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Prs Ps trr s r ss Prs P
❉✐❞✐❡r ❆✉ss❡❧
▲❛❜✳ Pr♦♠❡s ❯P❘ ❈◆❘❙ ✽✺✷✶✱ ❯♥✐✈❡rs✐t② ♦❢ P❡r♣✐❣♥❛♥✱ ❋r❛♥❝❡
❆▲❖P ❛✉t✉♠♥ s❝❤♦♦❧ ✲ ❖❝t♦❜❡r ✶✹t❤✱ ✷✵✷✵
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
Pr♦❢❡ss♦r ✐♥ ❆♣♣❧✐❡❞ ▼❛t❤❡♠❛t✐❝s ❛t ❯♥✐✈✳ ♦❢ P❡r♣✐❣♥❛♥ P❡r♣✐❣♥❛♥✱ ❋r❛♥❝❡
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
❙♦♠❡ ♣❡rs♦♥❛❧ ❞❛t❛ Pr♦❢❡ss♦r ✐♥ ❆♣♣❧✐❡❞ ▼❛t❤❡♠❛t✐❝s ❛t ❯♥✐✈✳ ♦❢ P❡r♣✐❣♥❛♥ ❘❡s❡❛r❝❤ t♦♣✐❝s✿ ✲ ❇✐❧❡✈❡❧ ♣r♦❣r❛♠♠✐♥❣✱ ◆❛s❤ ❣❛♠❡s ❛♥❞ ✐♥ ♣❛rt✐❝✉❧❛r ▼✉❧t✐✲❧❡❛❞❡r✲❢♦❧❧♦✇❡r ❣❛♠❡s ✲ ❊♥❡r❣② ♠❛♥❛❣❡♠❡♥t✿
❊❧❡❝tr✐❝✐t② ♠❛r❦❡ts ■♥❞✉str✐❛❧ ❊❝♦✲P❛r❦s ✭■❊P✮ ❉❡♠❛♥❞✲s✐❞❡ ♠❛♥❛❣❡♠❡♥t
✲ ❱❛r✐❛t✐♦♥❛❧ ❛♥❞ q✉❛s✐✲✈❛r✐❛t✐♦♥❛❧ ✐♥❡q✉❛❧✐t✐❡s ✲ ◗✉❛s✐❝♦♥✈❡① ♦♣t✐♠✐③❛t✐♦♥ ❘❡s❡❛r❝❤ ❧❛❜✳✿ P❘❖▼❊❙ ✭❈◆❘❙✮
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
❙♦♠❡ ♣❡rs♦♥❛❧ ❞❛t❛ Pr♦❢❡ss♦r ✐♥ ❆♣♣❧✐❡❞ ▼❛t❤❡♠❛t✐❝s ❛t ❯♥✐✈✳ ♦❢ P❡r♣✐❣♥❛♥ ❘❡s❡❛r❝❤ t♦♣✐❝s✿ ✲ ❇✐❧❡✈❡❧ ♣r♦❣r❛♠♠✐♥❣✱ ◆❛s❤ ❣❛♠❡s ❛♥❞ ✐♥ ♣❛rt✐❝✉❧❛r ▼✉❧t✐✲❧❡❛❞❡r✲❢♦❧❧♦✇❡r ❣❛♠❡s ✲ ❊♥❡r❣② ♠❛♥❛❣❡♠❡♥t✿
❊❧❡❝tr✐❝✐t② ♠❛r❦❡ts ■♥❞✉str✐❛❧ ❊❝♦✲P❛r❦s ✭■❊P✮ ❉❡♠❛♥❞✲s✐❞❡ ♠❛♥❛❣❡♠❡♥t
✲ ❱❛r✐❛t✐♦♥❛❧ ❛♥❞ q✉❛s✐✲✈❛r✐❛t✐♦♥❛❧ ✐♥❡q✉❛❧✐t✐❡s ✲ ◗✉❛s✐❝♦♥✈❡① ♦♣t✐♠✐③❛t✐♦♥ ❘❡s❡❛r❝❤ ❧❛❜✳✿ P❘❖▼❊❙ ✭❈◆❘❙✮
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
❙♦♠❡ ♣❡rs♦♥❛❧ ❞❛t❛ Pr♦❢❡ss♦r ✐♥ ❆♣♣❧✐❡❞ ▼❛t❤❡♠❛t✐❝s ❛t ❯♥✐✈✳ ♦❢ P❡r♣✐❣♥❛♥ ❘❡s❡❛r❝❤ t♦♣✐❝s✿ ✲ ❇✐❧❡✈❡❧ ♣r♦❣r❛♠♠✐♥❣✱ ◆❛s❤ ❣❛♠❡s ❛♥❞ ✐♥ ♣❛rt✐❝✉❧❛r ▼✉❧t✐✲❧❡❛❞❡r✲❢♦❧❧♦✇❡r ❣❛♠❡s ✲ ❊♥❡r❣② ♠❛♥❛❣❡♠❡♥t✿
