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pr s p s t r r s

Prs Ps - PowerPoint PPT Presentation

Oct 12, 2022 •261 likes •2.08k views

Prs Ps trr s r ss Prs P

  1. ❲❀❛t ❞♩ ■ ✇♩r❊ ♊♥❄ ❇✐❧❡✈❡❧ Pr♊❜❧❡♠s✱ â–ŒP❈❈s✱ ❛♥❞ ▌✉❧t✐✲▲❡❛❞❡r✲❋♊❧❧♊✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

  2. ❲❀❛t ❞♩ ■ ✇♩r❊ ♊♥❄ ❇✐❧❡✈❡❧ Pr♊❜❧❡♠s✱ â–ŒP❈❈s✱ ❛♥❞ ▌✉❧t✐✲▲❡❛❞❡r✲❋♊❧❧♊✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

  3. ❲❀❛t ❞♩ ■ ✇♩r❊ ♊♥❄ ❇✐❧❡✈❡❧ Pr♊❜❧❡♠s✱ â–ŒP❈❈s✱ ❛♥❞ ▌✉❧t✐✲▲❡❛❞❡r✲❋♊❧❧♊✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

  4. ❲❀❛t ❞♩ ■ ✇♩r❊ ♊♥❄ ❇✐❧❡✈❡❧ Pr♊❜❧❡♠s✱ â–ŒP❈❈s✱ ❛♥❞ ▌✉❧t✐✲▲❡❛❞❡r✲❋♊❧❧♊✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

  5. ❛♥ ❛❞✈❡rt✐s❡♠❡♥t ❆ s❀♊rt st❛t❡ ♊❢ ❛rt ♊♥ ▌✉❧t✐✲▲❡❛❞❡r✲❋♊❧❧♊✇❡r ❣❛♠❡s ✱ ❉✳❆✳ ❛♥❞ ❆✳ ❙✈❡♥ss♊♥✱ ✐♥ ❛ ❜♩♩❩ ❞❡❞✐❝❛t❡❞ t♩ ❙t❛❝❊❡❧❜❡r❣✱ ❡❞✐t♩rs ❆✳ ❩❡♠❊♊❀♊ ❛♥❞ ❙✳ ❉❡♠♣❡✱ ❙♣r✐♥❣❡r ❊❞✳ ✭✷✵✶✟✮ ❇✐❧❡✈❡❧ Pr♊❜❧❡♠s✱ â–ŒP❈❈s✱ ❛♥❞ ▌✉❧t✐✲▲❡❛❞❡r✲❋♊❧❧♊✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

  6. ❇✐❧❡✈❡❧✿ s♊♠❡ ❣❡♥❡r❛❧ ❝♊♠♠❡♥ts ❇✐❧❡✈❡❧ Pr♊❜❧❡♠s✱ â–ŒP❈❈s✱ ❛♥❞ ▌✉❧t✐✲▲❡❛❞❡r✲❋♊❧❧♊✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

  7. ❇▲✿ ❛ ✜rst ❞❡✜♥✐t✐♊♥ ❆ ❇✐❧❡✈❡❧ Pr♊❜❧❡♠ ❝♊♥s✐sts ✐♥ ❛♥ ✉♣♣❡r✲❧❡✈❡❧✎❧❡❛❞❡r✬s ♣r♊❜❧❡♠ ✏ min x ∈ R n ✑ F ( x, y ) ï¿œ x ∈ X s✳t✳ y ∈ S ( x ) ✇❀❡r❡ ∅ ï¿œ = X ⊂ R n ❛♥❞ S ( x ) st❛♥❞s ❢♊r t❀❡ s♩❧✉t✐♊♥ s❡t ♊❢ ✐ts ❧♊✇❡r✲❧❡✈❡❧✎❢♊❧❧♊✇❡r✬s ♣r♊❜❧❡♠ min y ∈ R m f ( x, y ) s✳t g ( x, y ) ≀ 0 ❇✐❧❡✈❡❧ Pr♊❜❧❡♠s✱ â–ŒP❈❈s✱ ❛♥❞ ▌✉❧t✐✲▲❡❛❞❡r✲❋♊❧❧♊✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

  8. ❆ tr✐✈✐❛❧ ❡①❛♠♣❧❡ ❈♊♥s✐❞❡r t❀❡ ❢♊❧❧♊✇✐♥❣ s✐♠♣❧❡ ❜✐❧❡✈❡❧ ♣r♊❜❧❡♠ ✏ min x ∈ R ✑ x ï¿œ x ∈ [ − 1 , 1] s✳t✳ y ∈ S ( x ) ✇✐t❀ S ( x ) = ✏ y s♊❧✈✐♥❣ min y ∈ R − xy x 2 ( y 2 − 1) ≀ 0 ✑ s✳t ❇✐❧❡✈❡❧ Pr♊❜❧❡♠s✱ â–ŒP❈❈s✱ ❛♥❞ ▌✉❧t✐✲▲❡❛❞❡r✲❋♊❧❧♊✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

  9. ❆ tr✐✈✐❛❧ ❡①❛♠♣❧❡ ▲♊✇❡r ❧❡✈❡❧ ♣r♊❜❧❡♠✿ min y ∈ R − x.y x 2 ( y 2 − 1) ≀ 0 ✑ s✳t ◆♩t❡ t❀❛t t❀❡ s♩❧✉t✐♊♥ ♠❛♣ ♊❢ t❀✐s ❝♊♥✈❡① ♣r♊❜❧❡♠ ✐s  { 1 } x < 0  {− 1 } x > 0 S ( x ) := x = 0 R  ❚❀✉s ❢♊r ❡❛❝❀ x ï¿œ = 0 t❀❡r❡ ✐s ❛ ✉♥✐q✉❡ ❛ss♩❝✐❛t❡❞ s♩❧✉t✐♊♥ ♊❢ t❀❡ ❧♊✇❡r ❧❡✈❡❧ ♣r♊❜❧❡♠ ❇✐❧❡✈❡❧ Pr♊❜❧❡♠s✱ â–ŒP❈❈s✱ ❛♥❞ ▌✉❧t✐✲▲❡❛❞❡r✲❋♊❧❧♊✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

  10. ❆ tr✐✈✐❛❧ ❡①❛♠♣❧❡ ▲♊✇❡r ❧❡✈❡❧ ♣r♊❜❧❡♠✿ min y ∈ R − xy x 2 ( y 2 − 1) ≀ 0 ✑ s✳t ◆♩t❡ t❀❛t t❀❡ s♩❧✉t✐♊♥ ♠❛♣ ♊❢ t❀✐s ❝♊♥✈❡① ♣r♊❜❧❡♠ ✐s ② 1 ① − 1 −∇ F ❇✐❧❡✈❡❧ Pr♊❜❧❡♠s✱ â–ŒP❈❈s✱ ❛♥❞ ▌✉❧t✐✲▲❡❛❞❡r✲❋♊❧❧♊✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

  11. ❆ tr✐✈✐❛❧ ❡①❛♠♣❧❡ ❈♊♥s✐❞❡r t❀❡ ❢♊❧❧♊✇✐♥❣ s✐♠♣❧❡ ❜✐❧❡✈❡❧ ♣r♊❜❧❡♠ ✏ min x ∈ R ✑ − x.y ï¿œ x ∈ [ − 1 , 1] s✳t✳ y ∈ S ( x ) ✇✐t❀ S ( x ) = ✏ y s♊❧✈✐♥❣  { 1 } x < 0  S ( x ) := {− 1 } x > 0 x = 0  R ❇✐❧❡✈❡❧ Pr♊❜❧❡♠s✱ â–ŒP❈❈s✱ ❛♥❞ ▌✉❧t✐✲▲❡❛❞❡r✲❋♊❧❧♊✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

  12. ❆♠❜✐❣✉✐t②✿ ❖♣t✐♠✐st✐❝ ❛♣♣r♊❛❝❀ ❆♥ ❖♣t✐♠✐st✐❝ ❇✐❧❡✈❡❧ Pr♊❜❧❡♠ ❝♊♥s✐sts ✐♥ ❛♥ ✉♣♣❡r✲❧❡✈❡❧✎❧❡❛❞❡r✬s ♣r♊❜❧❡♠ min x ∈ R n min y ∈ R m F ( x, y ) ï¿œ x ∈ X s✳t✳ y ∈ S ( x ) ✇❀❡r❡ ∅ ï¿œ = X ⊂ R n ❛♥❞ S ( x ) st❛♥❞s ❢♊r t❀❡ s♩❧✉t✐♊♥ s❡t ♊❢ ✐ts ❧♊✇❡r✲❧❡✈❡❧✎❢♊❧❧♊✇❡r✬s ♣r♊❜❧❡♠ min y ∈ R m f ( x, y ) s✳t g ( x, y ) ≀ 0 ❇✐❧❡✈❡❧ Pr♊❜❧❡♠s✱ â–ŒP❈❈s✱ ❛♥❞ ▌✉❧t✐✲▲❡❛❞❡r✲❋♊❧❧♊✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

  13. ❆♠❜✐❣✉✐t②✿ P❡ss✐♠✐st✐❝ ❛♣♣r♊❛❝❀ ❆♥ P❡ss✐♠✐st✐❝ ❇✐❧❡✈❡❧ Pr♊❜❧❡♠ ❝♊♥s✐sts ✐♥ ❛♥ ✉♣♣❡r✲❧❡✈❡❧✎❧❡❛❞❡r✬s ♣r♊❜❧❡♠ min x ∈ R n max y ∈ R m F ( x, y ) ï¿œ x ∈ X s✳t✳ y ∈ S ( x ) ✇❀❡r❡ ∅ ï¿œ = X ⊂ R n ❛♥❞ S ( x ) st❛♥❞s ❢♊r t❀❡ s♩❧✉t✐♊♥ s❡t ♊❢ ✐ts ❧♊✇❡r✲❧❡✈❡❧✎❢♊❧❧♊✇❡r✬s ♣r♊❜❧❡♠ min y ∈ R m f ( x, y ) s✳t g ( x, y ) ≀ 0 ❇✐❧❡✈❡❧ Pr♊❜❧❡♠s✱ â–ŒP❈❈s✱ ❛♥❞ ▌✉❧t✐✲▲❡❛❞❡r✲❋♊❧❧♊✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

  14. ❋♩r ❡①❛♠♣❧❡ ✇❀❡♥ ❢♊r ❛♥② ✐s q✉❛s✐❝♊♥✈❡① ❛♥❞ ✐s str✐❝t❧② ❝♊♥✈❡①✳ ❆♠❜✐❣✉✐t②✿ t❀❡ ♠♩st s✐♠♣❧❡ ❆♥❞ ♊❢ ❝♩✉rs❡ t❀❡ ✧❝♊♥❢♊rt❛❜❧❡ s✐t✉❛t✐♊♥✧ ❝♩rr❡s♣♊♥❞s t♩ t❀❡ ❝❛s❡ ♊❢ ❛ ✉♥✐q✉❡ r❡s♣♊♥s❡ ∀ x ∈ X, S ( x ) = { y ( x ) } . ❚❀❡♥ min x ∈ R n F ( x, y ( x )) ï¿œ x ∈ X s✳t✳ ❇✐❧❡✈❡❧ Pr♊❜❧❡♠s✱ â–ŒP❈❈s✱ ❛♥❞ ▌✉❧t✐✲▲❡❛❞❡r✲❋♊❧❧♊✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

  15. ❆♠❜✐❣✉✐t②✿ t❀❡ ♠♩st s✐♠♣❧❡ ❆♥❞ ♊❢ ❝♩✉rs❡ t❀❡ ✧❝♊♥❢♊rt❛❜❧❡ s✐t✉❛t✐♊♥✧ ❝♩rr❡s♣♊♥❞s t♩ t❀❡ ❝❛s❡ ♊❢ ❛ ✉♥✐q✉❡ r❡s♣♊♥s❡ ∀ x ∈ X, S ( x ) = { y ( x ) } . ❚❀❡♥ min x ∈ R n F ( x, y ( x )) ï¿œ x ∈ X s✳t✳ ❋♩r ❡①❛♠♣❧❡ ✇❀❡♥ ❢♊r ❛♥② x, g ( x, · ) ✐s q✉❛s✐❝♊♥✈❡① ❛♥❞ f ( x, · ) ✐s str✐❝t❧② ❝♊♥✈❡①✳ ❇✐❧❡✈❡❧ Pr♊❜❧❡♠s✱ â–ŒP❈❈s✱ ❛♥❞ ▌✉❧t✐✲▲❡❛❞❡r✲❋♊❧❧♊✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

  16. ❆♠❜✐❣✉✐t②✿ ❙❡❧❡❝t✐♊♥ ❛♣♣r♊❛❝❀ ❆♥ ✧❙❡❧❡❝t✐♊♥✲t②♣❡✧ ❇✐❧❡✈❡❧ Pr♊❜❧❡♠ ❝♊♥s✐sts ✐♥ ❛♥ ✉♣♣❡r✲❧❡✈❡❧✎❧❡❛❞❡r✬s ♣r♊❜❧❡♠ min x ∈ R n F ( x, y ( x )) ï¿œ x ∈ X s✳t✳ y ( x ) ✐s ❛ ✉♥✐q✉❡❧② ❞❡t❡r♠✐♥❡❞ s❡❧❡❝t✐♊♥ ♊❢ S ( x ) ❏✳ ❊s❝♩❜❛r ✫ ❆✳ ❏♊❢ré✱ ❊q✉✐❧✐❜r✐✉♠ ❆♥❛❧②s✐s ♊❢ ❊❧❡❝tr✐❝✐t② ❆✉❝t✐♊♥s ✭✷✵✶✶✮ ❇✐❧❡✈❡❧ Pr♊❜❧❡♠s✱ â–ŒP❈❈s✱ ❛♥❞ ▌✉❧t✐✲▲❡❛❞❡r✲❋♊❧❧♊✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

  17. ❆♠❜✐❣✉✐t②✿ ❚❀❡ ♥❡✇ ♣r♩❜❛❜✐❧✐st✐❝ ❛♣♣r♊❛❝❀ ■♥ ♊♥❡ ♊❢ t❀❡ ❊❧❡✈❛t♩r ♣✐t❝❀❡s ✭▌♊♥❞❛②✮✱ ❉✳❙❛❧❛s ❛♥❞ ❆✳ ❙✈❡♥ss♊♥ ♣r♊♣♊s❡❞ ❛ ♣r♩❜❛❜✐❧✐st✐❝ ❛♣♣r♊❛❝❀ ✿ ❈♊♥s✐❞❡r ❛ ♣r♩❜❛❜✐❧✐t② ♊♥ t❀❡ ❞✐✛❡r❡♥t ♣♊ss✐❜❧❡ ❢♊❧❧♊✇❡r✬s r❡❛❝t✐♊♥s ▌✐♥✐♠✐③❡ t❀❡ ❡①♣❡❝t❛t✐♊♥ ♊❢ t❀❡ ❧❡❛❞❡r✭s✮ ❇✐❧❡✈❡❧ Pr♊❜❧❡♠s✱ â–ŒP❈❈s✱ ❛♥❞ ▌✉❧t✐✲▲❡❛❞❡r✲❋♊❧❧♊✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

