❆♥ ✐♠♣♦ss✐❜✐❧✐t② r❡s✉❧t ❢♦r s♣♦rt r❛♥❦✐♥❣s ▲ás③❧ó ❈s❛tó ❧❛s③❧♦✳❝s❛t♦❅✉♥✐✲❝♦r✈✐♥✉s✳❤✉ ■♥st✐t✉t❡ ❢♦r ❈♦♠♣✉t❡r ❙❝✐❡♥❝❡ ❛♥❞ ❈♦♥tr♦❧✱ ❍✉♥❣❛r✐❛♥ ❆❝❛❞❡♠② ♦❢ ❙❝✐❡♥❝❡s ✭▼❚❆ ❙❩❚❆❑■✮ ▲❛❜♦r❛t♦r② ♦♥ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ▼❛♥❛❣❡♠❡♥t ■♥t❡❧❧✐❣❡♥❝❡✱ ❘❡s❡❛r❝❤ ●r♦✉♣ ♦❢ ❖♣❡r❛t✐♦♥s ❘❡s❡❛r❝❤ ❛♥❞ ❉❡❝✐s✐♦♥ ❙②st❡♠s ❈♦r✈✐♥✉s ❯♥✐✈❡rs✐t② ♦❢ ❇✉❞❛♣❡st ✭❇❈❊✮ ❉❡♣❛rt♠❡♥t ♦❢ ❖♣❡r❛t✐♦♥s ❘❡s❡❛r❝❤ ❛♥❞ ❆❝t✉❛r✐❛❧ ❙❝✐❡♥❝❡s ❇✉❞❛♣❡st✱ ❍✉♥❣❛r② ▼❛t❤❙♣♦rt ■♥t❡r♥❛t✐♦♥❛❧ ✷✵✶✼ ❈♦♥❢❡r❡♥❝❡ ❯♥✐✈❡rs✐t② ♦❢ P❛❞✉❛✱ P❛❞✉❛ ✭■t❛❧②✮ ✷✽ ❏✉♥❡ ✷✵✶✼
❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ ✶ ❘❛♥❦✐♥❣ ♣r♦❜❧❡♠s ✷ ❆①✐♦♠s ✸ ❆♥ ✐♠♣♦ss✐❜✐❧✐t② t❤❡♦r❡♠ ✹ ❍♦✇ t♦ ❛✈♦✐❞ ✐♠♣♦ss✐❜✐❧✐t②❄ ✺ ❉♦♠❛✐♥ r❡str✐❝t✐♦♥s ❲❡❛❦❡♥✐♥❣ ♦❢ t❤❡ ❛①✐♦♠s ❈♦♥❝❧✉s✐♦♥s ✻ ▲ás③❧ó ❈s❛tó ✭▼❚❆ ❙❩❚❆❑■✮ ❆♥ ✐♠♣♦ss✐❜✐❧✐t② r❡s✉❧t ❢♦r s♣♦rt r❛♥❦✐♥❣s ▼❛t❤❙♣♦rt ✷✵✶✼ ✷ ✴ ✷✵
■♥tr♦❞✉❝t✐♦♥ ✷✵✶✼ ❯❊❋❆ ❊✉r♦♣❡❛♥ ❯♥❞❡r✲✷✶ ❈❤❛♠♣✐♦♥s❤✐♣ ■✳ ❖♥❡ ♦❢ t❤❡ ❧❛st ♠❛t❝❤ ♦❢ t❤❡ ❣r♦✉♣ st❛❣❡ ✐♥ ●r♦✉♣ ❈ ♦♥ ✷✹ ❏✉♥❡✿ ●❡r♠❛♥② ✈s✳ ■t❛❧② ❇❡❢♦r❡ ●r♦✉♣ ●♦❛❧s ❢♦r ●♦❛❧s ❛❣❛✐♥st P♦✐♥ts ●❡r♠❛♥② ❈ ✺ ✵ ✻ ■t❛❧② ❈ ✸ ✸ ✸ ❙❧♦✈❛❦✐❛ ❆ ✻ ✸ ✻ ❲✐♥♥❡rs ♦❢ ●r♦✉♣s ❆✱ ❇ ❛♥❞ ❈ ❛s ✇❡❧❧ ❛s t❤❡ ❜❡st r✉♥♥❡r✲✉♣ q✉❛❧✐✜❡s ❢♦r t❤❡ s❡♠✐✜♥❛❧s✳ ❚✐❡✲❜r❡❛❦✐♥❣ r✉❧❡s✿ ◮ ●r♦✉♣✿ ♣♦✐♥ts ✐♥ ❤❡❛❞✲t♦✲❤❡❛❞ ♠❛t❝❤❡s ❛♠♦♥❣ t✐❡❞ t❡❛♠s ◮ ❇❡st r✉♥♥❡r✲✉♣✿ ♣♦✐♥ts ✴ ❣♦❛❧ ❞✐✛❡r❡♥❝❡ ✴ ❣♦❛❧s s❝♦r❡❞ ▲ás③❧ó ❈s❛tó ✭▼❚❆ ❙❩❚❆❑■✮ ❆♥ ✐♠♣♦ss✐❜✐❧✐t② r❡s✉❧t ❢♦r s♣♦rt r❛♥❦✐♥❣s ▼❛t❤❙♣♦rt ✷✵✶✼ ✸ ✴ ✷✵
