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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 ♦❢ m � max log v i i =1 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 z ( x ) = c T x min ✭✷✮ s✉❜❥❡❝t t♦ Ax = b, x ≥ 0 ▲❡t µ > 0 ❛♥❞ ❝♦♥s✐❞❡r   n � z ( x ) = c T x − µ min log x j   ✭✸✮ j =1 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 − A T y − s = − c T ✭✹✮ Xs = µe x, s > 0 ✇❤❡r❡ X = diag ( x 1 , ..., x n ) ✳ ❏♦❤❛♥ ❍❛❣❡♥❜❥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 x r ❜❡ ❛♥ ✐♥t❡r✐♦r ❢❡❛s✐❜❧❡ s♦❧✉t✐♦♥✳ ❚❤❡ ♠❡t❤♦❞ ❝r❡❛t❡s ❛♥ ❡❧❧✐♣s♦✐❞ ✐♥ R n ❛r♦✉♥❞ x r ❜② r❡♣❧❛❝✐♥❣ x ≥ 0 ✐♥t♦   � n � 2 x i − x r   � i x ∈ E r =  x : ≤ 1 ✭✺✮ x 2  i i =1 ❲❡ t❤❡♥ ♦❜t❛✐♥ t❤❡ ♣r♦❜❧❡♠ z ( x ) = c T x min s✉❜❥❡❝t t♦ Ax = b ✭✻✮ � n � 2 x i − x r � i ≤ 1 x 2 i i =1 ❏♦❤❛♥ ❍❛❣❡♥❜❥ör❦ ❈❤❛♣t❡r ✸

❚❤❡ ❆✣♥❡ ❙❝❛❧✐♥❣ ▼❡t❤♦❞ ✾ ❚❤❡ ✐♥t❡rs❡❝t✐♦♥ ♦❢ t❤❡ ❡❧❧✐♣s♦✐❞ ✇✐t❤ t❤❡ ❝♦♥str❛✐♥ts Ax = b ✐s ❛♥ ❡❧❧✐♣s♦✐❞ ¯ E r ✇✐t❤ ❝❡♥t❡r x r ✳ ❚❤❡ ♦♣t✐♠✉♠ ♦❢ t❤✐s ♣r♦❜❧❡♠ ❝❛♥ ❜❡ ❝♦♠♣✉t❡❞ ❛♥❛❧②t✐❝❛❧❧②✳ ❆♥ ❡❧❧✐♣s♦✐❞ ✐♥ R n ✐s t❤❡ s❡t ♦❢ x : ( x − x 0 ) T D ( x − x 0 ) ≤ ρ 2 � � E = , ✭✼✮ ✇❤❡r❡ D ✐s ❛ ♣♦s✐t✐✈❡ ✜♥✐t❡ ♠❛tr✐①✳ ❏♦❤❛♥ ❍❛❣❡♥❜❥ör❦ ❈❤❛♣t❡r ✸

❚❤❡ ❆✣♥❡ ❙❝❛❧✐♥❣ ▼❡t❤♦❞ ✶✵ ❈♦♥s✐❞❡r t❤❡ ♣r♦❜❧❡♠ z ( x ) = c T x min s✉❜❥❡❝t t♦ Ax = b ✭✽✮ ( x − x 0 ) T D ( x − x 0 ) ≤ ρ 2 ❋♦r t❤❡ s♣❡❝✐❛❧ ❝❛s❡ ✇❤❡r❡ t❤❡ ❡❧❧✐♣s♦✐❞ ✐s ❛ s♣❤❡r❡✱ D = I ✱ ✇❡ ❣❡t t❤❡ s♦❧✉t✐♦♥ x ⋆ = x 0 + ρ ( − c T ) / � c � ✭✾✮ ❏♦❤❛♥ ❍❛❣❡♥❜❥ör❦ ❈❤❛♣t❡r ✸