  18. ♊♥❡ ❝❛♥ ❞❡✜♥❡ t❀❡ ✭♊♣t✐♠✐st✐❝✮ ✈❛❧✉❡ ❢✉♥❝t✐♊♥ ✭✶✮ ❛♥❞ t❀❡ ❇❧ ♣r♊❜❧❡♠ ❜❡❝♊♠❡s s✳t✳ ❆♥ ❛❧t❡r♥❛t✐✈❡ ♣♊✐♥t ♊❢ ✈✐❡✇ ■♥st❡❛❞ ♊❢ ❝♊♥s✐❞❡r✐♥❣ t❀❡ ♣r❡✈✐♊✉s ✭♊♣t✐♠✐st✐❝✮ ❢♊r♠✉❧❛t✐♊♥ ♊❢ ❇▲✿ min x ∈ R n min y ∈ R m F ( x, y ) ï¿œ x ∈ X s✳t✳ y ∈ S ( x ) ❇✐❧❡✈❡❧ Pr♊❜❧❡♠s✱ â–ŒP❈❈s✱ ❛♥❞ ▌✉❧t✐✲▲❡❛❞❡r✲❋♊❧❧♊✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

  19. ❆♥ ❛❧t❡r♥❛t✐✈❡ ♣♊✐♥t ♊❢ ✈✐❡✇ ■♥st❡❛❞ ♊❢ ❝♊♥s✐❞❡r✐♥❣ t❀❡ ♣r❡✈✐♊✉s ✭♊♣t✐♠✐st✐❝✮ ❢♊r♠✉❧❛t✐♊♥ ♊❢ ❇▲✿ min x ∈ R n min y ∈ R m F ( x, y ) ï¿œ x ∈ X s✳t✳ y ∈ S ( x ) ♊♥❡ ❝❛♥ ❞❡✜♥❡ t❀❡ ✭♊♣t✐♠✐st✐❝✮ ✈❛❧✉❡ ❢✉♥❝t✐♊♥ y { F ( x, y ) : g ( x, y ) ≀ 0 } ϕ min ( x ) = min ✭✶✮ ❛♥❞ t❀❡ ❇❧ ♣r♊❜❧❡♠ ❜❡❝♊♠❡s min x ∈ R n ϕ min ( x ) x ∈ X s✳t✳ ❇✐❧❡✈❡❧ Pr♊❜❧❡♠s✱ â–ŒP❈❈s✱ ❛♥❞ ▌✉❧t✐✲▲❡❛❞❡r✲❋♊❧❧♊✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

  20. ♊♥❡ ❝❛♥ ❞❡✜♥❡ t❀❡ ✭♣❡ss✐♠✐st✐❝✮ ✈❛❧✉❡ ❢✉♥❝t✐♊♥ ✭✷✮ ❛♥❞ t❀❡ ❇❧ ♣r♊❜❧❡♠ ❜❡❝♊♠❡s s✳t✳ ❆♥ ❛❧t❡r♥❛t✐✈❡ ♣♊✐♥t ♊❢ ✈✐❡✇ ■♥st❡❛❞ ♊❢ ❝♊♥s✐❞❡r✐♥❣ t❀❡ ♣r❡✈✐♊✉s ✭♣❡ss✐♠✐st✐❝✮ ❢♊r♠✉❧❛t✐♊♥ ♊❢ ❇▲✿ min x ∈ R n max y ∈ R m F ( x, y ) ï¿œ x ∈ X s✳t✳ y ∈ S ( x ) ❇✐❧❡✈❡❧ Pr♊❜❧❡♠s✱ â–ŒP❈❈s✱ ❛♥❞ ▌✉❧t✐✲▲❡❛❞❡r✲❋♊❧❧♊✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

  21. ❆♥ ❛❧t❡r♥❛t✐✈❡ ♣♊✐♥t ♊❢ ✈✐❡✇ ■♥st❡❛❞ ♊❢ ❝♊♥s✐❞❡r✐♥❣ t❀❡ ♣r❡✈✐♊✉s ✭♣❡ss✐♠✐st✐❝✮ ❢♊r♠✉❧❛t✐♊♥ ♊❢ ❇▲✿ min x ∈ R n max y ∈ R m F ( x, y ) ï¿œ x ∈ X s✳t✳ y ∈ S ( x ) ♊♥❡ ❝❛♥ ❞❡✜♥❡ t❀❡ ✭♣❡ss✐♠✐st✐❝✮ ✈❛❧✉❡ ❢✉♥❝t✐♊♥ y { F ( x, y ) : g ( x, y ) ≀ 0 } ϕ max ( x ) = max ✭✷✮ ❛♥❞ t❀❡ ❇❧ ♣r♊❜❧❡♠ ❜❡❝♊♠❡s min x ∈ R n ϕ max ( x ) x ∈ X s✳t✳ ❇✐❧❡✈❡❧ Pr♊❜❧❡♠s✱ â–ŒP❈❈s✱ ❛♥❞ ▌✉❧t✐✲▲❡❛❞❡r✲❋♊❧❧♊✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

  22. ■t ✐♠♠❡❞✐❛t❡❧② r❛✐s❡s t❀❡ q✉❡st✐♊♥ ❲❀❛t ✐s ❛ s♩❧✉t✐♊♥❄❄ ❛♥ ♊♣t✐♠❛❧ ❂ ❧❡❛❞❡r✬s ♊♣t✐♠❛❧ str❛t❡❣②❄ ❛♥ ♊♣t✐♠❛❧ ❝♊✉♣❧❡ ❂ ❝♊✉♣❧❡ ♊❢ str❛t❡❣✐❡s ♊❢ ❧❡❛❞❡r ❛♥❞ ❢♊❧❧♊✇❡r❄ ❆♥ ❛❧t❡r♥❛t✐✈❡ ♣♊✐♥t ♊❢ ✈✐❡✇ ❚❀✐s ✐s t❀❡ ♣♊✐♥t ♊❢ ✈✐❡✇ ♣r❡s❡♥t❡❞ ✐♥ ❙t❡♣❀❛♥ ❉❡♠♣❡✬s ❜♊♊❊✿ min x ∈ R n min / max y ∈ R m F ( x, y ) min x ∈ R n ϕ min/max ( x ) ï¿œ x ∈ X ✈s s✳t✳ s✳t✳ x ∈ X y ∈ S ( x ) ❇✐❧❡✈❡❧ Pr♊❜❧❡♠s✱ â–ŒP❈❈s✱ ❛♥❞ ▌✉❧t✐✲▲❡❛❞❡r✲❋♊❧❧♊✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

  23. ❛♥ ♊♣t✐♠❛❧ ❂ ❧❡❛❞❡r✬s ♊♣t✐♠❛❧ str❛t❡❣②❄ ❛♥ ♊♣t✐♠❛❧ ❝♊✉♣❧❡ ❂ ❝♊✉♣❧❡ ♊❢ str❛t❡❣✐❡s ♊❢ ❧❡❛❞❡r ❛♥❞ ❢♊❧❧♊✇❡r❄ ❆♥ ❛❧t❡r♥❛t✐✈❡ ♣♊✐♥t ♊❢ ✈✐❡✇ ❚❀✐s ✐s t❀❡ ♣♊✐♥t ♊❢ ✈✐❡✇ ♣r❡s❡♥t❡❞ ✐♥ ❙t❡♣❀❛♥ ❉❡♠♣❡✬s ❜♊♊❊✿ min x ∈ R n min / max y ∈ R m F ( x, y ) min x ∈ R n ϕ min/max ( x ) ï¿œ x ∈ X ✈s s✳t✳ s✳t✳ x ∈ X y ∈ S ( x ) ■t ✐♠♠❡❞✐❛t❡❧② r❛✐s❡s t❀❡ q✉❡st✐♊♥ ❲❀❛t ✐s ❛ s♩❧✉t✐♊♥❄❄ ❇✐❧❡✈❡❧ Pr♊❜❧❡♠s✱ â–ŒP❈❈s✱ ❛♥❞ ▌✉❧t✐✲▲❡❛❞❡r✲❋♊❧❧♊✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

  24. ❆♥ ❛❧t❡r♥❛t✐✈❡ ♣♊✐♥t ♊❢ ✈✐❡✇ ❚❀✐s ✐s t❀❡ ♣♊✐♥t ♊❢ ✈✐❡✇ ♣r❡s❡♥t❡❞ ✐♥ ❙t❡♣❀❛♥ ❉❡♠♣❡✬s ❜♊♊❊✿ min x ∈ R n min / max y ∈ R m F ( x, y ) min x ∈ R n ϕ min/max ( x ) ï¿œ x ∈ X ✈s s✳t✳ s✳t✳ x ∈ X y ∈ S ( x ) ■t ✐♠♠❡❞✐❛t❡❧② r❛✐s❡s t❀❡ q✉❡st✐♊♥ ❲❀❛t ✐s ❛ s♩❧✉t✐♊♥❄❄ ❛♥ ♊♣t✐♠❛❧ x ❂ ❧❡❛❞❡r✬s ♊♣t✐♠❛❧ str❛t❡❣②❄ ❛♥ ♊♣t✐♠❛❧ ❝♊✉♣❧❡ ( x, y ) ❂ ❝♊✉♣❧❡ ♊❢ str❛t❡❣✐❡s ♊❢ ❧❡❛❞❡r ❛♥❞ ❢♊❧❧♊✇❡r❄ ❇✐❧❡✈❡❧ Pr♊❜❧❡♠s✱ â–ŒP❈❈s✱ ❛♥❞ ▌✉❧t✐✲▲❡❛❞❡r✲❋♊❧❧♊✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

  25. ❘❡❛❧ ❧✐❢❡✳✳✳ ❆❝t✉❛❧❧② ✉s✉❛❧❧② ✇❀❡♥ ❝♊♥s✐❞❡r✐♥❣ ❇▲ min x ∈ R n min y ∈ R m F ( x, y ) ï¿œ x ∈ X s✳t✳ y ∈ S ( x ) ♣❡♊♣❧❡ s❛② ❙t❡♣ ❆✿ t❀❡ ❧❡❛❞❡r ♣❧❛②s ✜rst ❙t❡♣ ❇✿ t❀❡ ❢♊❧❧♊✇❡r r❡❛❝ts ❇✉t ✐♥ r❡❛❧ ❧✐❢❡ ✐t✬s ❛ ❧✐tt❧❡ ❜✐t ♠♩r❡ ❝♊♠♣❧❡①✳✳✳✳ ❇✐❧❡✈❡❧ Pr♊❜❧❡♠s✱ â–ŒP❈❈s✱ ❛♥❞ ▌✉❧t✐✲▲❡❛❞❡r✲❋♊❧❧♊✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

  26. ■♥❞❡❡❞ t❀❡ ❧❡❛❞❡r ❀❛s ❛ ♠♊❞❡❧ ♊❢ t❀❡ ❢♊❧❧♊✇❡r✬s r❡❛❝t✐♊♥✿ ♊♣t✐♠✐st✐❝ ♩r ♣❡ss✐♠✐st✐❝ ❛♥❞ ❙t❡♣ ✶✿ ✇❡ ❝♊♠♣✉t❡ ❛ s♩❧✉t✐♊♥ ♩r ♊❢ t❀❡ ❇▲ ♠♊❞❡❧ ❙t❡♣ ✷✿ t❀❡ ❧❡❛❞❡r ♣❧❛②s ❙t❡♣ ✞✿ t❀❡ ❢♊❧❧♊✇❡r ❞❡❝✐❞❡s t♩ ♣❧❛②✳✳✳✇❀❛t❡✈❡r ❀❡ ✇❛♥ts✩✩✩ ❘❡❛❧ ❧✐❢❡✳✳✳ ❆❝t✉❛❧❧② ✐♥ r❡❛❧ ❧✐❢❡✱ ✇❀❡♥ ❝♊♥s✐❞❡r✐♥❣ ❇▲ min x ∈ R n min y ∈ R m F ( x, y ) ï¿œ x ∈ X s✳t✳ y ∈ S ( x ) ❲❡ ♊♥❧② ✇♩r❊ ❢♊r t❀❡ ❧❡❛❞❡r✩✩ ❇✐❧❡✈❡❧ Pr♊❜❧❡♠s✱ â–ŒP❈❈s✱ ❛♥❞ ▌✉❧t✐✲▲❡❛❞❡r✲❋♊❧❧♊✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

  27. ❘❡❛❧ ❧✐❢❡✳✳✳ ❆❝t✉❛❧❧② ✐♥ r❡❛❧ ❧✐❢❡✱ ✇❀❡♥ ❝♊♥s✐❞❡r✐♥❣ ❇▲ min x ∈ R n min y ∈ R m F ( x, y ) ï¿œ x ∈ X s✳t✳ y ∈ S ( x ) ❲❡ ♊♥❧② ✇♩r❊ ❢♊r t❀❡ ❧❡❛❞❡r✩✩ ■♥❞❡❡❞ t❀❡ ❧❡❛❞❡r ❀❛s ❛ ♠♊❞❡❧ ♊❢ t❀❡ ❢♊❧❧♊✇❡r✬s r❡❛❝t✐♊♥✿ ♊♣t✐♠✐st✐❝ ♩r ♣❡ss✐♠✐st✐❝ ❛♥❞ ❙t❡♣ ✶✿ ✇❡ ❝♊♠♣✉t❡ ❛ s♩❧✉t✐♊♥ x ♩r ( x, y ) ♊❢ t❀❡ ❇▲ ♠♊❞❡❧ ❙t❡♣ ✷✿ t❀❡ ❧❡❛❞❡r ♣❧❛②s x ❙t❡♣ ✞✿ t❀❡ ❢♊❧❧♊✇❡r ❞❡❝✐❞❡s t♩ ♣❧❛②✳✳✳✇❀❛t❡✈❡r ❀❡ ✇❛♥ts✩✩✩ ❇✐❧❡✈❡❧ Pr♊❜❧❡♠s✱ â–ŒP❈❈s✱ ❛♥❞ ▌✉❧t✐✲▲❡❛❞❡r✲❋♊❧❧♊✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

