Prs Ps - PowerPoint PPT Presentation

❲❤❛t ❞♦ ■ ✇♦r❦ ♦♥❄ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

❲❤❛t ❞♦ ■ ✇♦r❦ ♦♥❄ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

❲❤❛t ❞♦ ■ ✇♦r❦ ♦♥❄ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

❲❤❛t ❞♦ ■ ✇♦r❦ ♦♥❄ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

❛♥ ❛❞✈❡rt✐s❡♠❡♥t ❆ s❤♦rt st❛t❡ ♦❢ ❛rt ♦♥ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ❣❛♠❡s ✱ ❉✳❆✳ ❛♥❞ ❆✳ ❙✈❡♥ss♦♥✱ ✐♥ ❛ ❜♦♦❦ ❞❡❞✐❝❛t❡❞ t♦ ❙t❛❝❦❡❧❜❡r❣✱ ❡❞✐t♦rs ❆✳ ❩❡♠❦♦❤♦ ❛♥❞ ❙✳ ❉❡♠♣❡✱ ❙♣r✐♥❣❡r ❊❞✳ ✭✷✵✶✾✮ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

❇✐❧❡✈❡❧✿ s♦♠❡ ❣❡♥❡r❛❧ ❝♦♠♠❡♥ts ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

❇▲✿ ❛ ✜rst ❞❡✜♥✐t✐♦♥ ❆ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠ ❝♦♥s✐sts ✐♥ ❛♥ ✉♣♣❡r✲❧❡✈❡❧✴❧❡❛❞❡r✬s ♣r♦❜❧❡♠ ✏ min x ∈ R n ✑ F ( x, y ) � x ∈ X s✳t✳ y ∈ S ( x ) ✇❤❡r❡ ∅ � = 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✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

❆ tr✐✈✐❛❧ ❡①❛♠♣❧❡ ❈♦♥s✐❞❡r t❤❡ ❢♦❧❧♦✇✐♥❣ s✐♠♣❧❡ ❜✐❧❡✈❡❧ ♣r♦❜❧❡♠ ✏ min x ∈ R ✑ x � 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✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

❆ 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 � = 0 t❤❡r❡ ✐s ❛ ✉♥✐q✉❡ ❛ss♦❝✐❛t❡❞ s♦❧✉t✐♦♥ ♦❢ t❤❡ ❧♦✇❡r ❧❡✈❡❧ ♣r♦❜❧❡♠ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

❆ 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✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

❆ tr✐✈✐❛❧ ❡①❛♠♣❧❡ ❈♦♥s✐❞❡r t❤❡ ❢♦❧❧♦✇✐♥❣ s✐♠♣❧❡ ❜✐❧❡✈❡❧ ♣r♦❜❧❡♠ ✏ min x ∈ R ✑ − x.y � 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✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

❆♠❜✐❣✉✐t②✿ ❖♣t✐♠✐st✐❝ ❛♣♣r♦❛❝❤ ❆♥ ❖♣t✐♠✐st✐❝ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠ ❝♦♥s✐sts ✐♥ ❛♥ ✉♣♣❡r✲❧❡✈❡❧✴❧❡❛❞❡r✬s ♣r♦❜❧❡♠ min x ∈ R n min y ∈ R m F ( x, y ) � x ∈ X s✳t✳ y ∈ S ( x ) ✇❤❡r❡ ∅ � = 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✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

❆♠❜✐❣✉✐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 ) � x ∈ X s✳t✳ y ∈ S ( x ) ✇❤❡r❡ ∅ � = 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✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

❋♦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 )) � x ∈ X s✳t✳ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

❆♠❜✐❣✉✐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 )) � x ∈ X s✳t✳ ❋♦r ❡①❛♠♣❧❡ ✇❤❡♥ ❢♦r ❛♥② x, g ( x, · ) ✐s q✉❛s✐❝♦♥✈❡① ❛♥❞ f ( x, · ) ✐s str✐❝t❧② ❝♦♥✈❡①✳ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

❆♠❜✐❣✉✐t②✿ ❙❡❧❡❝t✐♦♥ ❛♣♣r♦❛❝❤ ❆♥ ✧❙❡❧❡❝t✐♦♥✲t②♣❡✧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠ ❝♦♥s✐sts ✐♥ ❛♥ ✉♣♣❡r✲❧❡✈❡❧✴❧❡❛❞❡r✬s ♣r♦❜❧❡♠ min x ∈ R n F ( x, y ( x )) � 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✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