❊❧❡❝tr✐❝✐t② ♠❛r❦❡ts ■♥❞✉str✐❛❧ ❊❝♦✲P❛r❦s ✭■❊P✮ ❉❡♠❛♥❞✲s✐❞❡ ♠❛♥❛❣❡♠❡♥t
✲ ❱❛r✐❛t✐♦♥❛❧ ❛♥❞ q✉❛s✐✲✈❛r✐❛t✐♦♥❛❧ ✐♥❡q✉❛❧✐t✐❡s ✲ ◗✉❛s✐❝♦♥✈❡① ♦♣t✐♠✐③❛t✐♦♥ ❘❡s❡❛r❝❤ ❧❛❜✳✿ P❘❖▼❊❙ ✭❈◆❘❙✮
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
❙♦♠❡ ♣❡rs♦♥❛❧ ❞❛t❛ Pr♦❢❡ss♦r ✐♥ ❆♣♣❧✐❡❞ ▼❛t❤❡♠❛t✐❝s ❛t ❯♥✐✈✳ ♦❢ P❡r♣✐❣♥❛♥ ❘❡s❡❛r❝❤ t♦♣✐❝s✿ ✲ ❇✐❧❡✈❡❧ ♣r♦❣r❛♠♠✐♥❣✱ ◆❛s❤ ❣❛♠❡s ❛♥❞ ✐♥ ♣❛rt✐❝✉❧❛r ▼✉❧t✐✲❧❡❛❞❡r✲❢♦❧❧♦✇❡r ❣❛♠❡s ✲ ❊♥❡r❣② ♠❛♥❛❣❡♠❡♥t✿
❊❧❡❝tr✐❝✐t② ♠❛r❦❡ts ■♥❞✉str✐❛❧ ❊❝♦✲P❛r❦s ✭■❊P✮ ❉❡♠❛♥❞✲s✐❞❡ ♠❛♥❛❣❡♠❡♥t
✲ ❱❛r✐❛t✐♦♥❛❧ ❛♥❞ q✉❛s✐✲✈❛r✐❛t✐♦♥❛❧ ✐♥❡q✉❛❧✐t✐❡s ✲ ◗✉❛s✐❝♦♥✈❡① ♦♣t✐♠✐③❛t✐♦♥ ❘❡s❡❛r❝❤ ❧❛❜✳✿ P❘❖▼❊❙ ✭❈◆❘❙✮
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
❙♦♠❡ ♣❡rs♦♥❛❧ ❞❛t❛ Pr♦❢❡ss♦r ✐♥ ❆♣♣❧✐❡❞ ▼❛t❤❡♠❛t✐❝s ❛t ❯♥✐✈✳ ♦❢ P❡r♣✐❣♥❛♥ ❘❡s❡❛r❝❤ t♦♣✐❝s✿ ✲ ❇✐❧❡✈❡❧ ♣r♦❣r❛♠♠✐♥❣✱ ◆❛s❤ ❣❛♠❡s ❛♥❞ ✐♥ ♣❛rt✐❝✉❧❛r ▼✉❧t✐✲❧❡❛❞❡r✲❢♦❧❧♦✇❡r ❣❛♠❡s ✲ ❊♥❡r❣② ♠❛♥❛❣❡♠❡♥t✿
❊❧❡❝tr✐❝✐t② ♠❛r❦❡ts ■♥❞✉str✐❛❧ ❊❝♦✲P❛r❦s ✭■❊P✮ ❉❡♠❛♥❞✲s✐❞❡ ♠❛♥❛❣❡♠❡♥t
✲ ❱❛r✐❛t✐♦♥❛❧ ❛♥❞ q✉❛s✐✲✈❛r✐❛t✐♦♥❛❧ ✐♥❡q✉❛❧✐t✐❡s ✲ ◗✉❛s✐❝♦♥✈❡① ♦♣t✐♠✐③❛t✐♦♥ ❘❡s❡❛r❝❤ ❧❛❜✳✿ P❘❖▼❊❙ ✭❈◆❘❙✮
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
❙♦♠❡ ♣❡rs♦♥❛❧ ❞❛t❛ Pr♦❢❡ss♦r ✐♥ ❆♣♣❧✐❡❞ ▼❛t❤❡♠❛t✐❝s ❛t ❯♥✐✈✳ ♦❢ P❡r♣✐❣♥❛♥ ❘❡s❡❛r❝❤ t♦♣✐❝s✿ ✲ ❇✐❧❡✈❡❧ ♣r♦❣r❛♠♠✐♥❣✱ ◆❛s❤ ❣❛♠❡s ❛♥❞ ✐♥ ♣❛rt✐❝✉❧❛r ▼✉❧t✐✲❧❡❛❞❡r✲❢♦❧❧♦✇❡r ❣❛♠❡s ✲ ❊♥❡r❣② ♠❛♥❛❣❡♠❡♥t✿