  28. ❆♥ ❡①✐st❡♥❝❡ r❡s✉❧t ✭♊♣t✐♠✐st✐❝✮ ❉❡✜♥✐t✐♊♥ ❚❀❡ ▌❛♥❣❛s❛r✐❛♥✲❋r♩♠♩✈✐t③ ❝♊♥str❛✐♥t q✉❛❧✐✜❝❛t✐♊♥ ✭▌❋❈◗✮ ✐s s❛t✐s✜❡❞ ❛t ( x, y ) ✇✐t❀ y ❢❡❛s✐❜❧❡ ♣♊✐♥t ♊❢ t❀❡ ♣r♊❜❧❡♠ min y { f ( x, y ) : g ( x, y ) ≀ 0 } ✐❢ t❀❡ s②st❡♠ ∇ y g i ( x, y ) d < 0 ∀ i ∈ I ( x, y ) := { j : g j ( x, y ) = 0 } ❀❛s ❛ s♩❧✉t✐♊♥✳ ❇✐❧❡✈❡❧ Pr♊❜❧❡♠s✱ â–ŒP❈❈s✱ ❛♥❞ ▌✉❧t✐✲▲❡❛❞❡r✲❋♊❧❧♊✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

  29. ❆♥ ❡①✐st❡♥❝❡ r❡s✉❧t ✭❝♊♥t✳✮ ❆ss✉♠❡ t❀❛t X = { x ∈ R n : G ( x ) ≀ 0 } ❚❀❡♊r❡♠ ✭❇❛♥❊✱ ●✉❞❞❛t✱ ❑❧❛tt❡✱ ❑✉♠♠❡r✱ ❚❛♠♠❡r ✭✜✞✮✮ ▲❡t x ✇✐t❀ G ( x ) ≀ 0 ❜❡ ✜①❡❞✳ t❀❡ s❡t { ( x, y ) : g ( x, y ) ≀ 0 } ✐s ♥♊t ❡♠♣t② ❛♥❞ ❝♊♠♣❛❝t❀ ❛t ❡❛❝❀ ♣♊✐♥t ( x, y ) ∈ gph S ✇✐t❀ G ( x ) ≀ 0 ✱ ❛ss✉♠♣t✐♊♥ ✭▌❋❈◗✮ ✐s s❛t✐s✜❡❞❀ t❀❡♥✱ t❀❡ s❡t✲✈❛❧✉❡❞ ♠❛♣ S ( · ) ✐s ✉♣♣❡r s❡♠✐❝♊♥t✐♥✉♊✉s ❛t ( x, y ) ❛♥❞ t❀❡ ❢✉♥❝t✐♊♥ ϕ o ( · ) ✐s ❝♊♥t✐♥✉♊✉s ❛t x ✳ ❇✐❧❡✈❡❧ Pr♊❜❧❡♠s✱ â–ŒP❈❈s✱ ❛♥❞ ▌✉❧t✐✲▲❡❛❞❡r✲❋♊❧❧♊✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

  30. ❆♥ ❡①✐st❡♥❝❡ r❡s✉❧t ✭❝♊♥t✳✮ ❚❀❡♊r❡♠ ❆ss✉♠❡ t❀❛t t❀❡ s❡t { ( x, y ) : g ( x, y ) ≀ 0 } ✐s ♥♊t ❡♠♣t② ❛♥❞ ❝♊♠♣❛❝t❀ ❛t ❡❛❝❀ ♣♊✐♥t ( x, y ) ∈ gph S ✇✐t❀ G ( x ) ≀ 0 ✱ ❛ss✉♠♣t✐♊♥s ✭▌❋❈◗✮ ✐s s❛t✐s✜❡❞❀ t❀❡ s❡t { x : G ( x ) ≀ 0 } ✐s ♥♊t ❡♠♣t② ❛♥❞ ❝♊♠♣❛❝t✱ t❀❡♥ ♊♣t✐♠✐st✐❝ ❜✐❧❡✈❡❧ ♣r♊❜❧❡♠ ❀❛s ❛ ✭❣❧♊❜❛❧✮ ♊♣t✐♠❛❧ s♩❧✉t✐♊♥✳ ❇✐❧❡✈❡❧ Pr♊❜❧❡♠s✱ â–ŒP❈❈s✱ ❛♥❞ ▌✉❧t✐✲▲❡❛❞❡r✲❋♊❧❧♊✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

  31. ❇✐❧❡✈❡❧ ♣r♊❜❧❡♠s ❛♥❞ â–ŒP❈❈ r❡❢♊r♠✉❧❛t✐♊♥ ❇✐❧❡✈❡❧ Pr♊❜❧❡♠s✱ â–ŒP❈❈s✱ ❛♥❞ ▌✉❧t✐✲▲❡❛❞❡r✲❋♊❧❧♊✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

  32. ❲❡ ❝♊♥s✐❞❡r ❛ ❇✐❧❡✈❡❧ Pr♊❜❧❡♠ ❝♊♥s✐st✐♥❣ ✐♥ ❛♥ ✉♣♣❡r✲❧❡✈❡❧ ✮ ❧❡❛❞❡r✬s ♣r♊❜❧❡♠ ✏ min x ∈ R n ✑ F ( x, y ) s✳t✳ y ∈ S ( x ) , x ∈ X ✇❀❡r❡ ∅ ï¿œ = X ⊂ R n ✱ ❛♥❞ S ( x ) st❛♥❞s ❢♊r t❀❡ s♩❧✉t✐♊♥ ♊❢ ✐ts ❧♊✇❡r✲❧❡✈❡❧ ✮ ❢♊❧❧♊✇❡r✬s ♣r♊❜❧❡♠ y ∈ R m f ( x, y ) min s✳t g ( x, y ) ≀ 0 ✇❀✐❝❀ ✇❡ ❛ss✉♠❡ t♩ ❜❡ ❝♊♥✈❡① ❛♥❞ s♠♩♩t❀✱ ✐✳❡✳ ∀ x ∈ X, t❀❡ ❢✉♥❝t✐♊♥s f ( x, · ) ❛♥❞ g i ( x, · ) ❛r❡ s♠♩♩t❀ ❝♊♥✈❡① ❢✉♥❝t✐♊♥s✱ ❛♥❞ t❀❡ ❣r❛❞✐❡♥ts ∇ y g i , ∇ y f ❛r❡ ❝♊♥t✐♥✉♊✉s✱ i = 1 , ..., p ✳ ❇✐❧❡✈❡❧ Pr♊❜❧❡♠s✱ â–ŒP❈❈s✱ ❛♥❞ ▌✉❧t✐✲▲❡❛❞❡r✲❋♊❧❧♊✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

  33. â–ŒP❈❈ r❡❢♊r♠✉❧❛t✐♊♥ ❘❡♣❧❛❝✐♥❣ t❀❡ ❧♊✇❡r✲❧❡✈❡❧ ♣r♊❜❧❡♠ ❜② ✐ts ❑❑❚ ❝♊♥❞✐t✐♊♥s✱ ❣✐✈❡s ♣❧❛❝❡ t♩ ❛ ▌❛t❀❡♠❛t✐❝❛❧ Pr♊❣r❛♠ ✇✐t❀ ❈♊♠♣❧❡♠❡♥t❛r✐t② ❈♊♥str❛✐♥ts✳ ❇✐❧❡✈❡❧ â–ŒP❈❈ ✏ min x ∈ X ✑ F ( x, y ) ✏ min x ∈ X ✑ F ( x, y ) s✳t✳ y ∈ S ( x ) s✳t✳ ( y, u ) ∈ KKT ( x ) ✇✐t❀ S ( x ) = ✏ y s♊❧✈✐♥❣ ✇✐t❀ KKT ( x ) = ✏ ( y, u ) s♊❧✈✐♥❣ ∇ y f ( x, y ) + u T ∇ y g ( x, y ) = 0 y ∈ R m f ( x, y ) min ï¿œ 0 ≀ u ⊥ − g ( x, y ) ≥ 0 ✑ s✳t g ( x, y ) ≀ 0 ✑ ❲❡ ✇r✐t❡ Λ( x, y ) ❢♊r t❀❡ s❡t ♊❢ u s❛t✐s❢②✐♥❣ ( y, u ) ∈ KKT ( x ) ✳ ❇✐❧❡✈❡❧ Pr♊❜❧❡♠s✱ â–ŒP❈❈s✱ ❛♥❞ ▌✉❧t✐✲▲❡❛❞❡r✲❋♊❧❧♊✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

  34. ❊①❛♠♣❧❡ ✶ ❈♊♥s✐❞❡r t❀❡ ❢♊❧❧♊✇✐♥❣ ❇✐❧❡✈❡❧ ♣r♊❜❧❡♠ ❛♥❞ ✐ts â–ŒP❈❈ r❡❢♊r♠✉❧❛t✐♊♥ ❇✐❧❡✈❡❧ â–ŒP❈❈ ✏ min x ∈ [ − 1 , 1] ✑ x ✏ min x ∈ [ − 1 , 1] ✑ x s✳t✳ y ∈ S ( x ) s✳t✳ ( y, u ) ∈ KKT ( x ) ✇✐t❀ S ( x ) = ✏ y s♊❧✈✐♥❣ ✇✐t❀ KKT ( x ) = ✏ ( y, u ) s♊❧✈✐♥❣ ï¿œ x + u · 2 yx 2 = 0 min xy y ∈ R 0 ≀ u ⊥ − x 2 ( y 2 − 1) ≥ 0 ✑ s✳t x 2 ( y 2 − 1) ≀ 0 ✑ (0 , − 1 , u ) ✐s ❛ ❧♩❝❛❧ s♩❧✉t✐♊♥ ♊❢ ✏▌P❈❈✑✱ ❢♊r ❛♥② u ∈ Λ(0 , − 1) = R + ✶ (0 , − 1) ✐s ◆❖❚ ❛ ❧♩❝❛❧ s♩❧✉t✐♊♥ ♊❢ ✏❇✐❧❡✈❡❧✑ ✷ ❇✐❧❡✈❡❧ Pr♊❜❧❡♠s✱ â–ŒP❈❈s✱ ❛♥❞ ▌✉❧t✐✲▲❡❛❞❡r✲❋♊❧❧♊✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

  35. ✉ ② S ( · ) KKT ( · ) 1 ② ① − 1 ① −∇ F −∇ F ✭❛✮ (0 , − 1 , u ) ✐s ❛ ✭❜✮ (0 , − 1) ✐s♥✬t ❛ ❧♩❝❛❧ s♩❧✉t✐♊♥ ♊❢ ❧♩❝❛❧ s♩❧✉t✐♊♥ ♊❢ t❀❡ â–ŒP❈❈✱ ∀ u ∈ R + ✳ ❇✐❧❡✈❡❧ ♣r♊❜❧❡♠✳ ❇✐❧❡✈❡❧ Pr♊❜❧❡♠s✱ â–ŒP❈❈s✱ ❛♥❞ ▌✉❧t✐✲▲❡❛❞❡r✲❋♊❧❧♊✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

  36. ❚❀❡ ♊♣t✐♠✐st✐❝ â–ŒP❈❈ ✭❖▌P❈❈✮✿ s✳t✳ ❚❀❡ ♣❡ss✐♠✐st✐❝ â–ŒP❈❈ ✭Pâ–ŒP❈❈✮✿ s✳t✳ ❖♣t✐♠✐st✐❝ ❛♥❞ P❡ss✐♠✐st✐❝ ❛♣♣r♊❛❝❀❡s ❚❀❡ ♊♣t✐♠✐st✐❝ ❇✐❧❡✈❡❧ ✭❖❇✮ ✐s min x min F ( x, y ) y s✳t✳ y ∈ S ( x ) , x ∈ X. ❚❀❡ ♣❡ss✐♠✐st✐❝ ❇✐❧❡✈❡❧ ✭P❇✮ ✐s min x max F ( x, y ) y s✳t✳ y ∈ S ( x ) , x ∈ X. ❇✐❧❡✈❡❧ Pr♊❜❧❡♠s✱ â–ŒP❈❈s✱ ❛♥❞ ▌✉❧t✐✲▲❡❛❞❡r✲❋♊❧❧♊✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

  37. ❖♣t✐♠✐st✐❝ ❛♥❞ P❡ss✐♠✐st✐❝ ❛♣♣r♊❛❝❀❡s ❚❀❡ ♊♣t✐♠✐st✐❝ ❇✐❧❡✈❡❧ ✭❖❇✮ ✐s ❚❀❡ ♊♣t✐♠✐st✐❝ â–ŒP❈❈ ✭❖▌P❈❈✮✿ min x min F ( x, y ) min x min F ( x, y ) y y s✳t✳ y ∈ S ( x ) , x ∈ X. s✳t✳ ( y, u ) ∈ KKT ( x ) , x ∈ X. ❚❀❡ ♣❡ss✐♠✐st✐❝ ❇✐❧❡✈❡❧ ✭P❇✮ ✐s ❚❀❡ ♣❡ss✐♠✐st✐❝ â–ŒP❈❈ ✭Pâ–ŒP❈❈✮✿ min x max F ( x, y ) min x max F ( x, y ) y y s✳t✳ y ∈ S ( x ) , x ∈ X. s✳t✳ ( y, u ) ∈ KKT ( x ) , x ∈ X. ❇✐❧❡✈❡❧ Pr♊❜❧❡♠s✱ â–ŒP❈❈s✱ ❛♥❞ ▌✉❧t✐✲▲❡❛❞❡r✲❋♊❧❧♊✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

  38. ❖♣t✐♠✐st✐❝ ❛♣♣r♊❛❝❀ ■s ❜✐❧❡✈❡❧ ♣r♊❣r❛♠♠✐♥❣ ❛ s♣❡❝✐❛❧ ❝❛s❡ ♊❢ ❛ â–ŒP❈❈❄ ❙✳ ❉❡♠♣❡ ✲❏✳ ❉✉tt❛ ✭✷✵✶✷ ▌❛t❀✳ Pr♊❣✳✮ min x min F ( x, y ) y s✳t✳ y ∈ S ( x ) , x ∈ X. ❇✐❧❡✈❡❧ Pr♊❜❧❡♠s✱ â–ŒP❈❈s✱ ❛♥❞ ▌✉❧t✐✲▲❡❛❞❡r✲❋♊❧❧♊✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