■♥tr♦❞✉❝t✐♦♥ ✷✵✶✼ ❯❊❋❆ ❊✉r♦♣❡❛♥ ❯♥❞❡r✲✷✶ ❈❤❛♠♣✐♦♥s❤✐♣ ■■✳ ❲✐♥♥❡rs ♦❢ ●r♦✉♣s ❆✱ ❇ ❛♥❞ ❈ ❛s ✇❡❧❧ ❛s t❤❡ ❜❡st r✉♥♥❡r✲✉♣ q✉❛❧✐✜❡s ❢♦r s❡♠✐✜♥❛❧s✳ ❚✐❡✲❜r❡❛❦✐♥❣ r✉❧❡s✿ ◮ ●r♦✉♣✿ ♣♦✐♥ts ✐♥ ❤❡❛❞✲t♦✲❤❡❛❞ ♠❛t❝❤❡s ❛♠♦♥❣ t✐❡❞ t❡❛♠s ◮ ❇❡st r✉♥♥❡r✲✉♣✿ ♣♦✐♥ts ✴ ❣♦❛❧ ❞✐✛❡r❡♥❝❡ ✴ ❣♦❛❧s s❝♦r❡❞ ❙✉♣♣♦s❡ t❤❛t ●❡r♠❛♥② ✈s✳ ■t❛❧② ✵✲✶ ❆❢t❡r ●r♦✉♣ ●♦❛❧s ❢♦r ●♦❛❧s ❛❣❛✐♥st P♦✐♥ts ■t❛❧② ❈ ✹ ✸ ✻ ●❡r♠❛♥② ❈ ✺ ✶ ✻ ❙❧♦✈❛❦✐❛ ❆ ✻ ✸ ✻ ❇✉t ✐❢ ❣♦❛❧ ❞✐✛❡r❡♥❝❡ ✇♦✉❧❞ ❜❡ t❤❡ ❣r♦✉♣ t✐❡✲❜r❡❛❦✐♥❣ r✉❧❡✱ t❤❡♥ ❙❧♦✈❛❦✐❛ ✇♦✉❧❞ q✉❛❧✐❢② ❢♦r t❤❡ s❡♠✐✜♥❛❧✳ ▲ás③❧ó ❈s❛tó ✭▼❚❆ ❙❩❚❆❑■✮ ❆♥ ✐♠♣♦ss✐❜✐❧✐t② r❡s✉❧t ❢♦r s♣♦rt r❛♥❦✐♥❣s ▼❛t❤❙♣♦rt ✷✵✶✼ ✹ ✴ ✷✵
■♥tr♦❞✉❝t✐♦♥ Pr♦❜❧❡♠✱ ♠♦t✐✈❛t✐♦♥ ❘❛♥❦✐♥❣ ❜❛s❡❞ ♦♥ ♣❛✐r❡❞ ❝♦♠♣❛r✐s♦♥s ◮ ❙♣♦rt t♦✉r♥❛♠❡♥ts ◮ Ps②❝❤♦❧♦❣② ◮ ❏♦✉r♥❛❧ ❝✐t❛t✐♦♥s ◮ ❱♦t✐♥❣ ♦♥ ❛❧t❡r♥❛t✐✈❡s ●❡♥❡r❛❧✐③❡❞ t♦✉r♥❛♠❡♥t ◮ ❚✐❡s ❛♥❞ ❞✐✛❡r❡♥t ♠❛r❣✐♥s ♦❢ ✈✐❝t♦r② ✭♣r❡❢❡r❡♥❝❡ ✐♥t❡♥s✐t✐❡s✮ ◮ ❆r❜✐tr❛r② ♥✉♠❜❡r ♦❢ ♠❛t❝❤❡s ❜❡t✇❡❡♥ ♣❧❛②❡rs✿ ✐♥❝♦♠♣❧❡t❡ ✭♠✐ss✐♥❣✮ ❛♥❞ ✐♠❜❛❧❛♥❝❡❞ ❞❛t❛ ❆✐♠s ◮ ❇❡tt❡r ✉♥❞❡rst❛♥❞✐♥❣ ♦❢ r❛♥❦✐♥❣ t❤r♦✉❣❤ ❛♥ ❛①✐♦♠❛t✐❝ ❛♣♣r♦❛❝❤ ◮ ❊①♣❧♦r❡ tr❛❞❡✲♦✛s ❛♠♦♥❣ ❞✐✛❡r❡♥t ♣r♦♣❡rt✐❡s ▲ás③❧ó ❈s❛tó ✭▼❚❆ ❙❩❚❆❑■✮ ❆♥ ✐♠♣♦ss✐❜✐❧✐t② r❡s✉❧t ❢♦r s♣♦rt r❛♥❦✐♥❣s ▼❛t❤❙♣♦rt ✷✵✶✼ ✺ ✴ ✷✵