❊❧❡❝tr✐❝✐t② ♠❛r❦❡ts ■♥❞✉str✐❛❧ ❊❝♦✲P❛r❦s ✭■❊P✮ ❉❡♠❛♥❞✲s✐❞❡ ♠❛♥❛❣❡♠❡♥t
✲ ❱❛r✐❛t✐♦♥❛❧ ❛♥❞ q✉❛s✐✲✈❛r✐❛t✐♦♥❛❧ ✐♥❡q✉❛❧✐t✐❡s ✲ ◗✉❛s✐❝♦♥✈❡① ♦♣t✐♠✐③❛t✐♦♥ ❘❡s❡❛r❝❤ ❧❛❜✳✿ P❘❖▼❊❙ ✭❈◆❘❙✮
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
❙♦♠❡ ♣❡rs♦♥❛❧ ❞❛t❛ Pr♦❢❡ss♦r ✐♥ ❆♣♣❧✐❡❞ ▼❛t❤❡♠❛t✐❝s ❛t ❯♥✐✈✳ ♦❢ P❡r♣✐❣♥❛♥ ❘❡s❡❛r❝❤ t♦♣✐❝s✿ ✲ ❇✐❧❡✈❡❧ ♣r♦❣r❛♠♠✐♥❣✱ ◆❛s❤ ❣❛♠❡s ❛♥❞ ✐♥ ♣❛rt✐❝✉❧❛r ▼✉❧t✐✲❧❡❛❞❡r✲❢♦❧❧♦✇❡r ❣❛♠❡s ✲ ❊♥❡r❣② ♠❛♥❛❣❡♠❡♥t✿
❊❧❡❝tr✐❝✐t② ♠❛r❦❡ts ■♥❞✉str✐❛❧ ❊❝♦✲P❛r❦s ✭■❊P✮ ❉❡♠❛♥❞✲s✐❞❡ ♠❛♥❛❣❡♠❡♥t
✲ ❱❛r✐❛t✐♦♥❛❧ ❛♥❞ q✉❛s✐✲✈❛r✐❛t✐♦♥❛❧ ✐♥❡q✉❛❧✐t✐❡s ✲ ◗✉❛s✐❝♦♥✈❡① ♦♣t✐♠✐③❛t✐♦♥ ❘❡s❡❛r❝❤ ❧❛❜✳✿ P❘❖▼❊❙ ✭❈◆❘❙✮
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
❙♦♠❡ ♣❡rs♦♥❛❧ ❞❛t❛ Pr♦❢❡ss♦r ✐♥ ❆♣♣❧✐❡❞ ▼❛t❤❡♠❛t✐❝s ❛t ❯♥✐✈✳ ♦❢ P❡r♣✐❣♥❛♥ ❘❡s❡❛r❝❤ t♦♣✐❝s✿ ✲ ❇✐❧❡✈❡❧ ♣r♦❣r❛♠♠✐♥❣✱ ◆❛s❤ ❣❛♠❡s ❛♥❞ ✐♥ ♣❛rt✐❝✉❧❛r ▼✉❧t✐✲❧❡❛❞❡r✲❢♦❧❧♦✇❡r ❣❛♠❡s ✲ ❊♥❡r❣② ♠❛♥❛❣❡♠❡♥t✿
❊❧❡❝tr✐❝✐t② ♠❛r❦❡ts ■♥❞✉str✐❛❧ ❊❝♦✲P❛r❦s ✭■❊P✮ ❉❡♠❛♥❞✲s✐❞❡ ♠❛♥❛❣❡♠❡♥t
✲ ❱❛r✐❛t✐♦♥❛❧ ❛♥❞ q✉❛s✐✲✈❛r✐❛t✐♦♥❛❧ ✐♥❡q✉❛❧✐t✐❡s ✲ ◗✉❛s✐❝♦♥✈❡① ♦♣t✐♠✐③❛t✐♦♥ ❘❡s❡❛r❝❤ ❧❛❜✳✿ P❘❖▼❊❙ ✭❈◆❘❙✮
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
❆ s❤♦rt st❛t❡ ♦❢ ❛rt ♦♥ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ❣❛♠❡s✱ ❉✳❆✳ ❛♥❞ ❆✳ ❙✈❡♥ss♦♥✱ ✐♥ ❛ ❜♦♦❦ ❞❡❞✐❝❛t❡❞ t♦ ❙t❛❝❦❡❧❜❡r❣✱ ❡❞✐t♦rs ❆✳ ❩❡♠❦♦❤♦ ❛♥❞ ❙✳ ❉❡♠♣❡✱ ❙♣r✐♥❣❡r ❊❞✳ ✭✷✵✶✾✮
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