  39. ▲♊❝❛❧ s♩❧✉t✐♊♥s ❢♊r ✐♥ ♊♣t✐♠✐st✐❝ ❛♣♣r♊❛❝❀ ❉❡✜♥✐t✐♊♥ ❆ ❧♩❝❛❧ ✭r❡s♣✳ ❣❧♊❜❛❧✮ s♩❧✉t✐♊♥ ♊❢ ✭❖❇✮ ✐s ❛ ♣♊✐♥t y ) ✭r❡s♣✳ U = R n × R m ✮ (¯ x, ¯ y ) ∈ Gr ( S ) ✐❢ t❀❡r❡ ❡①✐sts U ∈ N (¯ x, ¯ s✉❝❀ t❀❛t F (¯ x, ¯ y ) ≀ F ( x, y ) , ∀ ( x, y ) ∈ U ∩ Gr ( S ) . ❉❡✜♥✐t✐♊♥ ❆ ❧♩❝❛❧ ✭r❡s♣✳ ❣❧♊❜❛❧✮ s♩❧✉t✐♊♥ ❢♊r ✭❖▌P❈❈✮ ✐s ❛ tr✐♣❧❡t (¯ x, ¯ y, ¯ u ) ∈ Gr ( KKT ) s✉❝❀ t❀❛t t❀❡r❡ ❡①✐sts U ∈ N (¯ x, ¯ y, ¯ u ) ✭r❡s♣✳ U = R n × R m × R p ✮ ✇✐t❀ F (¯ x, ¯ y ) ≀ F ( x, y ) , ∀ ( x, y, u ) ∈ U ∩ Gr ( KKT ) . ❇✐❧❡✈❡❧ Pr♊❜❧❡♠s✱ â–ŒP❈❈s✱ ❛♥❞ ▌✉❧t✐✲▲❡❛❞❡r✲❋♊❧❧♊✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

  40. ❘❡s✉❧ts ❢♊r t❀❡ ♊♣t✐♠✐st✐❝ ❝❛s❡ ■♥ ❉❡♠♣❡✲❉✉tt❛ ✐t ✇❛s ❝♊♥s✐❞❡r❡❞ t❀❡ ❙❧❛t❡r t②♣❡ ❝♊♥str❛✐♥t q✉❛❧✐✜❝❛t✐♊♥ ❢♊r ❛ ♣❛r❛♠❡t❡r x ∈ X ✿ ❙❧❛t❡r✿ ∃ y ( x ) ∈ R m s✳t✳ g i ( x, y ( x )) < 0 ✱ ∀ i = 1 , .., p. ❇✐❧❡✈❡❧ Pr♊❜❧❡♠s✱ â–ŒP❈❈s✱ ❛♥❞ ▌✉❧t✐✲▲❡❛❞❡r✲❋♊❧❧♊✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

  41. ❘❡s✉❧ts ❢♊r t❀❡ ♊♣t✐♠✐st✐❝ ❝❛s❡ ❚❀❡♊r❡♠ ✶ ❉❡♠♣❡✲❉✉tt❛ ✭✷✵✶✷✮ ❆ss✉♠❡ t❀❡ ❝♊♥✈❡①✐t② ❝♊♥❞✐t✐♊♥ ❛♥❞ ❙❧❛t❡r✬s ❈◗ ❛t ¯ x ✳ ✶ ■❢ (¯ x, ¯ y ) ✐s ❛ ❧♩❝❛❧ s♩❧✉t✐♊♥ ❢♊r ✭❖❇✮✱ t❀❡♥ ❢♊r ❡❛❝❀ ¯ u ∈ Λ(¯ x, ¯ y ) ✱ (¯ x, ¯ y, ¯ u ) ✐s ❛ ❧♩❝❛❧ s♩❧✉t✐♊♥ ❢♊r ✭❖▌P❈❈✮✳ ✷ ❈♊♥✈❡rs❡❧②✱ ❛ss✉♠❡ t❀❛t ❙❧❛t❡r✬s ❈◗ ❀♊❧❞s ♊♥ ❛ y ) ï¿œ = ∅ ✱ ❛♥❞ (¯ ♥❡✐❣❀❜♊✉r❀♊♊❞ ♊❢ ¯ x ✱ Λ(¯ x, ¯ x, ¯ y, u ) ✐s ❛ ❧♩❝❛❧ s♩❧✉t✐♊♥ ♊❢ ✭❖▌P❈❈✮ ❢♊r ❡✈❡r② u ∈ Λ(¯ x, ¯ y ) ✳ ❚❀❡♥ (¯ x, ¯ y ) ✐s ❛ ❧♩❝❛❧ s♩❧✉t✐♊♥ ♊❢ ✭❖❇✮✳ ❇✐❧❡✈❡❧ Pr♊❜❧❡♠s✱ â–ŒP❈❈s✱ ❛♥❞ ▌✉❧t✐✲▲❡❛❞❡r✲❋♊❧❧♊✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

  42. ❯♥❞❡r t❀❡ ❝♊♥✈❡①✐t② ❛ss✉♠♣t✐♊♥ ❛♥❞ s♊♠❡ ❈◗ ❡♥s✉r✐♥❣ KKT ( x ) ï¿œ = ∅ , ∀ x ∈ X ✿ ∀ ¯ u ∈ Λ(¯ x, ¯ y ) (¯ x, ¯ y ) s♩❧ (¯ x, ¯ y, ¯ u ) s♩❧ ♊❢ ✭❖❇✮ ♊❢ ✭❖▌P❈❈✮ ❋✐❣✉r❡✿ ●❧♩❜❛❧ s♩❧✉t✐♊♥ ❝♊♠♣❛r✐s♊♥ ✐♥ ♊♣t✐♠✐st✐❝ ❛♣♣r♊❛❝❀ ∀ ¯ u ∈ Λ(¯ x, ¯ y ) ✱ (¯ x, ¯ y ) ❧♩❝❛❧ (¯ x, ¯ y, ¯ u ) s♩❧ ♊❢ ✭❖❇✮ ❧♩❝❛❧ s♩❧ ♊❢ ✐❢ ❙❧❛t❡r✬s ❈◗ ✭❖▌P❈❈✮ ❀♊❧❞s ❛r♊✉♥❞ ¯ x ❋✐❣✉r❡✿ ▲♊❝❛❧ s♩❧✉t✐♊♥ ❝♊♠♣❛r✐s♊♥ ✐♥ ♊♣t✐♠✐st✐❝ ❛♣♣r♊❛❝❀ ❇✐❧❡✈❡❧ Pr♊❜❧❡♠s✱ â–ŒP❈❈s✱ ❛♥❞ ▌✉❧t✐✲▲❡❛❞❡r✲❋♊❧❧♊✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

  43. ❊①❛♠♣❧❡ ✶ ✭♊♣t✐♠✐st✐❝✮ ❈♊♥s✐❞❡r t❀❡ ❢♊❧❧♊✇✐♥❣ ♊♣t✐♠✐st✐❝ ❇✐❧❡✈❡❧ ♣r♊❜❧❡♠ x ∈ [ − 1 , 1] min min x y s✳t✳ y ∈ S ( x ) , x ∈ R ✇✐t❀ ❧♊✇❡r✲❧❡✈❡❧ − xy min y s✳t x 2 ( y 2 − 1) ≀ 0 . ✶ (0 , − 1 , u ) ✐s ❛ ❧♩❝❛❧ s♩❧✉t✐♊♥ ♊❢ ✭❖▌P❈❈✮✱ ❢♊r ❛♥② u ∈ Λ(0 , − 1) = R + ✷ (0 , − 1) ✐s ◆❖❚ ❛ ❧♩❝❛❧ s♩❧✉t✐♊♥ ♊❢ ✭❖❇✮✳ ❇✐❧❡✈❡❧ Pr♊❜❧❡♠s✱ â–ŒP❈❈s✱ ❛♥❞ ▌✉❧t✐✲▲❡❛❞❡r✲❋♊❧❧♊✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

  44. P❡ss✐♠✐st✐❝ ❆♣♣r♊❛❝❀ ■s ❜✐❧❡✈❡❧ ♣r♊❣r❛♠♠✐♥❣ ❛ s♣❡❝✐❛❧ ❝❛s❡ ♊❢ ❛ ✭▌P❈❈✮❄ ❆✉ss❡❧ ✲ ❙✈❡♥ss♊♥ ✭✷✵✶✟ ✲ ❏✳ ❖♣t✐♠✳ ❚❀❡♊r② ❆♣♣❧✳✮ min x max F ( x, y ) y s✳t✳ y ∈ S ( x ) , x ∈ X. ❇✐❧❡✈❡❧ Pr♊❜❧❡♠s✱ â–ŒP❈❈s✱ ❛♥❞ ▌✉❧t✐✲▲❡❛❞❡r✲❋♊❧❧♊✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

  45. ❉❡✜♥✐t✐♊♥ ❆ ♣❛✐r (¯ x, ¯ y ) ✐s s❛✐❞ t♩ ❜❡ ❛ ❧♩❝❛❧ ✭r❡s♣✳ ❣❧♊❜❛❧✮ s♩❧✉t✐♊♥ ❢♊r ✭P❇✮✱ ✐❢ (¯ x, ¯ y ) ∈ Gr ( S p ) ❛♥❞ ∃ U ∈ N (¯ x, ¯ y ) s✉❝❀ t❀❛t F (¯ x, ¯ y ) ≀ F ( x, y ) , ∀ ( x, y ) ∈ U ∩ Gr ( S p ) . ✭✞✮ ✇❀❡r❡ S p ( x ) := argmax y { F ( x, y ) | y ∈ S ( x ) } . ❉❡✜♥✐t✐♊♥ ❆ tr✐♣❧❡t (¯ x, ¯ y, ¯ u ) ✐s s❛✐❞ t♩ ❜❡ ❛ ❧♩❝❛❧ ✭r❡s♣✳ ❣❧♊❜❛❧✮ s♩❧✉t✐♊♥ ❢♊r ✭Pâ–ŒP❈❈✮✱ ✐❢ (¯ x, ¯ y, ¯ u ) ∈ Gr ( KKT p ) ❛♥❞ ∃ U ∈ N (¯ x, ¯ y, ¯ u ) s✉❝❀ t❀❛t F (¯ x, ¯ y ) ≀ F ( x, y ) , ∀ ( x, y, u ) ∈ U ∩ Gr ( KKT p ) . ✭✹✮ ✇❀❡r❡ KKT p ( x ) := argmax y,u { F ( x, y ) | ( y, u ) ∈ KKT ( x ) } . ❇✐❧❡✈❡❧ Pr♊❜❧❡♠s✱ â–ŒP❈❈s✱ ❛♥❞ ▌✉❧t✐✲▲❡❛❞❡r✲❋♊❧❧♊✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

  46. ❘❡s✉❧ts ❢♊r t❀❡ ♣❡ss✐♠✐st✐❝ ❝❛s❡ ❚❀❡♊r❡♠ ✷ ❆ss✉♠❡ t❀❡ ❝♊♥✈❡①✐t② ❝♊♥❞✐t✐♊♥ ❛♥❞ t❀❛t KKT ( x ) ï¿œ = ∅ , ∀ x ∈ X ✳ ✶ ■❢ (¯ x, ¯ y ) ✐s ❛ ❧♩❝❛❧ s♩❧✉t✐♊♥ ❢♊r ✭P❇✮✱ t❀❡♥ ❢♊r ❡❛❝❀ ¯ u ∈ Λ(¯ x, ¯ y ) ✱ (¯ x, ¯ y, ¯ u ) ✐s ❛ ❧♩❝❛❧ s♩❧✉t✐♊♥ ❢♊r ✭Pâ–ŒP❈❈✮✳ ✷ ❈♊♥✈❡rs❡❧②✱ ❛ss✉♠❡ t❀❛t ♊♥❡ ♊❢ t❀❡ ❢♊❧❧♊✇✐♥❣ ❝♊♥❞✐t✐♊♥ ❛r❡ s❛t✐s✜❡❞✿ ✶ ❚❀❡ ♠✉❧t✐❢✉♥❝t✐♊♥ KKT p ✐s ▲❙❈ ❛r♊✉♥❞ (¯ x, ¯ y, ¯ u ) ❛♥❞ (¯ x, ¯ y, ¯ u ) ✐s ❛ ❧♩❝❛❧ s♩❧✉t✐♊♥ ♊❢ ✭P❇✮✳ ✷ ❙❧❛t❡r✬s ❈◗ ❀♊❧❞s ♊♥ ❛ ♥❡✐❣❀❜♊✉r❀♊♊❞ ♊❢ ¯ x ✱ Λ(¯ x, ¯ y ) ï¿œ = ∅ ✱ ❛♥❞ ❢♊r ❡✈❡r② u ∈ Λ(¯ x, ¯ y ) ✱ (¯ x, ¯ y, u ) ✐s ❛ ❧♩❝❛❧ s♩❧✉t✐♊♥ ♊❢ ✭Pâ–ŒP❈❈✮✳ ❚❀❡♥ (¯ x, ¯ y ) ✐s ❛ ❧♩❝❛❧ s♩❧✉t✐♊♥ ♊❢ ✭P❇✮✳ ❇✐❧❡✈❡❧ Pr♊❜❧❡♠s✱ â–ŒP❈❈s✱ ❛♥❞ ▌✉❧t✐✲▲❡❛❞❡r✲❋♊❧❧♊✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

  47. ❊①❛♠♣❧❡ ✶ ✭♣❡ss✐♠✐st✐❝✮ ❈♊♥s✐❞❡r t❀❡ ❢♊❧❧♊✇✐♥❣ ♣❡ss✐♠✐st✐❝ ❇✐❧❡✈❡❧ ♣r♊❜❧❡♠ x ∈ [ − 1 , 1] max min x y s✳t✳ y ∈ S ( x ) , x ∈ R ✇✐t❀ ❧♊✇❡r✲❧❡✈❡❧ − xy min y s✳t x 2 ( y 2 − 1) ≀ 0 . ✶ (0 , − 1 , u ) ✐s ❛ ❧♩❝❛❧ s♩❧✉t✐♊♥ ♊❢ ✭Pâ–ŒP❈❈✮✱ ❢♊r ❛♥② u ∈ Λ(0 , − 1) = R + ✷ (0 , − 1) ✐s ◆❖❚ ❛ ❧♩❝❛❧ s♩❧✉t✐♊♥ ♊❢ ✭P❇✮✳ ❇✐❧❡✈❡❧ Pr♊❜❧❡♠s✱ â–ŒP❈❈s✱ ❛♥❞ ▌✉❧t✐✲▲❡❛❞❡r✲❋♊❧❧♊✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

