r ✸

❖♣t✐♠❛❧✐t② ❈♦♥❞✐t✐♦♥s

✷✸

❙♦❧✈✐♥❣ t❤❡ ▲P ✐s ❡q✉✐✈❛❧❡♥t t♦ ✜♥❞✐♥❣ ❛ s♦❧✉t✐♦♥ (x, s) ≥ 0 t♦ t❤❡ ❢♦❧❧♦✇✐♥❣ s②st❡♠ ♦❢ (2n + m) ❡q✉❛t✐♦♥s ✇✐t❤ (2n + m) ✉♥❦♥♦✇♥✱ ✐❢ t❤❡ ❝♦♥str❛✐♥t ♠❛①tr✐① A ✐s m × n✳ F(x, y, s) =   AT y + s − c Ax − b XSe   = 0 ✭✷✵✮ ✇❤✐❝❤ ✐s ❛ ♥♦♥✲❧✐♥❡❛r s②st❡♠ ❜❡❝❛✉s❡ ♦❢ t❤❡ ❧❛st ❡q✉❛t✐♦♥✳

❏♦❤❛♥ ❍❛❣❡♥❜❥ör❦ ❈❤❛♣t❡r ✸

❚❤❡ ❈❡♥tr❛❧ P❛t❤

✷✹

❚❤❡ ❝❡♥tr❛❧ ♣❛t❤ ✐s ♣❛r❛♠❡tr✐③❡❞ ✉s✐♥❣ µ > 0✳ ❋♦r ❡❛❝❤ µ > 0 t❤❡ ♣♦✐♥t (xµ, yµ, sµ) ∈ C s❛t✐s✜❡s (xµ, sµ) > 0 ❛♥❞ AT yµ + sµ = cT Axµ = b xµ

j sµ j = µ, ∀j = 1, ..., n

■❢ µ = 0 t❤✐s ❞❡✜♥❡s t❤❡ ♦♣t✐♠❛❧✐t② ❝♦♥❞✐t✐♦♥s✳

❏♦❤❛♥ ❍❛❣❡♥❜❥ör❦ ❈❤❛♣t❡r ✸

❖♣t✐♠❛❧✐t② ❈♦♥❞✐t✐♦♥s

✷✺

❋♦❧❧♦✇✐♥❣ t❤❡ ❝❡♥tr❛❧ ♣❛t❤ ✇❤✐❧❡ ❞❡❝r❡❛s✐♥❣ µ t♦ ✵ ✐♠♣❧✐❡s t❤❛t ❛❧❧ ❝♦♠♣❧❡♠❡♥t❛r② ♣r♦❞✉❝ts xjsj ❛r❡ ❡q✉❛❧✳ ❚❤✉s µ ✐s ❛ ♠❡❛s✉r❡ ♦❢ ❝❧♦s❡♥❡ss t♦ ♦♣t✐♠❛❧✐t②✳ ❍♦✇❡✈❡r✱ t✇♦ ❞✐✣❝✉❧t✐❡s ❛r✐s❡ ❋✐♥❞✐♥❣ ❛ st❛rt✐♥❣ ♣♦✐♥t ♦♥ C ✇✐t❤ ❛❧❧ xjsj ❡q✉❛❧✳ C ✐s ❛ ♥♦♥✲❧✐♥❡❛r ❝✉r✈❡✳