❆ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠ ❝♦♥s✐sts ✐♥ ❛♥ ✉♣♣❡r✲❧❡✈❡❧✴❧❡❛❞❡r✬s ♣r♦❜❧❡♠ ✏minx∈Rn✑ F(x, y) s✳t✳ x ∈ X y ∈ S(x) ✇❤❡r❡ ∅ = X ⊂ Rn ❛♥❞ S(x) st❛♥❞s ❢♦r t❤❡ s♦❧✉t✐♦♥ s❡t ♦❢ ✐ts ❧♦✇❡r✲❧❡✈❡❧✴❢♦❧❧♦✇❡r✬s ♣r♦❜❧❡♠ miny∈Rm f(x, y) s✳t g(x, y) ≤ 0
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
❆ tr✐✈✐❛❧ ❡①❛♠♣❧❡
❈♦♥s✐❞❡r t❤❡ ❢♦❧❧♦✇✐♥❣ s✐♠♣❧❡ ❜✐❧❡✈❡❧ ♣r♦❜❧❡♠
✏minx∈R✑ x s✳t✳ x ∈ [−1, 1] y ∈ S(x)
✇✐t❤ S(x) = ✏y s♦❧✈✐♥❣
miny∈R −xy s✳t x2(y2 − 1) ≤ 0 ✑
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
▲♦✇❡r ❧❡✈❡❧ ♣r♦❜❧❡♠✿ miny∈R −x.y s✳t x2(y2 − 1) ≤ 0 ✑ ◆♦t❡ t❤❛t t❤❡ s♦❧✉t✐♦♥ ♠❛♣ ♦❢ t❤✐s ❝♦♥✈❡① ♣r♦❜❧❡♠ ✐s S(x) := {1} x < 0 {−1} x > 0 R x = 0 ❚❤✉s ❢♦r ❡❛❝❤ x = 0 t❤❡r❡ ✐s ❛ ✉♥✐q✉❡ ❛ss♦❝✐❛t❡❞ s♦❧✉t✐♦♥ ♦❢ t❤❡ ❧♦✇❡r ❧❡✈❡❧ ♣r♦❜❧❡♠
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
▲♦✇❡r ❧❡✈❡❧ ♣r♦❜❧❡♠✿ miny∈R −xy s✳t x2(y2 − 1) ≤ 0 ✑ ◆♦t❡ t❤❛t t❤❡ s♦❧✉t✐♦♥ ♠❛♣ ♦❢ t❤✐s ❝♦♥✈❡① ♣r♦❜❧❡♠ ✐s
② ① −∇F −1 1
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
❆ tr✐✈✐❛❧ ❡①❛♠♣❧❡
❈♦♥s✐❞❡r t❤❡ ❢♦❧❧♦✇✐♥❣ s✐♠♣❧❡ ❜✐❧❡✈❡❧ ♣r♦❜❧❡♠
✏minx∈R✑ −x.y s✳t✳ x ∈ [−1, 1] y ∈ S(x)
✇✐t❤ S(x) = ✏y s♦❧✈✐♥❣
S(x) := {1} x < 0 {−1} x > 0 R x = 0
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
❆♥ ❖♣t✐♠✐st✐❝ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠ ❝♦♥s✐sts ✐♥ ❛♥ ✉♣♣❡r✲❧❡✈❡❧✴❧❡❛❞❡r✬s ♣r♦❜❧❡♠ minx∈Rn miny∈Rm F(x, y) s✳t✳ x ∈ X y ∈ S(x) ✇❤❡r❡ ∅ = X ⊂ Rn ❛♥❞ S(x) st❛♥❞s ❢♦r t❤❡ s♦❧✉t✐♦♥ s❡t ♦❢ ✐ts ❧♦✇❡r✲❧❡✈❡❧✴❢♦❧❧♦✇❡r✬s ♣r♦❜❧❡♠ miny∈Rm f(x, y) s✳t g(x, y) ≤ 0