  48. ❊①❛♠♣❧❡ ✷ ❈♊♥s✐❞❡r t❀❡ ❢♊❧❧♊✇✐♥❣ ❇✐❧❡✈❡❧ ♣r♊❜❧❡♠ ✏ min x ✑ x y ∈ S ( x ) s.t. ✇✐t❀ S ( x ) t❀❡ s♩❧✉t✐♊♥ ♊❢ t❀❡ ❧♊✇❡r✲❧❡✈❡❧ ♣r♊❜❧❡♠ min {− y | x + y ≀ 0 , y ≀ 0 } y ❊✈❡♥ t❀♊✉❣❀ ❙❧❛t❡r✬s ❈◗ ❀♊❧❞s✱ ✇❡ ❀❛✈❡ ✶ (0 , 0 , u 1 , u 2 ) ✇✐t❀ ( u 1 , u 2 ) ∈ Λ(0 , 0) = { ( λ, 1 − λ ) | λ ∈ [0 , 1] } ✐s ❛ ❧♩❝❛❧ s♩❧✉t✐♊♥ ♊❢ ✏✭▌P❈❈✮✑✱ ✐✛ u 1 ï¿œ = 0 ✱ ✷ (0 , 0) ✐s ◆❖❚ ❛ ❧♩❝❛❧ s♩❧✉t✐♊♥ ❢♊r ✏✭❇✮✑✳ ❇✐❧❡✈❡❧ Pr♊❜❧❡♠s✱ â–ŒP❈❈s✱ ❛♥❞ ▌✉❧t✐✲▲❡❛❞❡r✲❋♊❧❧♊✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

  49. ❯♥❞❡r t❀❡ ❝♊♥✈❡①✐t② ❛ss✉♠♣t✐♊♥ ❛♥❞ s♊♠❡ ✭❈◗✮ ❡♥s✉r✐♥❣ KKT ( x ) ï¿œ = ∅ , ∀ x ∈ X ✿ ∀ ¯ u ∈ Λ(¯ x, ¯ y ) (¯ x, ¯ y ) s♩❧ (¯ x, ¯ y, ¯ u ) s♩❧ ♊❢ ✭P❇✮ ♊❢ ✭Pâ–ŒP❈❈✮ ❋✐❣✉r❡✿ ●❧♩❜❛❧ s♩❧✉t✐♊♥ ❝♊♠♣❛r✐s♊♥ ✐♥ ♣❡ss✐♠✐st✐❝ ❛♣♣r♊❛❝❀ ∀ ¯ u ∈ Λ(¯ x, ¯ y ) ✱ (¯ x, ¯ y ) ❧♩❝❛❧ (¯ x, ¯ y, ¯ u ) s♩❧ ♊❢ ✭P❇✮ ❧♩❝❛❧ s♩❧ ♊❢ ❙❧❛t❡r✬s ❈◗ ❢♊r ✭Pâ–ŒP❈❈✮ ❛❧❧ x ❛r♊✉♥❞ ¯ x ❋✐❣✉r❡✿ ▲♊❝❛❧ s♩❧✉t✐♊♥s ❝♊♠♣❛r✐s♊♥ ✐♥ ♣❡ss✐♠✐st✐❝ ❛♣♣r♊❛❝❀ ❇✐❧❡✈❡❧ Pr♊❜❧❡♠s✱ â–ŒP❈❈s✱ ❛♥❞ ▌✉❧t✐✲▲❡❛❞❡r✲❋♊❧❧♊✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

  50. ❆♥ ✐♥tr♩❞✉❝t✐♊♥ t♩ ▌▲❋● ❇✐❧❡✈❡❧ Pr♊❜❧❡♠s✱ â–ŒP❈❈s✱ ❛♥❞ ▌✉❧t✐✲▲❡❛❞❡r✲❋♊❧❧♊✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

  51. ❉❡♥♊t❡ ❜② t❀❡ ♥✉♠❜❡r ♊❢ ♣❧❛②❡rs ❛♥❞ ❡❛❝❀ ♣❧❛②❡r ❝♊♥tr♩❧s ✈❛r✐❛❜❧❡s ✳ ❚❀❡ ✏t♩t❛❧ str❛t❡❣② ✈❡❝t♩r✑ ✐s ✇❀✐❝❀ ✇✐❧❧ ❜❡ ♊❢t❡♥ ❞❡♥♊t❡❞ ❜② ✇❀❡r❡ ✐s t❀❡ str❛t❡❣② ✈❡❝t♩r ♊❢ t❀❡ ♩t❀❡r ♣❧❛②❡rs ✳ ◆♩t❛t✐♊♥s ❛♥❞ ❞❡✜♥✐t✐♊♥s✿ ◆❛s❀ ♣r♊❜❧❡♠s ❆ ◆❛s❀ ❡q✉✐❧✐❜r✐✉♠ ♣r♊❜❧❡♠ ✐s ❛ ♥♊♥❝♊♊♣❡r❛t✐✈❡ ❣❛♠❡ ✐♥ ✇❀✐❝❀ t❀❡ ❞❡❝✐s✐♊♥ ❢✉♥❝t✐♊♥ ✭❝♩st✎❜❡♥❡✜t✮ ♊❢ ❡❛❝❀ ♣❧❛②❡r ❞❡♣❡♥❞s ♊♥ t❀❡ ❞❡❝✐s✐♊♥ ♊❢ t❀❡ ♩t❀❡r ♣❧❛②❡rs✳ ❇✐❧❡✈❡❧ Pr♊❜❧❡♠s✱ â–ŒP❈❈s✱ ❛♥❞ ▌✉❧t✐✲▲❡❛❞❡r✲❋♊❧❧♊✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

  52. ◆♩t❛t✐♊♥s ❛♥❞ ❞❡✜♥✐t✐♊♥s✿ ◆❛s❀ ♣r♊❜❧❡♠s ❆ ◆❛s❀ ❡q✉✐❧✐❜r✐✉♠ ♣r♊❜❧❡♠ ✐s ❛ ♥♊♥❝♊♊♣❡r❛t✐✈❡ ❣❛♠❡ ✐♥ ✇❀✐❝❀ t❀❡ ❞❡❝✐s✐♊♥ ❢✉♥❝t✐♊♥ ✭❝♩st✎❜❡♥❡✜t✮ ♊❢ ❡❛❝❀ ♣❧❛②❡r ❞❡♣❡♥❞s ♊♥ t❀❡ ❞❡❝✐s✐♊♥ ♊❢ t❀❡ ♩t❀❡r ♣❧❛②❡rs✳ ❉❡♥♊t❡ ❜② N t❀❡ ♥✉♠❜❡r ♊❢ ♣❧❛②❡rs ❛♥❞ ❡❛❝❀ ♣❧❛②❡r i ❝♊♥tr♩❧s ✈❛r✐❛❜❧❡s x i ∈ R n i ✳ ❚❀❡ ✏t♩t❛❧ str❛t❡❣② ✈❡❝t♩r✑ ✐s x ✇❀✐❝❀ ✇✐❧❧ ❜❡ ♊❢t❡♥ ❞❡♥♊t❡❞ ❜② x = ( x i , x − i ) . ✇❀❡r❡ x − i ✐s t❀❡ str❛t❡❣② ✈❡❝t♩r ♊❢ t❀❡ ♩t❀❡r ♣❧❛②❡rs ✳ ❇✐❧❡✈❡❧ Pr♊❜❧❡♠s✱ â–ŒP❈❈s✱ ❛♥❞ ▌✉❧t✐✲▲❡❛❞❡r✲❋♊❧❧♊✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

  53. ●✐✈❡♥ t❀❡ str❛t❡❣✐❡s ♊❢ t❀❡ ♩t❀❡r ♣❧❛②❡rs✱ t❀❡ ❛✐♠ ♊❢ ♣❧❛②❡r ✐s t♩ ❝❀♊♊s❡ ❛ str❛t❡❣② s♊❧✈✐♥❣ s✳t✳ ✇❀❡r❡ ✐s t❀❡ ❞❡❝✐s✐♊♥ ❢✉♥❝t✐♊♥ ❢♊r ♣❧❛②❡r ✳ ❆ ✈❡❝t♩r ✐s ❛ ◆❛s❀ ❊q✉✐❧✐❜r✐✉♠ ✐❢ ❢♊r ❛♥② s♊❧✈❡s ◆❛s❀ ❊q✉✐❧✐❜r✐✉♠ Pr♊❜❧❡♠ ✭◆❊P✮ ❚❀❡ str❛t❡❣② ♊❢ ♣❧❛②❡r i ❜❡❧♊♥❣s t♩ ❛ str❛t❡❣② s❡t x i ∈ X i ❇✐❧❡✈❡❧ Pr♊❜❧❡♠s✱ â–ŒP❈❈s✱ ❛♥❞ ▌✉❧t✐✲▲❡❛❞❡r✲❋♊❧❧♊✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

  54. ❆ ✈❡❝t♩r ✐s ❛ ◆❛s❀ ❊q✉✐❧✐❜r✐✉♠ ✐❢ ❢♊r ❛♥② s♊❧✈❡s ◆❛s❀ ❊q✉✐❧✐❜r✐✉♠ Pr♊❜❧❡♠ ✭◆❊P✮ ❚❀❡ str❛t❡❣② ♊❢ ♣❧❛②❡r i ❜❡❧♊♥❣s t♩ ❛ str❛t❡❣② s❡t x i ∈ X i ●✐✈❡♥ t❀❡ str❛t❡❣✐❡s x − i ♊❢ t❀❡ ♩t❀❡r ♣❧❛②❡rs✱ t❀❡ ❛✐♠ ♊❢ ♣❧❛②❡r i ✐s t♩ ❝❀♊♊s❡ ❛ str❛t❡❣② x i s♊❧✈✐♥❣ P i ( x − i ) Ξ i ( x i , x − i ) max x i ∈ X i s✳t✳ ✇❀❡r❡ Ξ i ( · , x − i ) : R n i → R ✐s t❀❡ ❞❡❝✐s✐♊♥ ❢✉♥❝t✐♊♥ ❢♊r ♣❧❛②❡r i ✳ ❇✐❧❡✈❡❧ Pr♊❜❧❡♠s✱ â–ŒP❈❈s✱ ❛♥❞ ▌✉❧t✐✲▲❡❛❞❡r✲❋♊❧❧♊✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

  55. ◆❛s❀ ❊q✉✐❧✐❜r✐✉♠ Pr♊❜❧❡♠ ✭◆❊P✮ ❚❀❡ str❛t❡❣② ♊❢ ♣❧❛②❡r i ❜❡❧♊♥❣s t♩ ❛ str❛t❡❣② s❡t x i ∈ X i ●✐✈❡♥ t❀❡ str❛t❡❣✐❡s x − i ♊❢ t❀❡ ♩t❀❡r ♣❧❛②❡rs✱ t❀❡ ❛✐♠ ♊❢ ♣❧❛②❡r i ✐s t♩ ❝❀♊♊s❡ ❛ str❛t❡❣② x i s♊❧✈✐♥❣ P i ( x − i ) Ξ i ( x i , x − i ) max x i ∈ X i s✳t✳ ✇❀❡r❡ Ξ i ( · , x − i ) : R n i → R ✐s t❀❡ ❞❡❝✐s✐♊♥ ❢✉♥❝t✐♊♥ ❢♊r ♣❧❛②❡r i ✳ ❆ ✈❡❝t♩r ¯ x ✐s ❛ ◆❛s❀ ❊q✉✐❧✐❜r✐✉♠ ✐❢ x i x − i ) . ❢♊r ❛♥② i, ¯ s♊❧✈❡s P i (¯ ❇✐❧❡✈❡❧ Pr♊❜❧❡♠s✱ â–ŒP❈❈s✱ ❛♥❞ ▌✉❧t✐✲▲❡❛❞❡r✲❋♊❧❧♊✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

  56. ●✐✈❡♥ t❀❡ str❛t❡❣✐❡s ♊❢ t❀❡ ♩t❀❡r ♣❧❛②❡rs✱ t❀❡ ❛✐♠ ♊❢ ♣❧❛②❡r ✐s t♩ ❝❀♊♊s❡ ❛ str❛t❡❣② s♊❧✈✐♥❣ s✳t✳ ✇❀❡r❡ ✐s t❀❡ ❞❡❝✐s✐♊♥ ❢✉♥❝t✐♊♥ ❢♊r ♣❧❛②❡r ✳ ❆ ✈❡❝t♩r ✐s ❛ ●❡♥❡r❛❧✐③❡❞ ◆❛s❀ ❊q✉✐❧✐❜r✐✉♠ ✐❢ ❢♊r ❛♥② s♊❧✈❡s ●❡♥❡r❛❧✐③❡❞ ◆❛s❀ ❊q✉✐❧✐❜r✐✉♠ Pr♊❜❧❡♠ ❆ ❣❡♥❡r❛❧✐③❡❞ ◆❛s❀ ❡q✉✐❧✐❜r✐✉♠ ♣r♊❜❧❡♠ ✭●◆❊P✮ ✐s ❛ ♥♊♥❝♊♊♣❡r❛t✐✈❡ ❣❛♠❡ ✐♥ ✇❀✐❝❀ t❀❡ ❞❡❝✐s✐♊♥ ❢✉♥❝t✐♊♥ ❛♥❞ str❛t❡❣② s❡t ♊❢ ❡❛❝❀ ♣❧❛②❡r ❞❡♣❡♥❞ ♊♥ t❀❡ ❞❡❝✐s✐♊♥ ♊❢ t❀❡ ♩t❀❡r ♣❧❛②❡rs✳ ❚❀❡ str❛t❡❣② ♊❢ ♣❧❛②❡r i ❜❡❧♊♥❣s t♩ ❛ str❛t❡❣② s❡t x i ∈ X i ( x − i ) ✇❀✐❝❀ ❞❡♣❡♥❞s ♊♥ t❀❡ ❞❡❝✐s✐♊♥ ✈❛r✐❛❜❧❡s ♊❢ t❀❡ ♩t❀❡r ♣❧❛②❡rs✳ ❇✐❧❡✈❡❧ Pr♊❜❧❡♠s✱ â–ŒP❈❈s✱ ❛♥❞ ▌✉❧t✐✲▲❡❛❞❡r✲❋♊❧❧♊✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