❚❤❡ ❆✣♥❡ ❙❝❛❧✐♥❣ ▼❡t❤♦❞ ✶✶ ❏♦❤❛♥ ❍❛❣❡♥❜❥ör❦ ❈❤❛♣t❡r ✸

❚❤❡ ❆✣♥❡ ❙❝❛❧✐♥❣ ▼❡t❤♦❞ ✶✷ ▲❡t x 0 ❜❡ ❛ ❣✐✈❡♥ ♣♦✐♥t ❛♥❞ ❝♦♥s✐❞❡r t❤❡ ♣r♦❜❧❡♠ z ( x ) = c T x min s✉❜❥❡❝t t♦ Ax = b ✭✶✵✮ ( x − x 0 ) T ( X 0 ) − 2 ( x − x 0 ) ≤ ρ 2 ✇❤✐❝❤ ❝❛♥ ❜❡ tr❛♥s❢♦r♠❡❞ ✐♥t♦ t❤❡ s♣❤❡r❡ ♣r♦❜❧❡♠ ✭✽✮ ❜② y T = e T + ( x − x 0 ) T ( X 0 ) − 1 ✭✶✶✮ ✇❤✐❝❤ ❣✐✈❡s ✉s t❤❡ ❡①♣r❡ss✐♦♥ ❢♦r t❤❡ ♣♦✐♥t x ❛s x T = ( y − e ) T X 0 ( x 0 ) T = y T X 0 ✭✶✷✮ ❏♦❤❛♥ ❍❛❣❡♥❜❥ör❦ ❈❤❛♣t❡r ✸

❚❤❡ ❆✣♥❡ ❙❝❛❧✐♥❣ ▼❡t❤♦❞ ✶✸ ❚❤✐s ❣✐✈❡s ✉s z ( x ) = c T x min ✭✶✸✮ ( y − e ) T ( y − e ) ≤ ρ 2 s✉❜❥❡❝t t♦ ✇✐t❤ t❤❡ ♦♣t✐♠❛❧ s♦❧✉t✐♦♥ x ⋆ = x 0 − ρ ( X 0 ) 2 c T � X 0 c T � ❏♦❤❛♥ ❍❛❣❡♥❜❥ör❦ ❈❤❛♣t❡r ✸

❚❤❡ ❆✣♥❡ ❙❝❛❧✐♥❣ ▼❡t❤♦❞ ✶✹ ◆♦✇ ❝♦♥s✐❞❡r t❤❡ ♣r♦❜❧❡♠ z ( x ) = c T x min s✉❜❥❡❝t t♦ Ax = b ✭✶✹✮ ( x − x 0 ) T ( x − x 0 ) ≤ ρ 2 ✇❤❡r❡ A ✐s ❛ ♠❛tr✐① ♦❢ ♦r❞❡r m × n ❛♥❞ ❢✉❧❧ r♦✇ r❛♥❦ m ✳ ❆♥❞ x 0 ✐s ❛ ♣♦✐♥t ✐♥ t❤❡ s♣❛❝❡ H = { x : Ax = b } ✳ ❉❡♥♦t✐♥❣ t❤❡ ❡❧❧✐♣s♦✐❞ B ✇❡ ❦♥♦✇ t❤❛t t❤❡ ❝❡♥t❡r x 0 ♦❢ B ✐s ✐♥ H ❛♥❞ t❤❛t G ∩ B ✐s ❛♥♦t❤❡r ❜❛❧❧ ✇❤✐❝❤ ❤❛s ❝❡♥t❡r x 0 ❛♥❞ r❛❞✐✉s ρ ❛♥❞ ✐s t♦t❛❧❧② ❝♦♥t❛✐♥❡❞ ✐♥ H ✳ ❏♦❤❛♥ ❍❛❣❡♥❜❥ör❦ ❈❤❛♣t❡r ✸

❚❤❡ ❆✣♥❡ ❙❝❛❧✐♥❣ ▼❡t❤♦❞ ✶✺ ❙✐♥❝❡ A ✐s ♦❢ ❢✉❧❧ r❛♥❦✱ t❤❡ ♦rt❤♦❣♦♥❛❧ ♣r♦❥❡❝t✐♦♥ ♦❢ c T ✐♥t♦ t❤❡ s✉❜s♣❛❝❡ { x : Ax = 0 } ✐s Pc T ✇❤❡r❡ P = I − A T ( AA T ) − 1 A ✐s t❤❡ ♣r♦❥❡❝t✐♦♥ ♠❛tr✐①✳ ❏♦❤❛♥ ❍❛❣❡♥❜❥ör❦ ❈❤❛♣t❡r ✸