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
❆♥ P❡ss✐♠✐st✐❝ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠ ❝♦♥s✐sts ✐♥ ❛♥ ✉♣♣❡r✲❧❡✈❡❧✴❧❡❛❞❡r✬s ♣r♦❜❧❡♠ minx∈Rn maxy∈Rm F(x, y) s✳t✳ x ∈ X y ∈ S(x) ✇❤❡r❡ ∅ = X ⊂ Rn ❛♥❞ S(x) st❛♥❞s ❢♦r t❤❡ s♦❧✉t✐♦♥ s❡t ♦❢ ✐ts ❧♦✇❡r✲❧❡✈❡❧✴❢♦❧❧♦✇❡r✬s ♣r♦❜❧❡♠ miny∈Rm f(x, y) s✳t g(x, y) ≤ 0
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
❆♥❞ ♦❢ ❝♦✉rs❡ t❤❡ ✧❝♦♥❢♦rt❛❜❧❡ s✐t✉❛t✐♦♥✧ ❝♦rr❡s♣♦♥❞s t♦ t❤❡ ❝❛s❡ ♦❢ ❛ ✉♥✐q✉❡ r❡s♣♦♥s❡ ∀ x ∈ X, S(x) = {y(x)}. ❚❤❡♥ minx∈Rn F(x, y(x)) s✳t✳
❋♦r ❡①❛♠♣❧❡ ✇❤❡♥ ❢♦r ❛♥② ✐s q✉❛s✐❝♦♥✈❡① ❛♥❞ ✐s str✐❝t❧② ❝♦♥✈❡①✳
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
❆♥❞ ♦❢ ❝♦✉rs❡ t❤❡ ✧❝♦♥❢♦rt❛❜❧❡ s✐t✉❛t✐♦♥✧ ❝♦rr❡s♣♦♥❞s t♦ t❤❡ ❝❛s❡ ♦❢ ❛ ✉♥✐q✉❡ r❡s♣♦♥s❡ ∀ x ∈ X, S(x) = {y(x)}. ❚❤❡♥ minx∈Rn F(x, y(x)) s✳t✳
❋♦r ❡①❛♠♣❧❡ ✇❤❡♥ ❢♦r ❛♥② x, g(x, ·) ✐s q✉❛s✐❝♦♥✈❡① ❛♥❞ f(x, ·) ✐s str✐❝t❧② ❝♦♥✈❡①✳
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
❆♥ ✧❙❡❧❡❝t✐♦♥✲t②♣❡✧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠ ❝♦♥s✐sts ✐♥ ❛♥ ✉♣♣❡r✲❧❡✈❡❧✴❧❡❛❞❡r✬s ♣r♦❜❧❡♠ minx∈Rn F(x, y(x)) s✳t✳ x ∈ X y(x) ✐s ❛ ✉♥✐q✉❡❧② ❞❡t❡r♠✐♥❡❞ s❡❧❡❝t✐♦♥ ♦❢ S(x) ❏✳ ❊s❝♦❜❛r ✫ ❆✳ ❏♦❢ré✱ ❊q✉✐❧✐❜r✐✉♠ ❆♥❛❧②s✐s ♦❢ ❊❧❡❝tr✐❝✐t② ❆✉❝t✐♦♥s ✭✷✵✶✶✮
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