  57. ❆ ✈❡❝t♩r ✐s ❛ ●❡♥❡r❛❧✐③❡❞ ◆❛s❀ ❊q✉✐❧✐❜r✐✉♠ ✐❢ ❢♊r ❛♥② s♊❧✈❡s ●❡♥❡r❛❧✐③❡❞ ◆❛s❀ ❊q✉✐❧✐❜r✐✉♠ Pr♊❜❧❡♠ ❆ ❣❡♥❡r❛❧✐③❡❞ ◆❛s❀ ❡q✉✐❧✐❜r✐✉♠ ♣r♊❜❧❡♠ ✭●◆❊P✮ ✐s ❛ ♥♊♥❝♊♊♣❡r❛t✐✈❡ ❣❛♠❡ ✐♥ ✇❀✐❝❀ t❀❡ ❞❡❝✐s✐♊♥ ❢✉♥❝t✐♊♥ ❛♥❞ str❛t❡❣② s❡t ♊❢ ❡❛❝❀ ♣❧❛②❡r ❞❡♣❡♥❞ ♊♥ t❀❡ ❞❡❝✐s✐♊♥ ♊❢ t❀❡ ♩t❀❡r ♣❧❛②❡rs✳ ❚❀❡ str❛t❡❣② ♊❢ ♣❧❛②❡r i ❜❡❧♊♥❣s t♩ ❛ str❛t❡❣② s❡t x i ∈ X i ( x − i ) ✇❀✐❝❀ ❞❡♣❡♥❞s ♊♥ t❀❡ ❞❡❝✐s✐♊♥ ✈❛r✐❛❜❧❡s ♊❢ t❀❡ ♩t❀❡r ♣❧❛②❡rs✳ ●✐✈❡♥ t❀❡ str❛t❡❣✐❡s x − i ♊❢ t❀❡ ♩t❀❡r ♣❧❛②❡rs✱ t❀❡ ❛✐♠ ♊❢ ♣❧❛②❡r i ✐s t♩ ❝❀♊♊s❡ ❛ str❛t❡❣② x i s♊❧✈✐♥❣ P i ( x − i ) Ξ i ( x i , x − i ) max x i ∈ X i ( x − i ) s✳t✳ ✇❀❡r❡ Ξ i ( · , x − i ) : R ni → R ✐s t❀❡ ❞❡❝✐s✐♊♥ ❢✉♥❝t✐♊♥ ❢♊r ♣❧❛②❡r i ✳ ❇✐❧❡✈❡❧ Pr♊❜❧❡♠s✱ â–ŒP❈❈s✱ ❛♥❞ ▌✉❧t✐✲▲❡❛❞❡r✲❋♊❧❧♊✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

  58. ●❡♥❡r❛❧✐③❡❞ ◆❛s❀ ❊q✉✐❧✐❜r✐✉♠ Pr♊❜❧❡♠ ❆ ❣❡♥❡r❛❧✐③❡❞ ◆❛s❀ ❡q✉✐❧✐❜r✐✉♠ ♣r♊❜❧❡♠ ✭●◆❊P✮ ✐s ❛ ♥♊♥❝♊♊♣❡r❛t✐✈❡ ❣❛♠❡ ✐♥ ✇❀✐❝❀ t❀❡ ❞❡❝✐s✐♊♥ ❢✉♥❝t✐♊♥ ❛♥❞ str❛t❡❣② s❡t ♊❢ ❡❛❝❀ ♣❧❛②❡r ❞❡♣❡♥❞ ♊♥ t❀❡ ❞❡❝✐s✐♊♥ ♊❢ t❀❡ ♩t❀❡r ♣❧❛②❡rs✳ ❚❀❡ str❛t❡❣② ♊❢ ♣❧❛②❡r i ❜❡❧♊♥❣s t♩ ❛ str❛t❡❣② s❡t x i ∈ X i ( x − i ) ✇❀✐❝❀ ❞❡♣❡♥❞s ♊♥ t❀❡ ❞❡❝✐s✐♊♥ ✈❛r✐❛❜❧❡s ♊❢ t❀❡ ♩t❀❡r ♣❧❛②❡rs✳ ●✐✈❡♥ t❀❡ str❛t❡❣✐❡s x − i ♊❢ t❀❡ ♩t❀❡r ♣❧❛②❡rs✱ t❀❡ ❛✐♠ ♊❢ ♣❧❛②❡r i ✐s t♩ ❝❀♊♊s❡ ❛ str❛t❡❣② x i s♊❧✈✐♥❣ P i ( x − i ) Ξ i ( x i , x − i ) max x i ∈ X i ( x − i ) s✳t✳ ✇❀❡r❡ Ξ i ( · , x − i ) : R ni → R ✐s t❀❡ ❞❡❝✐s✐♊♥ ❢✉♥❝t✐♊♥ ❢♊r ♣❧❛②❡r i ✳ ❆ ✈❡❝t♩r ¯ x ✐s ❛ ●❡♥❡r❛❧✐③❡❞ ◆❛s❀ ❊q✉✐❧✐❜r✐✉♠ ✐❢ x i x − i ) . ❢♊r ❛♥② i, ¯ s♊❧✈❡s P i (¯ ❇✐❧❡✈❡❧ Pr♊❜❧❡♠s✱ â–ŒP❈❈s✱ ❛♥❞ ▌✉❧t✐✲▲❡❛❞❡r✲❋♊❧❧♊✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

  59. ●❡♥❡r❛❧ ♠♊❞❡❧ ●❡♥❡r❛❧✐③❡❞ ◆❛s❀ ❣❛♠❡ ✭●◆❊P✮✿ min Ξ 1 ( x ) min Ξ n ( x ) x 1 ï¿œ x 1 ∈ X 1 ( x − 1 ) . . . x n ï¿œ x n ∈ X n ( x − n ) s✳t✳ s✳t✳ ❇✐❧❡✈❡❧ Pr♊❜❧❡♠s✱ â–ŒP❈❈s✱ ❛♥❞ ▌✉❧t✐✲▲❡❛❞❡r✲❋♊❧❧♊✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

  60. ❆ ❝❧❛ss✐❝❛❧ ❡①✐st❡♥❝❡ r❡s✉❧t ❚❀❡♊r❡♠ ✭■❝❀✐✐s❀✐✲◗✉✐♥③✐✐ ✶✟✜✞✮ ▲❡t ❛ ●◆❊P ❜❡ ❣✐✈❡♥ ❛♥❞ s✉♣♣♊s❡ t❀❛t ✶ ❋♩r ❡❛❝❀ Îœ = 1 , ..., N t❀❡r❡ ❡①✐st ❛ ♥♊♥❡♠♣t②✱ ❝♊♥✈❡① ❛♥❞ ❝♊♠♣❛❝t s❡t K Îœ ⊂ R n Îœ s✉❝❀ t❀❛t t❀❡ ♣♊✐♥t✲t♊✲s❡t ♠❛♣ X Îœ : K − Îœ ⇒ K Îœ ✱ ✐s ❜♩t❀ ✉♣♣❡r ❛♥❞ ❧♊✇❡r s❡♠✐❝♊♥t✐♥✉♊✉s ✇✐t❀ ♥♊♥❡♠♣t② ❝❧♩s❡❞ ❛♥❞ ❝♊♥✈❡① ✈❛❧✉❡s✱ ✇❀❡r❡ K − Îœ := ï¿œ Îœ ′ ï¿œ = Îœ K Îœ ✳ ✷ ❋♩r ❡✈❡r② ♣❧❛②❡r Îœ ✱ t❀❡ ❢✉♥❝t✐♊♥ Ξ Îœ ✐s ❝♊♥t✐♥✉♊✉s ❛♥❞ Ξ Îœ ( · , x − Îœ ) ✐s q✉❛s✐✲❝♊♥✈❡① ♊♥ X Îœ ( x − Îœ ) ✳ ❚❀❡♥ ❛ ❣❡♥❡r❛❧✐③❡❞ ◆❛s❀ ❡q✉✐❧✐❜r✐✉♠ ❡①✐sts✳ ◆♩t❡ t❀❛t ✐♥ ❆✉ss❡❧✲❉✉tt❛ ✭✷✵✵✜✮ ❛♥ ❛❧t❡r♥❛t✐✈❡ ♣r♊♊❢ ♊❢ ❡①✐st❡♥❝❡ ♊❢ ❡q✉✐❧✐❜r✐❛ ❀❛s ❜❡❡♥ ❣✐✈❡♥✱ ✉♥❞❡r t❀❡ ❛ss✉♠♣t✐♊♥ ♊❢ t❀❡ ❘♩s❡♥✬s ❧❛✇✱ ❜② ✉s✐♥❣ t❀❡ ♥♊r♠❛❧ ❛♣♣r♊❛❝❀ t❡❝❀♥✐q✉❡✳ ❇✐❧❡✈❡❧ Pr♊❜❧❡♠s✱ â–ŒP❈❈s✱ ❛♥❞ ▌✉❧t✐✲▲❡❛❞❡r✲❋♊❧❧♊✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

  61. ❙tr✉❝t✉r❡ ♊❢ t❀❡ s❡t ♊❢ ●◆❊Ps ❊①❛♠♣❧❡ ▲❡t x = ( x 1 , x 2 ) ∈ [0 , 4] 2 ❛♥❞ f Îœ ( x ) := d T Îœ ( x ) 2 ✱ ✇❀❡r❡ T 1 ✐s t❀❡ tr✐❛♥❣❧❡ ✇✐t❀ ✈❡rt✐❝❡s (0 , 0) ✱ (0 , 4) ❛♥❞ (1 , 2) ✱ ❛♥❞ T 2 ✐s t❀❡ tr✐❛♥❣❧❡ ✇❀♊s❡ ✈❡rt✐❝❡s ❛r❡ (0 , 0) ✱ (4 , 0) ❛♥❞ (2 , 1) ✳ ▲❡t f Îœ ( x 1 , x 2 ) | x Îœ ∈ [0 , 4] S Îœ ( x − Îœ ) := ❛r❣♠✐♥ x Îœ ï¿œ ï¿œ ✳ ❲❡ s❡❡ t❀❛t x 1 ∈ [0 , 4] | ( x 1 , x 2 ) ∈ T 1 ❢♊r x 2 ∈ [0 , 1] S 1 ( x 2 ) = ï¿œ ï¿œ S 1 ( x 2 ) = { 2 } ❢♊r ❛❧❧ x 2 ∈ (1 , 4]) x 2 ∈ [0 , 4] | ( x 1 , x 2 ) ∈ T 2 ❢♊r x 1 ∈ [0 , 1] S 2 ( x 1 ) = ï¿œ ï¿œ S 2 ( x 1 ) = { 2 } ❢♊r ❛❧❧ x 1 ∈ (1 , 4]) ✳ ❇✐❧❡✈❡❧ Pr♊❜❧❡♠s✱ â–ŒP❈❈s✱ ❛♥❞ ▌✉❧t✐✲▲❡❛❞❡r✲❋♊❧❧♊✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

  62. ❙tr✉❝t✉r❡ ♊❢ t❀❡ s❡t ♊❢ ●◆❊Ps ✭❝♊♥t✳✮ x 2 S 2 ( · ) S 1 ( · ) x 1 ❇✐❧❡✈❡❧ Pr♊❜❧❡♠s✱ â–ŒP❈❈s✱ ❛♥❞ ▌✉❧t✐✲▲❡❛❞❡r✲❋♊❧❧♊✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

  63. ❆ ♣❛rt✐❝✉❧❛r ❝❛s❡ ▌✉❧t✐✲▲❡❛❞❡r✲❋♊❧❧♊✇❡r✲●❛♠❡ ✭▌▲❋●✮✿ min Ξ 1 ( x, y ) min Ξ n ( x, y ) x 1 x n y 1 ,..,y p ï¿œ x 1 ∈ X 1 ( x − 1 ) y 1 ,..,y p ï¿œ x n ∈ X n ( x − n ) . . . s✳t✳ s✳t✳ y ∈ Y ( x ) y ∈ Y ( x ) ↓↑ ↓↑ min y 1 ,..,y p φ 1 ( x, y ) min y 1 ,..,y p φ p ( x, y ) ï¿œ y ∈ Y ( x ) ï¿œ y ∈ Y ( x ) . . . s✳t✳ s✳t✳ ❇✐❧❡✈❡❧ Pr♊❜❧❡♠s✱ â–ŒP❈❈s✱ ❛♥❞ ▌✉❧t✐✲▲❡❛❞❡r✲❋♊❧❧♊✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

  64. ❛♥❞ ❛♥♊t❀❡r ♣r♊❜❧❡♠ ❙✐♥❣❧❡✲▲❡❛❞❡r✲▌✉❧t✐✲❋♊❧❧♊✇❡r✲●❛♠❡ ✭❙▲▌❋●✮✿ min Ξ 1 ( x, y ) x y 1 ,..,y p ï¿œ x ∈ X s✳t✳ y ∈ Y ( x ) ↓↑ min y 1 ,..,y p φ 1 ( x, y ) min y 1 ,..,y p φ p ( x, y ) . . . ï¿œ y ∈ Y ( x ) ï¿œ y ∈ Y ( x ) s✳t✳ s✳t✳ ❇✐❧❡✈❡❧ Pr♊❜❧❡♠s✱ â–ŒP❈❈s✱ ❛♥❞ ▌✉❧t✐✲▲❡❛❞❡r✲❋♊❧❧♊✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

  65. ❆ ♣❛rt✐❝✉❧❛r ❝❛s❡ ▌✉❧t✐✲▲❡❛❞❡r✲❙✐♥❣❧❡✲❋♊❧❧♊✇❡r✲●❛♠❡ ✭▌▲❙❋●✮✿ min Ξ 1 ( x, y ) min Ξ n ( x, y ) x 1 x n y 1 ,..,y p ï¿œ x 1 ∈ X 1 ( x − 1 ) y 1 ,..,y p ï¿œ x n ∈ X n ( x − n ) . . . s✳t✳ s✳t✳ y ∈ Y ( x ) y ∈ Y ( x ) ↓↑ min y 1 ,..,y p φ 1 ( x, y ) ï¿œ y ∈ Y ( x ) s✳t✳ ❇✐❧❡✈❡❧ Pr♊❜❧❡♠s✱ â–ŒP❈❈s✱ ❛♥❞ ▌✉❧t✐✲▲❡❛❞❡r✲❋♊❧❧♊✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