■♥tr♦❞✉❝t✐♦♥ ◆❡✇❝♦♠❜❡ ❉❥♦❦♦✈✐❝ ◆❛st❛s❡ ◆❛❞❛❧ ❈♦♥♥♦rs ❘♦❞❞✐❝❦ ❇♦r❣ ❋❡❞❡r❡r ❋❡❞❡r❡r ❖♣♣♦♥❡♥t ❲✐♥ ▲♦ss ▼❝❊♥r♦❡ ❆❣❛ss✐ ✽ ✸ ❍❡✇✐tt ❉❥♦❦♦✈✐❝ ✶✻ ✶✺ ▲❡♥❞❧ ❋❡rr❡r♦ ✶✵ ✸ ❍❡✇✐tt ✶✽ ✽ ❋❡rr❡r♦ ❑❛❢❡❧♥✐❦♦✈ ✷ ✹ ❲✐❧❛♥❞❡r ❑✉❡rt❡♥ ✶ ✷ ▼♦②❛ ✼ ✵ ❙❛✜♥ ◆❛❞❛❧ ✶✵ ✷✷ ❊❞❜❡r❣ ❘❛❢t❡r ✵ ✸ ❘✐♦s ✷ ✵ ❑✉❡rt❡♥ ❘♦❞❞✐❝❦ ✷✶ ✸ ❇❡❝❦❡r ❙❛✜♥ ✶✵ ✷ ▼♦②❛ ❙❛♠♣r❛s ✶ ✵ ▼✉st❡r ❙✉♠ ✶✵✻ ✻✺ ❘✐♦s ❆❣❛ss✐ ❑❛❢❡❧♥✐❦♦✈ ❈♦✉r✐❡r ❘❛❢t❡r ❙❛♠♣r❛s ▲ás③❧ó ❈s❛tó ✭▼❚❆ ❙❩❚❆❑■✮ ❆♥ ✐♠♣♦ss✐❜✐❧✐t② r❡s✉❧t ❢♦r s♣♦rt r❛♥❦✐♥❣s ▼❛t❤❙♣♦rt ✷✵✶✼ ✻ ✴ ✷✵
❘❛♥❦✐♥❣ ♣r♦❜❧❡♠s ▼♦❞❡❧ s❡tt✐♥❣ ●❡♥❡r❛❧ r❛♥❦✐♥❣ ♣r♦❜❧❡♠ ( N , ❚ ) = ( N , T ( ✶ ) , T ( ✷ ) , . . . , T ( m ) ) ◮ ❙❡t ♦❢ ♣❧❛②❡rs✿ N = { X ✶ , X ✷ , . . . , X n } ◮ ❚♦✉r♥❛♠❡♥t ♠❛tr✐① ♦❢ r♦✉♥❞ p ✿ T ( p ) ∈ R n × n ✭ p = ✶ , ✷ , . . . , m ✮ ✶ t ( p ) + t ( p ) = ✶ ✐❢ ♣❧❛②❡rs X i ❛♥❞ X j ❤❛✈❡ ♣❧❛②❡❞ ✐♥ r♦✉♥❞ p ij ji ✷ t ( p ) = t ( p ) = ✵ ♦t❤❡r✇✐s❡ ij ji ◮ ❆❣❣r❡❣❛t❡❞ t♦✉r♥❛♠❡♥t ♠❛tr✐①✿ A = � m p = ✶ T ( p ) ❘❛♥❦✐♥❣ ♣r♦❜❧❡♠ ( N , A ) ♦r ( N , R , M ) ◮ ▼❛t❝❤❡s ♠❛tr✐① M = A + A ⊤ ✿ s②♠♠❡tr✐❝✱ m ii = ✵ ❢♦r ❛❧❧ X i p = ✶ t ( p ) + t ( p ) m ij = a ij + a ji = � m ij ji m ij = m ji ∈ N ✐s t❤❡ ♥✉♠❜❡r ♦❢ ♠❛t❝❤❡s ❜❡t✇❡❡♥ X i ❛♥❞ X j ◮ ◆✉♠❜❡r ♦❢ ♠❛t❝❤❡s ♦❢ ♣❧❛②❡r X i ✿ d i = � X j ∈ N m ij ◮ ❘❡s✉❧ts ♠❛tr✐① R = T − T ⊤ ✿ s❦❡✇✲s②♠♠❡tr✐❝✱ r ii = ✵ ❢♦r ❛❧❧ X i r ji = − r ij ❛♥❞ r ij ∈ [ − m ij , m ij ] ▲ás③❧ó ❈s❛tó ✭▼❚❆ ❙❩❚❆❑■✮ ❆♥ ✐♠♣♦ss✐❜✐❧✐t② r❡s✉❧t ❢♦r s♣♦rt r❛♥❦✐♥❣s ▼❛t❤❙♣♦rt ✷✵✶✼ ✼ ✴ ✷✵
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