❏♦❤❛♥ ❍❛❣❡♥❜❥ör❦ ❈❤❛♣t❡r ✸

❖♣t✐♠❛❧✐t② ❈♦♥❞✐t✐♦♥s

✷✻

❲❡ ✐♥tr♦❞✉❝❡ t❤❡ ♠❡❛s✉r❡ ♦❢ ❞❡✈✐❛t✐♦♥ ❢r♦♠ t❤❡ ❝❡♥tr❛❧ ♣❛t❤ C ❛s XSe − µe µ . ❯s✉❛❧❧② t❤❡ ✷✲♥♦r♠ ✐s ✉s❡❞ ✇✐t❤ ❛ ♣❛r❛♠❡t❡r θ XSe − µe ≤ µ, ✇❤❡r❡ 0 < θ < 1✱ ✇❤❡r❡ ✵✳✺ ✐s ❛ ❝♦♠♠♦♥ ✈❛❧✉❡✳ ❇② ❦❡❡♣✐♥❣ ❛❧❧ ✐t❡r❛t❡s ✇✐t❤✐♥ t❤✐s ❦✐♥❞ ♦❢ ♥❡✐❣❤❜♦✉r❤♦♦❞✱ ♣❛t❤✲❢♦❧❧♦✇✐♥❣ ♠❡t❤♦❞s r❡❞✉❝❡ ❛❧❧ xjsj t♦ ✵ ❛t ❛❜♦✉t t❤❡ s❛♠❡ r❛t❡✳

❏♦❤❛♥ ❍❛❣❡♥❜❥ör❦ ❈❤❛♣t❡r ✸

❙t❡♣

✷✼

▼♦✈✐♥❣ ❢r♦♠ t❤❡ ❝✉rr❡♥t ♣♦✐♥t✱ ✇❡ ♠❛② ♦♥❧② t❛❦❡ ❛ s♠❛❧❧ st❡♣✳ ❚❤✉s t❤❡ ♥♦♥✲♥❡❣❛t✐✈✐t② ❝♦♥str❛✐♥ts (x, s) ≥ 0 ❝❛♥ ❜❡ ✐❣♥♦r❡❞ ✇❤❡♥ ❝♦♠♣✉t✐♥❣ t❤✐s st❡♣✳ ❚♦ ✜♥❞ t❤❡ ❞✐r❡❝t✐♦♥✱ ✇❡ s♦❧✈❡   AT I A S X     ∆x ∆y ∆s   =   −XSe + σµe   ✭✷✶✮ ✇❤❡r❡ σ = 1 ✐s t❤❡ ❝❡♥t❡r✐♥❣ ❞✐r❡❝t✐♦♥✱ ✇❤✐❝❤ ✇✐❧❧ t❛❦❡ ✉s t♦✇❛r❞s t❤❡ ♣♦✐♥t (xµ, yµ, sµ) ∈ C✱ ❜✉t ♠❛② ♣r♦❞✉❝❡ ❛ s♠❛❧❧ ✐♠♣r♦✈❡♠❡♥t ✐♥ ♦❜❥❡❝t✐✈❡ ✈❛❧✉❡✳ ❯s✐♥❣ σ = 1 ✐s ❛ ♣✉r❡ ◆❡✇t♦♥ st❡♣✳ ❆❧❣♦r✐t❤♠s ✉s✉❛❧❧② ❝❤♦♦s❡ ❛ tr❛❞❡ ♦✛ ✐♥ t❤❡ ✐♥t❡r✈❛❧ (0, 1)✳

❏♦❤❛♥ ❍❛❣❡♥❜❥ör❦ ❈❤❛♣t❡r ✸

❙t❛rt✐♥❣ P♦✐♥t

✷✽

min 1 2xT x s✉❜❥❡❝t t♦ Ax = b ✭✷✷✮ min 1 2sT s s✉❜❥❡❝t t♦ AT y + s = c ✭✷✸✮ ❚❤❡ s♦❧✉t✐♦♥ ♦❢ t❤❡s❡ ♣r♦❜❧❡♠s ❝❛♥ ❜❡ ✇r✐tt❡♥ ❛s ¯ x = AT (AAT )−1b ¯ y = (AAT )−1Ac ¯ s = c − AT ¯ y