  66. ❊①❡r❝✐s❡✿ P❧❡❛s❡ ❛♥❛❧②s❡ t❀✐s s♠❛❧❧ ❡①❛♠♣❧❡✳✳✳ ▌▲❙❋ ❣❛♠❡✿ ✐❧❧✲♣♊s❡❞♥❡ss min x 1 ,y Ξ 1 ( x 1 , x 2 , y ) = x 1 .y min x 2 ,y Ξ 1 ( x 1 , x 2 , y ) = − x 2 .y ï¿œ ï¿œ x 1 ∈ [0 , 1] x 2 ∈ [0 , 1] s✳t✳ s✳t✳ y ∈ S ( x 1 , x 2 ) y ∈ S ( x 1 , x 2 ) ✇✐t❀ 3 y 3 − ( x 1 + x 2 ) 2 y f ( x 1 , x 2 , y ) = 1 min y s✳t✳ y ∈ R ❇✐❧❡✈❡❧ Pr♊❜❧❡♠s✱ â–ŒP❈❈s✱ ❛♥❞ ▌✉❧t✐✲▲❡❛❞❡r✲❋♊❧❧♊✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

  67. ▌▲❙❋ ❣❛♠❡✿ ✐❧❧✲♣♊s❡❞♥❡ss min x 1 ,y Ξ 1 ( x 1 , x 2 , y ) = x 1 .y min x 2 ,y Ξ 1 ( x 1 , x 2 , y ) = − x 2 .y ï¿œ ï¿œ x 1 ∈ [0 , 1] x 2 ∈ [0 , 1] s✳t✳ s✳t✳ y ∈ S ( x 1 , x 2 ) y ∈ S ( x 1 , x 2 ) ✇✐t❀ 3 y 3 − ( x 1 + x 2 ) 2 y f ( x 1 , x 2 , y ) = 1 min y s✳t✳ y ∈ R ❊①❡r❝✐s❡✿ P❧❡❛s❡ ❛♥❛❧②s❡ t❀✐s s♠❛❧❧ ❡①❛♠♣❧❡✳✳✳ ❇✐❧❡✈❡❧ Pr♊❜❧❡♠s✱ â–ŒP❈❈s✱ ❛♥❞ ▌✉❧t✐✲▲❡❛❞❡r✲❋♊❧❧♊✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

  68. ❚❀❡ s♩❧✉t✐♊♥ ♠❛♣ ♊❢ t❀✐s ❢♊❧❧♊✇❡r ♣r♊❜❧❡♠ ✐s ▌▲❙❋ ❣❛♠❡✿ ✐❧❧✲♣♊s❡❞♥❡ss ❚❀❡ ❢♊❧❧♊✇❡r ♣r♊❜❧❡♠ ✜rst 3 y 3 − ( x 1 + x 2 ) 2 y f ( x 1 , x 2 , y ) = 1 min y s✳t✳ y ∈ R ❇✐❧❡✈❡❧ Pr♊❜❧❡♠s✱ â–ŒP❈❈s✱ ❛♥❞ ▌✉❧t✐✲▲❡❛❞❡r✲❋♊❧❧♊✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

  69. ▌▲❙❋ ❣❛♠❡✿ ✐❧❧✲♣♊s❡❞♥❡ss ❚❀❡ ❢♊❧❧♊✇❡r ♣r♊❜❧❡♠ ✜rst 3 y 3 − ( x 1 + x 2 ) 2 y f ( x 1 , x 2 , y ) = 1 min y s✳t✳ y ∈ R ❚❀❡ s♩❧✉t✐♊♥ ♠❛♣ ♊❢ t❀✐s ❢♊❧❧♊✇❡r ♣r♊❜❧❡♠ ✐s S ( x 1 , x 2 ) = { y 1 = x 1 + x 2 , y 2 = − x 1 − x 2 } . ❇✐❧❡✈❡❧ Pr♊❜❧❡♠s✱ â–ŒP❈❈s✱ ❛♥❞ ▌✉❧t✐✲▲❡❛❞❡r✲❋♊❧❧♊✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

  70. ❚❀✉s t❀❡ r❡s♣♊♥s❡ ❢✉♥❝t✐♊♥ ♊❢ ♣❧❛②❡r ✶ ✐s ✐❢ ✇✐t❀ ❛ ♣❛②♊✛ ✐❢ ✇✐t❀ ❛ ♣❛②♊✛ ▌▲❙❋ ❣❛♠❡✿ ✐❧❧✲♣♊s❡❞♥❡ss ❚❀❡ s♩❧✉t✐♊♥ ♠❛♣ ♊❢ t❀✐s ❢♊❧❧♊✇❡r ♣r♊❜❧❡♠ ✐s S ( x 1 , x 2 ) = { y 1 = x 1 + x 2 , y 2 = − x 1 − x 2 } . ❚❀❡ ❧❡❛❞❡r ✶ ♣r♊❜❧❡♠ ï¿œ x 2 1 + x 1 .x 2 ✐❢ y = y 1 Ξ 1 ( x, y ) = x 1 .y = − x 2 1 − x 1 .x 2 ✐❢ y = y 2 ❇✐❧❡✈❡❧ Pr♊❜❧❡♠s✱ â–ŒP❈❈s✱ ❛♥❞ ▌✉❧t✐✲▲❡❛❞❡r✲❋♊❧❧♊✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

  71. ▌▲❙❋ ❣❛♠❡✿ ✐❧❧✲♣♊s❡❞♥❡ss ❚❀❡ s♩❧✉t✐♊♥ ♠❛♣ ♊❢ t❀✐s ❢♊❧❧♊✇❡r ♣r♊❜❧❡♠ ✐s S ( x 1 , x 2 ) = { y 1 = x 1 + x 2 , y 2 = − x 1 − x 2 } . ❚❀❡ ❧❡❛❞❡r ✶ ♣r♊❜❧❡♠ ï¿œ x 2 1 + x 1 .x 2 ✐❢ y = y 1 Ξ 1 ( x, y ) = x 1 .y = − x 2 1 − x 1 .x 2 ✐❢ y = y 2 ❚❀✉s t❀❡ r❡s♣♊♥s❡ ❢✉♥❝t✐♊♥ ♊❢ ♣❧❛②❡r ✶ ✐s ï¿œ { 0 } ✐❢ y = y 1 ✇✐t❀ ❛ ♣❛②♊✛ = 0 R 1 ( x 2 ) = { 1 } ✐❢ y = y 2 ✇✐t❀ ❛ ♣❛②♊✛ = − 1 − x 2 ❇✐❧❡✈❡❧ Pr♊❜❧❡♠s✱ â–ŒP❈❈s✱ ❛♥❞ ▌✉❧t✐✲▲❡❛❞❡r✲❋♊❧❧♊✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

  72. ❚❀✉s t❀❡ r❡s♣♊♥s❡ ❢✉♥❝t✐♊♥ ♊❢ ♣❧❛②❡r ✶ ✐s ✐❢ ✇✐t❀ ❛ ♣❛②♊✛ ✐❢ ✇✐t❀ ❛ ♣❛②♊✛ ▌▲❙❋ ❣❛♠❡✿ ✐❧❧✲♣♊s❡❞♥❡ss ❚❀❡ s♩❧✉t✐♊♥ ♠❛♣ ♊❢ t❀✐s ❢♊❧❧♊✇❡r ♣r♊❜❧❡♠ ✐s S ( x 1 , x 2 ) = { y 1 = x 1 + x 2 , y 2 = − x 1 − x 2 } . ❚❀❡ ❧❡❛❞❡r ✷ ♣r♊❜❧❡♠ ï¿œ − x 2 1 − x 1 .x 2 ✐❢ y = y 1 Ξ 1 ( x, y ) = − x 2 .y = x 2 1 + x 1 .x 2 ✐❢ y = y 2 ❇✐❧❡✈❡❧ Pr♊❜❧❡♠s✱ â–ŒP❈❈s✱ ❛♥❞ ▌✉❧t✐✲▲❡❛❞❡r✲❋♊❧❧♊✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

  73. ▌▲❙❋ ❣❛♠❡✿ ✐❧❧✲♣♊s❡❞♥❡ss ❚❀❡ s♩❧✉t✐♊♥ ♠❛♣ ♊❢ t❀✐s ❢♊❧❧♊✇❡r ♣r♊❜❧❡♠ ✐s S ( x 1 , x 2 ) = { y 1 = x 1 + x 2 , y 2 = − x 1 − x 2 } . ❚❀❡ ❧❡❛❞❡r ✷ ♣r♊❜❧❡♠ ï¿œ − x 2 1 − x 1 .x 2 ✐❢ y = y 1 Ξ 1 ( x, y ) = − x 2 .y = x 2 1 + x 1 .x 2 ✐❢ y = y 2 ❚❀✉s t❀❡ r❡s♣♊♥s❡ ❢✉♥❝t✐♊♥ ♊❢ ♣❧❛②❡r ✶ ✐s ï¿œ { 1 } ✐❢ y = y 1 ✇✐t❀ ❛ ♣❛②♊✛ = − 1 − x 1 R 2 ( x 1 ) = { 0 } ✐❢ y = y 2 ✇✐t❀ ❛ ♣❛②♊✛ = 0 ❇✐❧❡✈❡❧ Pr♊❜❧❡♠s✱ â–ŒP❈❈s✱ ❛♥❞ ▌✉❧t✐✲▲❡❛❞❡r✲❋♊❧❧♊✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

  74. ❙♩ t❀❡ ◆❛s❀ ❡q✉✐❧✐❜r✐✉♠ ✇✐❧❧ ❜❡ ❜✉t✳✳✳✳ ▌▲❙❋ ❣❛♠❡✿ ✐❧❧✲♣♊s❡❞♥❡ss ï¿œ { (0 , y = y 1 ) } ✇✐t❀ ❛ ♣❛②♊✛ = 0 R 1 ( x 2 ) = { (1 , y = y 2 ) } ✇✐t❀ ❛ ♣❛②♊✛ = − 1 − x 2 ï¿œ { (1 , y = y 1 ) } ✇✐t❀ ❛ ♣❛②♊✛ = − 1 − x 1 R 2 ( x 1 ) = { (0 , y = y 2 ) } ✇✐t❀ ❛ ♣❛②♊✛ = 0 ❇✐❧❡✈❡❧ Pr♊❜❧❡♠s✱ â–ŒP❈❈s✱ ❛♥❞ ▌✉❧t✐✲▲❡❛❞❡r✲❋♊❧❧♊✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

  75. ▌▲❙❋ ❣❛♠❡✿ ✐❧❧✲♣♊s❡❞♥❡ss ï¿œ { (0 , y = y 1 ) } ✇✐t❀ ❛ ♣❛②♊✛ = 0 R 1 ( x 2 ) = { (1 , y = y 2 ) } ✇✐t❀ ❛ ♣❛②♊✛ = − 1 − x 2 ï¿œ { (1 , y = y 1 ) } ✇✐t❀ ❛ ♣❛②♊✛ = − 1 − x 1 R 2 ( x 1 ) = { (0 , y = y 2 ) } ✇✐t❀ ❛ ♣❛②♊✛ = 0 ❙♩ t❀❡ ◆❛s❀ ❡q✉✐❧✐❜r✐✉♠ ✇✐❧❧ ❜❡ ( x 1 , x 2 ) = (1 , 1) ❜✉t✳✳✳✳ ❇✐❧❡✈❡❧ Pr♊❜❧❡♠s✱ â–ŒP❈❈s✱ ❛♥❞ ▌✉❧t✐✲▲❡❛❞❡r✲❋♊❧❧♊✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

  76. ❆ ✜♥❛❧ ♠♊❞❡❧ ❋♩r t❀❡ ❉❡♠❛♥❞✲s✐❞❡ ♠❛♥❛❣❡♠❡♥t✱ ✇❡ r❡❝❡♥t❧② ✐♥tr♊❞✉❝❡❞ t❀❡ ▌✉❧t✐✲▲❡❛❞❡r✲❉✐s❥♊✐♥t✲❋♊❧❧♊✇❡r ❣❛♠❡ ❇✐❧❡✈❡❧ Pr♊❜❧❡♠s✱ â–ŒP❈❈s✱ ❛♥❞ ▌✉❧t✐✲▲❡❛❞❡r✲❋♊❧❧♊✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

  77. ❏✉st ♊♥❡ ❡①❛♠♣❧❡ ❬P❛♥❣✲❋✉❊✉s❀✐♠❛ ✵✺❪ ▲❡t ✉s ❝♊♥s✐❞❡r ❛ ✷✲❧❡❛❞❡r✲s✐♥❣❧❡✲❢♊❧❧♊✇❡r ❣❛♠❡✿ 1 − 1 2 x 2 − y ♠✐♥ x 1 ,y 2 x 1 + y ♠✐♥ x 2 ,y ï¿œ ï¿œ x 1 ∈ [0 , 1] x 2 ∈ [0 , 1] y ∈ S ( x 1 , x 2 ) y ∈ S ( x 1 , x 2 ) ✇❀❡r❡ S ( x 1 , x 2 ) ✐s t❀❡ s♩❧✉t✐♊♥ ♠❛♣ ♊❢ ♠✐♥ y ≥ 0 y ( − 1 + x 1 + x 2 ) + 1 2 y 2 ❇✐❧❡✈❡❧ Pr♊❜❧❡♠s✱ â–ŒP❈❈s✱ ❛♥❞ ▌✉❧t✐✲▲❡❛❞❡r✲❋♊❧❧♊✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

  78. ❏✉st ♊♥❡ ❡①❛♠♣❧❡ ❬P❛♥❣✲❋✉❊✉s❀✐♠❛ ✵✺❪ ▲❡t ✉s ❝♊♥s✐❞❡r ❛ ✷✲❧❡❛❞❡r✲s✐♥❣❧❡✲❢♊❧❧♊✇❡r ❣❛♠❡✿ 1 − 1 2 x 2 − y 2 ♠✐♥ x 1 ,y 1 2 x 1 + y 1 ♠✐♥ x 2 ,y 2 ï¿œ ï¿œ x 1 ∈ [0 , 1] x 2 ∈ [0 , 1] y 1 ∈ S ( x 1 , x 2 ) y 2 ∈ S ( x 1 , x 2 ) ✇❀❡r❡ S ( x 1 , x 2 ) ✐s t❀❡ s♩❧✉t✐♊♥ ♠❛♣ ♊❢ ♠✐♥ y ≥ 0 y ( − 1 + x 1 + x 2 ) + 1 2 y 2 ❇✐❧❡✈❡❧ Pr♊❜❧❡♠s✱ â–ŒP❈❈s✱ ❛♥❞ ▌✉❧t✐✲▲❡❛❞❡r✲❋♊❧❧♊✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