❆♠❜✐❣✉✐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✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

♦♥❡ ❝❛♥ ❞❡✜♥❡ 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 ) � x ∈ X s✳t✳ y ∈ S ( x ) ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

❆♥ ❛❧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 ) � 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✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

♦♥❡ ❝❛♥ ❞❡✜♥❡ 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 ) � x ∈ X s✳t✳ y ∈ S ( x ) ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

❆♥ ❛❧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 ) � 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✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

■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 ) � x ∈ X ✈s s✳t✳ s✳t✳ x ∈ X y ∈ S ( x ) ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

❛♥ ♦♣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 ) � 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✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

❆♥ ❛❧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 ) � 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✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

❘❡❛❧ ❧✐❢❡✳✳✳ ❆❝t✉❛❧❧② ✉s✉❛❧❧② ✇❤❡♥ ❝♦♥s✐❞❡r✐♥❣ ❇▲ min x ∈ R n min y ∈ R m F ( x, y ) � 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✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

■♥❞❡❡❞ 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 ) � x ∈ X s✳t✳ y ∈ S ( x ) ❲❡ ♦♥❧② ✇♦r❦ ❢♦r t❤❡ ❧❡❛❞❡r✦✦ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

❘❡❛❧ ❧✐❢❡✳✳✳ ❆❝t✉❛❧❧② ✐♥ r❡❛❧ ❧✐❢❡✱ ✇❤❡♥ ❝♦♥s✐❞❡r✐♥❣ ❇▲ min x ∈ R n min y ∈ R m F ( x, y ) � 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✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

❆♥ ❡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✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

❆♥ ❡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✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

❆♥ ❡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✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

❇✐❧❡✈❡❧ ♣r♦❜❧❡♠s ❛♥❞ ▼P❈❈ r❡❢♦r♠✉❧❛t✐♦♥ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

❲❡ ❝♦♥s✐❞❡r ❛ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠ ❝♦♥s✐st✐♥❣ ✐♥ ❛♥ ✉♣♣❡r✲❧❡✈❡❧ ✴ ❧❡❛❞❡r✬s ♣r♦❜❧❡♠ ✏ min x ∈ R n ✑ F ( x, y ) s✳t✳ y ∈ S ( x ) , x ∈ X ✇❤❡r❡ ∅ � = 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✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

▼P❈❈ r❡❢♦r♠✉❧❛t✐♦♥ ❘❡♣❧❛❝✐♥❣ t❤❡ ❧♦✇❡r✲❧❡✈❡❧ ♣r♦❜❧❡♠ ❜② ✐ts ❑❑❚ ❝♦♥❞✐t✐♦♥s✱ ❣✐✈❡s ♣❧❛❝❡ t♦ ❛ ▼❛t❤❡♠❛t✐❝❛❧ Pr♦❣r❛♠ ✇✐t❤ ❈♦♠♣❧❡♠❡♥t❛r✐t② ❈♦♥str❛✐♥ts✳ ❇✐❧❡✈❡❧ ▼P❈❈ ✏ 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 � 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✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

❊①❛♠♣❧❡ ✶ ❈♦♥s✐❞❡r t❤❡ ❢♦❧❧♦✇✐♥❣ ❇✐❧❡✈❡❧ ♣r♦❜❧❡♠ ❛♥❞ ✐ts ▼P❈❈ r❡❢♦r♠✉❧❛t✐♦♥ ❇✐❧❡✈❡❧ ▼P❈❈ ✏ 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♦❧✈✐♥❣ � 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✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

✉ ② S ( · ) KKT ( · ) 1 ② ① − 1 ① −∇ F −∇ F ✭❛✮ (0 , − 1 , u ) ✐s ❛ ✭❜✮ (0 , − 1) ✐s♥✬t ❛ ❧♦❝❛❧ s♦❧✉t✐♦♥ ♦❢ ❧♦❝❛❧ s♦❧✉t✐♦♥ ♦❢ t❤❡ ▼P❈❈✱ ∀ u ∈ R + ✳ ❇✐❧❡✈❡❧ ♣r♦❜❧❡♠✳ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