❏♦❤❛♥ ❍❛❣❡♥❜❥ör❦ ❈❤❛♣t❡r ✸

❙t❛rt✐♥❣ P♦✐♥t

✷✾

❲❡ ♠✉st ♠♦❞✐❢② t❤❡ s♦❧✉t✐♦♥s t♦ ❛✈♦✐❞ ♥♦♥✲♥❡❣❛t✐✈❡ ✈❛❧✉❡s ♦❢ ¯ x, ¯ s✳ ˆ x = ¯ x + max(−3/2 min ¯ xi, 0)e+ ˆ s = ¯ s + max(−3/2 min ¯ si, 0)e ❲❡ t❤❡♥ s❝❛❧❛rs t♦ ❡♥s✉r❡ t❤❡ ❝♦♠♣♦♥❡♥ts ❛r❡ ♥♦t t♦♦ ❝❧♦s❡ t♦ ③❡r♦ x0 = ˆ x + 1 2 ˆ xT ˆ s eT ˆ xe s0 = ˆ s + 1 2 ˆ sT ˆ x eT ˆ xe ❚❤❡ ❝♦♠♣✉t❛t✐♦♥❛❧ ❝♦st ♦❢ ✜♥❞✐♥❣ t❤✐s st❛rt✐♥❣ ♣♦✐♥t ✐s ❛❜♦✉t t❤❡ s❛♠❡ ❛s ♦♥❡ st❡♣ ♦❢ t❤❡ ♣r✐♠❛❧✲❞✉❛❧ ♠❡t❤♦❞✳

❏♦❤❛♥ ❍❛❣❡♥❜❥ör❦ ❈❤❛♣t❡r ✸

❙♦❧✈✐♥❣ t❤❡ ▲✐♥❡❛r ❙②st❡♠

✸✵

▼♦st ❝♦♠♣✉t❛t✐♦♥❛❧ t✐♠❡ ✐s t❛❦❡♥ ✉♣ s♦❧✈✐♥❣ t❤❡ ❧✐♥❡❛r s②st❡♠ ✭✷✶✮✳ ❚❍❡ ♠❛tr✐① ✐♥ t❤❡s❡ s②st❡♠s ❛r❡ ✉s✉❛❧❧② ❧❛r❣❡ ❛♥❞ s♣❛rs❡✱ ❛♥❞ t❤❡ str✉❝t✉r❡ ❛❧❧♦✇s ✉s t♦ r❡❢♦r♠✉❧❛t❡ t❤❡♠ ❛s s②st❡♠s ✇✐t❤ ♠♦r❡ ❝♦♠♣❛❝t s②♠♠❡tr✐❝ ❝♦❡✣❝✐❡♥t ♠❛tr✐❝❡s ✇❤✐❝❤ ❛r❡ ❡❛s✐❡r t♦ ❢❛❝t♦r t❤❛♥ t❤❡ ♦r✐❣✐♥❛❧ ♦♥❡✳   AT I A S X     ∆x ∆y ∆s   =   −rc −rb −rxs   ✭✷✹✮ ❙✐♥❝❡ x ❛♥❞ s ❛r❡ str✐❝t❧② ♣♦s✐t✐✈❡✱ t❤❡ ❞✐❛❣♦♥❛❧ ♠❛tr✐❝❡s X ❛♥❞ S ❛r❡ ♥♦♥✲s✐♥❣✉❧❛r✳ ❲❡ ❝❛♥ ❡❧✐♠✐♥❛t❡ ∆s ❜② X∆s = −rxs − S∆x ❛♥❞ ❜② ♠✉❧t✐♣❧②✐♥❣ ✇✐t❤ −X−1 ♦❜t❛✐♥ ∆s = −X−1rxs − X−1S∆x

❏♦❤❛♥ ❍❛❣❡♥❜❥ör❦ ❈❤❛♣t❡r ✸

❙♦❧✈✐♥❣ t❤❡ ▲✐♥❡❛r ❙②st❡♠

✸✶

❈♦♥s✐❞❡r t❤❡ t❤✐r❞ ❡q✉❛t✐♦♥ S∆x + X∆s = −rxs ❆❞❞ −X−1 t✐♠❡s t❤❡ t❤✐r❞ ❡q✉❛t✐♦♥ t♦ t❤❡ ✜rst ❡q✉❛t✐♦♥✳ X−1S∆x + X−1X