  79. ❚❀❡♥ t❀❡ ❘❡s♣♊♥s❡ ♠❛♣s ❛r❡ ❛♥❞ ❛♥❞ t❀✉s t❀❡r❡ ✐s ♥♊ ◆❛s❀ ❡q✉✐❧✐❜r✐✉♠✳✳✳✳✳✳✳ ❆❝t✉❛❧❧② S ( x 1 , x 2 ) = max { 0 , 1 − x 1 − x 2 } t❀✉s t❀❡ ♣r♊❜❧❡♠ ❜❡❝♊♠❡s 1 − 1 ♠✐♥ x 1 ,y 1 2 x 1 + y 1 ♠✐♥ x 2 ,y 2 2 x 2 − y 2 ï¿œ ï¿œ x 1 ∈ [0 , 1] x 2 ∈ [0 , 1] y 1 = max { 0 , 1 − x 1 − x 2 } y 2 = max { 0 , 1 − x 1 − x 2 } ❇✐❧❡✈❡❧ Pr♊❜❧❡♠s✱ â–ŒP❈❈s✱ ❛♥❞ ▌✉❧t✐✲▲❡❛❞❡r✲❋♊❧❧♊✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

  80. ❆❝t✉❛❧❧② S ( x 1 , x 2 ) = max { 0 , 1 − x 1 − x 2 } t❀✉s t❀❡ ♣r♊❜❧❡♠ ❜❡❝♊♠❡s 1 − 1 ♠✐♥ x 1 ,y 1 2 x 1 + y 1 ♠✐♥ x 2 ,y 2 2 x 2 − y 2 ï¿œ ï¿œ x 1 ∈ [0 , 1] x 2 ∈ [0 , 1] y 1 = max { 0 , 1 − x 1 − x 2 } y 2 = max { 0 , 1 − x 1 − x 2 } ❚❀❡♥ t❀❡ ❘❡s♣♊♥s❡ ♠❛♣s ❛r❡  x 1 ∈ [0 , 1 { 0 } 2 [    x 1 = 1 R 1 ( x 2 ) = { 1 − x 2 } ❛♥❞ R 2 ( x 1 ) = { 0 , 1 } 2  x 1 ∈ ] 1  { 1 } 2 , 1]  ❛♥❞ t❀✉s t❀❡r❡ ✐s ♥♊ ◆❛s❀ ❡q✉✐❧✐❜r✐✉♠✳✳✳✳✳✳✳ ❇✐❧❡✈❡❧ Pr♊❜❧❡♠s✱ â–ŒP❈❈s✱ ❛♥❞ ▌✉❧t✐✲▲❡❛❞❡r✲❋♊❧❧♊✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

  81. t❀❛t ❝❛♥ ❜❡ ♣r♊✈❡❞ t♩ ❀❛✈❡ ❛ ✭✉♥✐q✉❡✮ ◆❛s❀ ❡q✉✐❧✐❜r✐✉♠ ♥❛♠❡❧② ✇✐t❀ ✩✩✩✩ ❇✉t ❧❡t ✉s ❝♊♥s✐❞❡r t❀❡ s❧✐❣❀t❧② ♠♊❞✐✜❡❞ ♣r♊❜❧❡♠✳✳✳✳✳✳✳ 1 − 1 2 x 1 + y 1 2 x 2 − y 2 ♠✐♥ x 1 ,y 1 ♠✐♥ x 2 ,y 2   x 1 ∈ [0 , 1] x 2 ∈ [0 , 1]   y 1 = max { 0 , 1 − x 1 − x 2 } y 1 = max { 0 , 1 − x 1 − x 2 } y 2 = max { 0 , 1 − x 1 − x 2 } y 2 = max { 0 , 1 − x 1 − x 2 }   ❇✐❧❡✈❡❧ Pr♊❜❧❡♠s✱ â–ŒP❈❈s✱ ❛♥❞ ▌✉❧t✐✲▲❡❛❞❡r✲❋♊❧❧♊✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

  82. ❇✉t ❧❡t ✉s ❝♊♥s✐❞❡r t❀❡ s❧✐❣❀t❧② ♠♊❞✐✜❡❞ ♣r♊❜❧❡♠✳✳✳✳✳✳✳ 1 − 1 2 x 1 + y 1 2 x 2 − y 2 ♠✐♥ x 1 ,y 1 ♠✐♥ x 2 ,y 2   x 1 ∈ [0 , 1] x 2 ∈ [0 , 1]   y 1 = max { 0 , 1 − x 1 − x 2 } y 1 = max { 0 , 1 − x 1 − x 2 } y 2 = max { 0 , 1 − x 1 − x 2 } y 2 = max { 0 , 1 − x 1 − x 2 }   t❀❛t ❝❛♥ ❜❡ ♣r♊✈❡❞ t♩ ❀❛✈❡ ❛ ✭✉♥✐q✉❡✮ ◆❛s❀ ❡q✉✐❧✐❜r✐✉♠ ♥❛♠❡❧② ( x 1 , x 2 ) = (0 , 1) ✇✐t❀ y 1 = y 2 = 0 ✩✩✩✩ ❇✐❧❡✈❡❧ Pr♊❜❧❡♠s✱ â–ŒP❈❈s✱ ❛♥❞ ▌✉❧t✐✲▲❡❛❞❡r✲❋♊❧❧♊✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

  83. ❚❀❡ ❊✐♥❞ ♊❢ ✏tr✐❝❩✑ ✐s ❝❛❧❧❡❞ ✏❆❧❧ ❊q✉✐❧✐❜r✐✉♠ ❛♣♣r♊❛❝❀✑ ❛♥❞ ❀❛s ❜❡❡♥ ✐♥tr♊❞✉❝❡❞ ✐♥ ❆✳❆✳ ❑✉❧❩❛r♥✐ & ❯✳❱✳ ❙❀❛♥❜❀❛❣✱ ❆ ❙❀❛r❡❞✲❈♊♥str❛✐♥t ❆♣♣r♊❛❝❀ t♩ ▌✉❧t✐✲▲❡❛❞❡r ▌✉❧t✐✲❋♊❧❧♊✇❡r ●❛♠❡s ✱ ❙❡t✲❱❛❧✉❡❞ ❱❛r✳ ❆♥❛❧ ✭✷✵✶✹✮✳ ❚❀❡② ♣r♊✈❡❞ t❀❛t ❡✈❡r② ◆❛s❀ ❡q✉✐❧✐❜✐r✉♠ ✭✐♥✐t✐❛❧ ♣r♊❜❧❡♠✮ ✐s ❛ ◆❛s❀ ❡q✉✐❧✐❜r✐✉♠ ❢♊r t❀❡ ✏❛❧❧ ❡q✉✐❧✐❜r✐✉♠✑ ❢♊r♠✉❧❛t✐♊♥✳ ■t ❝♩rr❡s♣♊♥❞s t♩ t❀❡ ❝❛s❡ ✇❀❡r❡ ❡❛❝❀ ❧❡❛❞❡r t❛❊❡s ✐♥t♩ ❛❝❝♊✉♥t t❀❡ ❝♊♥❥❡❝t✉r❡s r❡❣❛r❞✐♥❣ t❀❡ ❢♊❧❧♊✇❡r ❞❡❝✐s✐♊♥ ♠❛❞❡ ❜② ❛❧❧ ♩t❀❡r ❧❡❛❞❡rs✳✳✳✳ ❇✐❧❡✈❡❧ Pr♊❜❧❡♠s✱ â–ŒP❈❈s✱ ❛♥❞ ▌✉❧t✐✲▲❡❛❞❡r✲❋♊❧❧♊✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

  84. ❙♊♠❡ ♠♩t✐✈❛t✐♊♥ ❡①❛♠♣❧❡s ❊❧❡❝tr✐❝✐t② ♠❛r❊❡ts ❇✐❧❡✈❡❧ Pr♊❜❧❡♠s✱ â–ŒP❈❈s✱ ❛♥❞ ▌✉❧t✐✲▲❡❛❞❡r✲❋♊❧❧♊✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

  85. ❆ s❀♊rt ✐♥tr♩❞✉❝t✐♊♥ t♩ ❡❧❡❝tr✐❝✐t② ♠❛r❊❡ts ❇✐❧❡✈❡❧ Pr♊❜❧❡♠s✱ â–ŒP❈❈s✱ ❛♥❞ ▌✉❧t✐✲▲❡❛❞❡r✲❋♊❧❧♊✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

  86. ❇✐❞ s❝❀❡❞✉❧❡ ♊❢ t❀❡ s♣♊t ♠❛r❊❡t ❆ s❀♊rt ✐♥tr♩❞✉❝t✐♊♥ t♩ ❡❧❡❝tr✐❝✐t② ♠❛r❊❡ts ✭❝♊♥t✳✮ ❱♊❧✉♠❡ ♊❢ ❡①❝❀❛♥❣❡s ❇✐❧❡✈❡❧ Pr♊❜❧❡♠s✱ â–ŒP❈❈s✱ ❛♥❞ ▌✉❧t✐✲▲❡❛❞❡r✲❋♊❧❧♊✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

  87. ❆ s❀♊rt ✐♥tr♩❞✉❝t✐♊♥ t♩ ❡❧❡❝tr✐❝✐t② ♠❛r❊❡ts ✭❝♊♥t✳✮ ❱♊❧✉♠❡ ♊❢ ❡①❝❀❛♥❣❡s ❇✐❞ s❝❀❡❞✉❧❡ ♊❢ t❀❡ s♣♊t ♠❛r❊❡t ❇✐❧❡✈❡❧ Pr♊❜❧❡♠s✱ â–ŒP❈❈s✱ ❛♥❞ ▌✉❧t✐✲▲❡❛❞❡r✲❋♊❧❧♊✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

  88. s✐♥❝❡ ✶✟✟✵s✱ ●❡♥❡r❛❧✐③❡❞ ◆❛s❀ ❡q✉✐❧✐❜r✐✉♠ ♣r♊❜❧❡♠ ✐s t❀❡ ♠♩st ♣♊♣✉❧❛r ✇❛② ♊❢ ♠♊❞❡❧✐♥❣ s♣♊t ❡❧❡❝tr✐❝✐t② ♠❛r❊❡ts ♩r✱ ♠♩r❡ ♣r❡❝✐s❡❧②✱ ▌✉❧t✐✲❧❡❛❞❡r✲❝♊♠♠♊♥✲❢♊❧❧♊✇❡r ❣❛♠❡ ▌♊❞❡❧✐♥❣ ❛♥ ❊❧❡❝tr✐❝✐t② ▌❛r❊❡ts ❡❧❡❝tr✐❝✐t② ♠❛r❊❡t ❝♊♥s✐sts ♊❢ ✐✮ ❣❡♥❡r❛t♩rs✎❝♊♥s✉♠❡rs i ∈ N r❡s♣❡❝t t❀❡✐r ♊✇♥ ✐♥t❡r❡sts ✐♥ ❝♊♠♣❡t✐t✐♊♥ ✇✐t❀ ♩t❀❡rs ✐✐✮ ♠❛r❊❡t ♊♣❡r❛t♩r ✭■❙❖✮ ✇❀♊ ♠❛✐♥t❛✐♥ ❡♥❡r❣② ❣❡♥❡r❛t✐♊♥ ❛♥❞ ❧♩❛❞ ❜❛❧❛♥❝❡✱ ❛♥❞ ♣r♩t❡❝t ♣✉❜❧✐❝ ✇❡❧❢❛r❡ t❀❡ ■❙❖ ❀❛s t♩ ❝♊♥s✐❞❡r✿ ✐✐✮ q✉❛♥t✐t✐❡s q i ♊❢ ❣❡♥❡r❛t❡❞✎❝♊♥s✉♠❡❞ ❡❧❡❝tr✐❝✐t② ✐✐✐✮ ❡❧❡❝tr✐❝✐t② ❞✐s♣❛t❝❀ t e ✇✐t❀ r❡s♣❡❝t t♩ tr❛♥s♠✐ss✐♊♥ ❝❛♣❛❝✐t✐❡s ❇✐❧❡✈❡❧ Pr♊❜❧❡♠s✱ â–ŒP❈❈s✱ ❛♥❞ ▌✉❧t✐✲▲❡❛❞❡r✲❋♊❧❧♊✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

  89. ▌♊❞❡❧✐♥❣ ❛♥ ❊❧❡❝tr✐❝✐t② ▌❛r❊❡ts ❡❧❡❝tr✐❝✐t② ♠❛r❊❡t ❝♊♥s✐sts ♊❢ ✐✮ ❣❡♥❡r❛t♩rs✎❝♊♥s✉♠❡rs i ∈ N r❡s♣❡❝t t❀❡✐r ♊✇♥ ✐♥t❡r❡sts ✐♥ ❝♊♠♣❡t✐t✐♊♥ ✇✐t❀ ♩t❀❡rs ✐✐✮ ♠❛r❊❡t ♊♣❡r❛t♩r ✭■❙❖✮ ✇❀♊ ♠❛✐♥t❛✐♥ ❡♥❡r❣② ❣❡♥❡r❛t✐♊♥ ❛♥❞ ❧♩❛❞ ❜❛❧❛♥❝❡✱ ❛♥❞ ♣r♩t❡❝t ♣✉❜❧✐❝ ✇❡❧❢❛r❡ t❀❡ ■❙❖ ❀❛s t♩ ❝♊♥s✐❞❡r✿ ✐✐✮ q✉❛♥t✐t✐❡s q i ♊❢ ❣❡♥❡r❛t❡❞✎❝♊♥s✉♠❡❞ ❡❧❡❝tr✐❝✐t② ✐✐✐✮ ❡❧❡❝tr✐❝✐t② ❞✐s♣❛t❝❀ t e ✇✐t❀ r❡s♣❡❝t t♩ tr❛♥s♠✐ss✐♊♥ ❝❛♣❛❝✐t✐❡s s✐♥❝❡ ✶✟✟✵s✱ ●❡♥❡r❛❧✐③❡❞ ◆❛s❀ ❡q✉✐❧✐❜r✐✉♠ ♣r♊❜❧❡♠ ✐s t❀❡ ♠♩st ♣♊♣✉❧❛r ✇❛② ♊❢ ♠♊❞❡❧✐♥❣ s♣♊t ❡❧❡❝tr✐❝✐t② ♠❛r❊❡ts ♩r✱ ♠♩r❡ ♣r❡❝✐s❡❧②✱ ▌✉❧t✐✲❧❡❛❞❡r✲❝♊♠♠♊♥✲❢♊❧❧♊✇❡r ❣❛♠❡ ❇✐❧❡✈❡❧ Pr♊❜❧❡♠s✱ â–ŒP❈❈s✱ ❛♥❞ ▌✉❧t✐✲▲❡❛❞❡r✲❋♊❧❧♊✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

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