❚❤❡ ♦♣t✐♠✐st✐❝ ▼P❈❈ ✭❖▼P❈❈✮✿ s✳t✳ ❚❤❡ ♣❡ss✐♠✐st✐❝ ▼P❈❈ ✭P▼P❈❈✮✿ 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✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

❖♣t✐♠✐st✐❝ ❛♥❞ P❡ss✐♠✐st✐❝ ❛♣♣r♦❛❝❤❡s ❚❤❡ ♦♣t✐♠✐st✐❝ ❇✐❧❡✈❡❧ ✭❖❇✮ ✐s ❚❤❡ ♦♣t✐♠✐st✐❝ ▼P❈❈ ✭❖▼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✐❝ ▼P❈❈ ✭P▼P❈❈✮✿ 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✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

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

▲♦❝❛❧ 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✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

❘❡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✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

❘❡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 ) � = ∅ ✱ ❛♥❞ (¯ ♥❡✐❣❤❜♦✉r❤♦♦❞ ♦❢ ¯ x ✱ Λ(¯ x, ¯ x, ¯ y, u ) ✐s ❛ ❧♦❝❛❧ s♦❧✉t✐♦♥ ♦❢ ✭❖▼P❈❈✮ ❢♦r ❡✈❡r② u ∈ Λ(¯ x, ¯ y ) ✳ ❚❤❡♥ (¯ x, ¯ y ) ✐s ❛ ❧♦❝❛❧ s♦❧✉t✐♦♥ ♦❢ ✭❖❇✮✳ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

❯♥❞❡r t❤❡ ❝♦♥✈❡①✐t② ❛ss✉♠♣t✐♦♥ ❛♥❞ s♦♠❡ ❈◗ ❡♥s✉r✐♥❣ KKT ( x ) � = ∅ , ∀ 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✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

❊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✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

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✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

❉❡✜♥✐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 ✭P▼P❈❈✮✱ ✐❢ (¯ 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✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

❘❡s✉❧ts ❢♦r t❤❡ ♣❡ss✐♠✐st✐❝ ❝❛s❡ ❚❤❡♦r❡♠ ✷ ❆ss✉♠❡ t❤❡ ❝♦♥✈❡①✐t② ❝♦♥❞✐t✐♦♥ ❛♥❞ t❤❛t KKT ( x ) � = ∅ , ∀ x ∈ X ✳ ✶ ■❢ (¯ x, ¯ y ) ✐s ❛ ❧♦❝❛❧ s♦❧✉t✐♦♥ ❢♦r ✭P❇✮✱ t❤❡♥ ❢♦r ❡❛❝❤ ¯ u ∈ Λ(¯ x, ¯ y ) ✱ (¯ x, ¯ y, ¯ u ) ✐s ❛ ❧♦❝❛❧ s♦❧✉t✐♦♥ ❢♦r ✭P▼P❈❈✮✳ ✷ ❈♦♥✈❡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 ) � = ∅ ✱ ❛♥❞ ❢♦r ❡✈❡r② u ∈ Λ(¯ x, ¯ y ) ✱ (¯ x, ¯ y, u ) ✐s ❛ ❧♦❝❛❧ s♦❧✉t✐♦♥ ♦❢ ✭P▼P❈❈✮✳ ❚❤❡♥ (¯ x, ¯ y ) ✐s ❛ ❧♦❝❛❧ s♦❧✉t✐♦♥ ♦❢ ✭P❇✮✳ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

❊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✐♦♥ ♦❢ ✭P▼P❈❈✮✱ ❢♦r ❛♥② u ∈ Λ(0 , − 1) = R + ✷ (0 , − 1) ✐s ◆❖❚ ❛ ❧♦❝❛❧ s♦❧✉t✐♦♥ ♦❢ ✭P❇✮✳ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

❊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 � = 0 ✱ ✷ (0 , 0) ✐s ◆❖❚ ❛ ❧♦❝❛❧ s♦❧✉t✐♦♥ ❢♦r ✏✭❇✮✑✳ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