=I

∆s = −X−1rxs   AT I A X−1S −I     ∆x ∆y ∆s   =   −rc −rb X−1rxs  

❏♦❤❛♥ ❍❛❣❡♥❜❥ör❦ ❈❤❛♣t❡r ✸

❙♦❧✈✐♥❣ t❤❡ ▲✐♥❡❛r ❙②st❡♠

✸✷

−D−2 AT A ∆x ∆y

• =

−rc + X−1rxs −rb

• ✭✷✺✮

✇❤❡r❡ D = S−1/2X1/2✳ ❚❤✐s ✐s ❦♥♦✇♥ ❛s t❤❡ ❛❣✉♠❡♥t❡❞ s②st❡♠✳ ❲❡ ❝❛♥ ❣♦ ❢✉rt❤❡r ❜② ❡❧✐♠✐♥❛t✐♥❣ ∆x ❛♥❞ ❛❞❞✐♥❣ AD2 t✐♠❡s t❤❡ ✜rst ❡q✉❛t✐♦♥ t♦ t❤❡ s❡❝♦♥❞ ✐♥ ♦r❞❡r t♦ ❝❛♥❝❡❧ ♦✉t t❤❡ t❡r♠ A∆x✳ −AD2D−2 AD2AT A ∆x ∆y

• =

−AD2rc + AD2X−1rxs −rb

• ✉s✐♥❣ t❤❡ ❞❡✜♥✐t✐♦♥ D = S−1/2X1/2

−A AD2AT A ∆x ∆y

• =

−AXS−1rc + AS−1rxs −rb

• ❏♦❤❛♥ ❍❛❣❡♥❜❥ör❦

❈❤❛♣t❡r ✸

❙♦❧✈✐♥❣ t❤❡ ▲✐♥❡❛r ❙②st❡♠

✸✸

❆❞❞✐♥❣ ❡q✉❛t✐♦♥ ♦♥❡ t♦ ❡q✉❛t✐♦♥ t✇♦ ②✐❡❧❞s −A AD2AT AD2AT ∆x ∆y

• =
• −AXS−1rc + AS−1rxs

−rb − AXS−1rc + AS−1rxs

• D2AT ∆y = −rb − AXS−1rc + AS−1rxs

∆s = −rc + AT ∆y ∆x = −S−1rxs − XS−1∆s ✭✷✻✮ ✇❤❡r❡ t❤❡ ❡①♣r❡ss✐♦♥s ❢♦r ∆s ❛♥❞ ∆x ❛r❡ ♦❜t❛✐♥❡❞ ❢r♦♠ t❤❡ ♦r✐❣✐♥❛❧ s②st❡♠ ✭✷✹✮✳

❏♦❤❛♥ ❍❛❣❡♥❜❥ör❦ ❈❤❛♣t❡r ✸

❙♦❧✈✐♥❣ t❤❡ ▲✐♥❡❛r ❙②st❡♠

✸✹

▼♦st ✐♠♣❧❡♠❡♥t❛t✐♦♥s ♦❢ ♣r✐♠❛❧ ❞✉❛❧ ✐♥t❡r✐♦r ♣♦✐♥t s♦❧✈❡rs ❛r❡ ❜❛s❡❞ ♦♥ ❢♦r♠✉❧❛t✐♦♥s ❧✐❦❡ ✭✷✻✮✱ ✇❤✐❝❤ ❛r❡ ❝❛❧❧❡❞ t❤❡ ♥♦r♠❛❧ ❡q✉❛t✐♦♥s✳