❯♥❞❡r t❤❡ ❝♦♥✈❡①✐t② ❛ss✉♠♣t✐♦♥ ❛♥❞ s♦♠❡ ✭❈◗✮ ❡♥s✉r✐♥❣ KKT ( x ) � = ∅ , ∀ x ∈ X ✿ ∀ ¯ u ∈ Λ(¯ x, ¯ y ) (¯ x, ¯ y ) s♦❧ (¯ x, ¯ y, ¯ u ) s♦❧ ♦❢ ✭P❇✮ ♦❢ ✭P▼P❈❈✮ ❋✐❣✉r❡✿ ●❧♦❜❛❧ s♦❧✉t✐♦♥ ❝♦♠♣❛r✐s♦♥ ✐♥ ♣❡ss✐♠✐st✐❝ ❛♣♣r♦❛❝❤ ∀ ¯ u ∈ Λ(¯ x, ¯ y ) ✱ (¯ x, ¯ y ) ❧♦❝❛❧ (¯ x, ¯ y, ¯ u ) s♦❧ ♦❢ ✭P❇✮ ❧♦❝❛❧ s♦❧ ♦❢ ❙❧❛t❡r✬s ❈◗ ❢♦r ✭P▼P❈❈✮ ❛❧❧ x ❛r♦✉♥❞ ¯ x ❋✐❣✉r❡✿ ▲♦❝❛❧ s♦❧✉t✐♦♥s ❝♦♠♣❛r✐s♦♥ ✐♥ ♣❡ss✐♠✐st✐❝ ❛♣♣r♦❛❝❤ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

❆♥ ✐♥tr♦❞✉❝t✐♦♥ t♦ ▼▲❋● ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

❉❡♥♦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✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

◆♦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✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

●✐✈❡♥ 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✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

❆ ✈❡❝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 ) θ i ( x i , x − i ) max x i ∈ X i s✳t✳ ✇❤❡r❡ θ i ( · , x − i ) : R n i → R ✐s t❤❡ ❞❡❝✐s✐♦♥ ❢✉♥❝t✐♦♥ ❢♦r ♣❧❛②❡r i ✳ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

◆❛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 ) θ i ( x i , x − i ) max x i ∈ X i s✳t✳ ✇❤❡r❡ θ 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✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

●✐✈❡♥ 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✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

❆ ✈❡❝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 ) θ i ( x i , x − i ) max x i ∈ X i ( x − i ) s✳t✳ ✇❤❡r❡ θ i ( · , x − i ) : R ni → R ✐s t❤❡ ❞❡❝✐s✐♦♥ ❢✉♥❝t✐♦♥ ❢♦r ♣❧❛②❡r i ✳ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

●❡♥❡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 ) θ i ( x i , x − i ) max x i ∈ X i ( x − i ) s✳t✳ ✇❤❡r❡ θ 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✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

●❡♥❡r❛❧ ♠♦❞❡❧ ●❡♥❡r❛❧✐③❡❞ ◆❛s❤ ❣❛♠❡ ✭●◆❊P✮✿ min θ 1 ( x ) min θ n ( x ) x 1 � x 1 ∈ X 1 ( x − 1 ) . . . x n � x n ∈ X n ( x − n ) s✳t✳ s✳t✳ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

❆ ❝❧❛ss✐❝❛❧ ❡①✐st❡♥❝❡ r❡s✉❧t ❚❤❡♦r❡♠ ✭■❝❤✐✐s❤✐✲◗✉✐♥③✐✐ ✶✾✽✸✮ ▲❡t ❛ ●◆❊P ❜❡ ❣✐✈❡♥ ❛♥❞ s✉♣♣♦s❡ t❤❛t ✶ ❋♦r ❡❛❝❤ ν = 1 , ..., N t❤❡r❡ ❡①✐st ❛ ♥♦♥❡♠♣t②✱ ❝♦♥✈❡① ❛♥❞ ❝♦♠♣❛❝t s❡t K ν ⊂ R n ν s✉❝❤ t❤❛t t❤❡ ♣♦✐♥t✲t♦✲s❡t ♠❛♣ X ν : K − ν ⇒ K ν ✱ ✐s ❜♦t❤ ✉♣♣❡r ❛♥❞ ❧♦✇❡r s❡♠✐❝♦♥t✐♥✉♦✉s ✇✐t❤ ♥♦♥❡♠♣t② ❝❧♦s❡❞ ❛♥❞ ❝♦♥✈❡① ✈❛❧✉❡s✱ ✇❤❡r❡ K − ν := � ν ′ � = ν K ν ✳ ✷ ❋♦r ❡✈❡r② ♣❧❛②❡r ν ✱ t❤❡ ❢✉♥❝t✐♦♥ θ ν ✐s ❝♦♥t✐♥✉♦✉s ❛♥❞ θ ν ( · , x − ν ) ✐s q✉❛s✐✲❝♦♥✈❡① ♦♥ X ν ( x − ν ) ✳ ❚❤❡♥ ❛ ❣❡♥❡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✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

❙tr✉❝t✉r❡ ♦❢ t❤❡ s❡t ♦❢ ●◆❊Ps ❊①❛♠♣❧❡ ▲❡t x = ( x 1 , x 2 ) ∈ [0 , 4] 2 ❛♥❞ f ν ( x ) := d T ν ( 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 ν ( x 1 , x 2 ) | x ν ∈ [0 , 4] S ν ( x − ν ) := ❛r❣♠✐♥ x ν � � ✳ ❲❡ s❡❡ t❤❛t x 1 ∈ [0 , 4] | ( x 1 , x 2 ) ∈ T 1 ❢♦r x 2 ∈ [0 , 1] S 1 ( x 2 ) = � � 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 ) = � � S 2 ( x 1 ) = { 2 } ❢♦r ❛❧❧ x 1 ∈ (1 , 4]) ✳ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

❙tr✉❝t✉r❡ ♦❢ t❤❡ s❡t ♦❢ ●◆❊Ps ✭❝♦♥t✳✮ x 2 S 2 ( · ) S 1 ( · ) x 1 ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

❆ ♣❛rt✐❝✉❧❛r ❝❛s❡ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r✲●❛♠❡ ✭▼▲❋●✮✿ min θ 1 ( x, y ) min θ n ( x, y ) x 1 x n y 1 ,..,y p � x 1 ∈ X 1 ( x − 1 ) y 1 ,..,y p � 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 ) � y ∈ Y ( x ) � y ∈ Y ( x ) . . . s✳t✳ s✳t✳ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

❛♥❞ ❛♥♦t❤❡r ♣r♦❜❧❡♠ ❙✐♥❣❧❡✲▲❡❛❞❡r✲▼✉❧t✐✲❋♦❧❧♦✇❡r✲●❛♠❡ ✭❙▲▼❋●✮✿ min θ 1 ( x, y ) x y 1 ,..,y p � 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 ) . . . � y ∈ Y ( x ) � y ∈ Y ( x ) s✳t✳ s✳t✳ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

❆ ♣❛rt✐❝✉❧❛r ❝❛s❡ ▼✉❧t✐✲▲❡❛❞❡r✲❙✐♥❣❧❡✲❋♦❧❧♦✇❡r✲●❛♠❡ ✭▼▲❙❋●✮✿ min θ 1 ( x, y ) min θ n ( x, y ) x 1 x n y 1 ,..,y p � x 1 ∈ X 1 ( x − 1 ) y 1 ,..,y p � 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 ) � y ∈ Y ( x ) s✳t✳ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

❊①❡r❝✐s❡✿ P❧❡❛s❡ ❛♥❛❧②s❡ t❤✐s s♠❛❧❧ ❡①❛♠♣❧❡✳✳✳ ▼▲❙❋ ❣❛♠❡✿ ✐❧❧✲♣♦s❡❞♥❡ss min x 1 ,y θ 1 ( x 1 , x 2 , y ) = x 1 .y min x 2 ,y θ 1 ( x 1 , x 2 , y ) = − x 2 .y � � 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✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

▼▲❙❋ ❣❛♠❡✿ ✐❧❧✲♣♦s❡❞♥❡ss min x 1 ,y θ 1 ( x 1 , x 2 , y ) = x 1 .y min x 2 ,y θ 1 ( x 1 , x 2 , y ) = − x 2 .y � � 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✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

❚❤❡ 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✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

▼▲❙❋ ❣❛♠❡✿ ✐❧❧✲♣♦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✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

❚❤✉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♦❜❧❡♠ � x 2 1 + x 1 .x 2 ✐❢ y = y 1 θ 1 ( x, y ) = x 1 .y = − x 2 1 − x 1 .x 2 ✐❢ y = y 2 ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

▼▲❙❋ ❣❛♠❡✿ ✐❧❧✲♣♦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♦❜❧❡♠ � x 2 1 + x 1 .x 2 ✐❢ y = y 1 θ 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 � { 0 } ✐❢ y = y 1 ✇✐t❤ ❛ ♣❛②♦✛ = 0 R 1 ( x 2 ) = { 1 } ✐❢ y = y 2 ✇✐t❤ ❛ ♣❛②♦✛ = − 1 − x 2 ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