• ❡♥❡r❛❧ ♣✉r♣♦s❡ ❈❤♦❧❡s❦❡② s♦❢t✇❛r❡ ❝❛♥ ❜❡ ❛♣♣❧✐❡❞ t♦ AD2AT ❜✉t

♠♦❞✐✜❝❛t✐♦♥s ❛r❡ ♥❡❡❞❡❞ ❜❡❝❛✉s❡ t❤❡ ♠❛tr✐① ♠❛② ❜❡ ✐❧❧✲❝♦♥❞✐t✐♦♥❡❞ ♦r s✐♥❣✉❧❛r✳

❏♦❤❛♥ ❍❛❣❡♥❜❥ör❦ ❈❤❛♣t❡r ✸

❈♦♠♣❧❡①✐t②

✸✺

❆❧❧ ♣❛t❤✲❢♦❧❧♦✇✐♥❣ ♠❡t❤♦❞s ❤❛✈❡ ❜❡❡♥ s❤♦✇♥ t♦ ❜❡ ♣♦❧②♥♦♠✐❛❧ t✐♠❡ ❛❧❣♦r✐t❤♠s ✭✇❤❡r❡ s✐♠♣❧❡① ✐s ✇♦rst ❝❛s❡ ❡①♣♦♥❡♥t✐❛❧✮✳ ❊❛❝❤ st❡♣ r❡q✉✐r❡s ❛ ❢✉❧❧ ♠❛tr✐① ✐♥✈❡rs✐♦♥✱ ❛ r❛t❤❡r ❡①♣❡♥s✐✈❡ t❛s❦ ❢♦r ❛ ❧❛r❣❡ s❝❛❧❡ ♣r♦❜❧❡♠✳ ❍♦✇❡✈❡r t❤❡ ♥✉♠❜❡r ♦❢ st❡♣s ❛r❡ s♠❛❧❧❡r ✐♥ ✐♥t❡r✐♦r ♣♦✐♥t ♠❡t❤♦❞s t❤❛♥ ✐♥ t❤❡ s✐♠♣❧❡① ♠❡t❤♦❞✳

❏♦❤❛♥ ❍❛❣❡♥❜❥ör❦ ❈❤❛♣t❡r ✸

❆❞✈❛♥t❛❣❡s ❛♥❞ ❉✐s❛❞✈❛♥t❛❣❡s

✸✻

❈♦♠♣❛r❡❞ t♦ t❤❡ s✐♠♣❧❡① ♠❡t❤♦❞ ✰ ❇❡tt❡r ✇♦rst ❝❛s❡ ❝♦♠♣❧❡①✐t②✳ ✰ ❊❛s✐❡r t♦ ♣r♦❣r❛♠✳ ✲ ❈❛♥♥♦t t❛❦❡ ❛❞✈❛♥t❛❣❡ ♦❢ ✇❛r♠ st❛rt ✐♥❢♦r♠❛t✐♦♥ ❛s ❣♦♦❞✳

❏♦❤❛♥ ❍❛❣❡♥❜❥ör❦ ❈❤❛♣t❡r ✸

❇✐❜❧✐♦❣r❛♣❤② ■

❑❛tt❛ ● ▼✉rt②✳ ❖♣t✐♠✐③❛t✐♦♥ ❢♦r ❞❡❝✐s✐♦♥ ♠❛❦✐♥❣✳ ❙♣r✐♥❣❡r✱ ✷✵✵✾✳ ❏♦r❣❡ ◆♦❝❡❞❛❧ ❛♥❞ ❙t❡♣❤❡♥ ❲r✐❣❤t✳ ◆✉♠❡r✐❝❛❧ ♦♣t✐♠✐③❛t✐♦♥✳ ❙♣r✐♥❣❡r ❙❝✐❡♥❝❡ ✫ ❇✉s✐♥❡ss ▼❡❞✐❛✱ ✷✵✵✻✳

❏♦❤❛♥ ❍❛❣❡♥❜❥ör❦ ❈❤❛♣t❡r ✸