❚❤✉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♦❜❧❡♠ � − x 2 1 − x 1 .x 2 ✐❢ y = y 1 θ 1 ( x, y ) = − x 2 .y = x 2 1 + x 1 .x 2 ✐❢ y = y 2 ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

▼▲❙❋ ❣❛♠❡✿ ✐❧❧✲♣♦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♦❜❧❡♠ � − x 2 1 − x 1 .x 2 ✐❢ y = y 1 θ 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 � { 1 } ✐❢ y = y 1 ✇✐t❤ ❛ ♣❛②♦✛ = − 1 − x 1 R 2 ( x 1 ) = { 0 } ✐❢ y = y 2 ✇✐t❤ ❛ ♣❛②♦✛ = 0 ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

❙♦ t❤❡ ◆❛s❤ ❡q✉✐❧✐❜r✐✉♠ ✇✐❧❧ ❜❡ ❜✉t✳✳✳✳ ▼▲❙❋ ❣❛♠❡✿ ✐❧❧✲♣♦s❡❞♥❡ss � { (0 , y = y 1 ) } ✇✐t❤ ❛ ♣❛②♦✛ = 0 R 1 ( x 2 ) = { (1 , y = y 2 ) } ✇✐t❤ ❛ ♣❛②♦✛ = − 1 − x 2 � { (1 , y = y 1 ) } ✇✐t❤ ❛ ♣❛②♦✛ = − 1 − x 1 R 2 ( x 1 ) = { (0 , y = y 2 ) } ✇✐t❤ ❛ ♣❛②♦✛ = 0 ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

▼▲❙❋ ❣❛♠❡✿ ✐❧❧✲♣♦s❡❞♥❡ss � { (0 , y = y 1 ) } ✇✐t❤ ❛ ♣❛②♦✛ = 0 R 1 ( x 2 ) = { (1 , y = y 2 ) } ✇✐t❤ ❛ ♣❛②♦✛ = − 1 − x 2 � { (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✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

❆ ✜♥❛❧ ♠♦❞❡❧ ❋♦r t❤❡ ❉❡♠❛♥❞✲s✐❞❡ ♠❛♥❛❣❡♠❡♥t✱ ✇❡ r❡❝❡♥t❧② ✐♥tr♦❞✉❝❡❞ t❤❡ ▼✉❧t✐✲▲❡❛❞❡r✲❉✐s❥♦✐♥t✲❋♦❧❧♦✇❡r ❣❛♠❡ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

❏✉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 � � 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✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

❏✉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 � � 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✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

❚❤❡♥ 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 � � 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✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

❆❝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 � � 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✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

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✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

❇✉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✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

❚❤❡ ❦✐♥❞ ♦❢ ✏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✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

❙♦♠❡ ♠♦t✐✈❛t✐♦♥ ❡①❛♠♣❧❡s ❊❧❡❝tr✐❝✐t② ♠❛r❦❡ts ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

❆ s❤♦rt ✐♥tr♦❞✉❝t✐♦♥ t♦ ❡❧❡❝tr✐❝✐t② ♠❛r❦❡ts ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

❇✐❞ s❝❤❡❞✉❧❡ ♦❢ t❤❡ s♣♦t ♠❛r❦❡t ❆ s❤♦rt ✐♥tr♦❞✉❝t✐♦♥ t♦ ❡❧❡❝tr✐❝✐t② ♠❛r❦❡ts ✭❝♦♥t✳✮ ❱♦❧✉♠❡ ♦❢ ❡①❝❤❛♥❣❡s ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

❆ s❤♦rt ✐♥tr♦❞✉❝t✐♦♥ t♦ ❡❧❡❝tr✐❝✐t② ♠❛r❦❡ts ✭❝♦♥t✳✮ ❱♦❧✉♠❡ ♦❢ ❡①❝❤❛♥❣❡s ❇✐❞ s❝❤❡❞✉❧❡ ♦❢ t❤❡ s♣♦t ♠❛r❦❡t ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

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✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧

▼♦❞❡❧✐♥❣ ❛♥ ❊❧❡❝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✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s ❉✐❞✐❡r ❆✉ss❡❧