■♥ ♦♥❡ ♦❢ 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✮
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
■♥st❡❛❞ ♦❢ ❝♦♥s✐❞❡r✐♥❣ t❤❡ ♣r❡✈✐♦✉s ✭♦♣t✐♠✐st✐❝✮ ❢♦r♠✉❧❛t✐♦♥ ♦❢ ❇▲✿ minx∈Rn miny∈Rm F(x, y) s✳t✳ x ∈ X y ∈ S(x) ♦♥❡ ❝❛♥ ❞❡✜♥❡ t❤❡ ✭♦♣t✐♠✐st✐❝✮ ✈❛❧✉❡ ❢✉♥❝t✐♦♥ ✭✶✮ ❛♥❞ t❤❡ ❇❧ ♣r♦❜❧❡♠ ❜❡❝♦♠❡s s✳t✳
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
■♥st❡❛❞ ♦❢ ❝♦♥s✐❞❡r✐♥❣ t❤❡ ♣r❡✈✐♦✉s ✭♦♣t✐♠✐st✐❝✮ ❢♦r♠✉❧❛t✐♦♥ ♦❢ ❇▲✿ minx∈Rn miny∈Rm F(x, y) s✳t✳ x ∈ X y ∈ S(x) ♦♥❡ ❝❛♥ ❞❡✜♥❡ t❤❡ ✭♦♣t✐♠✐st✐❝✮ ✈❛❧✉❡ ❢✉♥❝t✐♦♥ ϕmin(x) = min
y {F(x, y) : g(x, y) ≤ 0}
✭✶✮ ❛♥❞ t❤❡ ❇❧ ♣r♦❜❧❡♠ ❜❡❝♦♠❡s minx∈Rn ϕmin(x) s✳t✳ x ∈ X
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
■♥st❡❛❞ ♦❢ ❝♦♥s✐❞❡r✐♥❣ t❤❡ ♣r❡✈✐♦✉s ✭♣❡ss✐♠✐st✐❝✮ ❢♦r♠✉❧❛t✐♦♥ ♦❢ ❇▲✿ minx∈Rn maxy∈Rm F(x, y) s✳t✳ x ∈ X y ∈ S(x) ♦♥❡ ❝❛♥ ❞❡✜♥❡ t❤❡ ✭♣❡ss✐♠✐st✐❝✮ ✈❛❧✉❡ ❢✉♥❝t✐♦♥ ✭✷✮ ❛♥❞ t❤❡ ❇❧ ♣r♦❜❧❡♠ ❜❡❝♦♠❡s s✳t✳
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
■♥st❡❛❞ ♦❢ ❝♦♥s✐❞❡r✐♥❣ t❤❡ ♣r❡✈✐♦✉s ✭♣❡ss✐♠✐st✐❝✮ ❢♦r♠✉❧❛t✐♦♥ ♦❢ ❇▲✿ minx∈Rn maxy∈Rm F(x, y) s✳t✳ x ∈ X y ∈ S(x) ♦♥❡ ❝❛♥ ❞❡✜♥❡ t❤❡ ✭♣❡ss✐♠✐st✐❝✮ ✈❛❧✉❡ ❢✉♥❝t✐♦♥ ϕmax(x) = max
y {F(x, y) : g(x, y) ≤ 0}
✭✷✮ ❛♥❞ t❤❡ ❇❧ ♣r♦❜❧❡♠ ❜❡❝♦♠❡s minx∈Rn ϕmax(x) s✳t✳ x ∈ X
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
❚❤✐s ✐s t❤❡ ♣♦✐♥t ♦❢ ✈✐❡✇ ♣r❡s❡♥t❡❞ ✐♥ ❙t❡♣❤❛♥ ❉❡♠♣❡✬s ❜♦♦❦✿ minx∈Rn min / maxy∈Rm F(x, y) s✳t✳
y ∈ S(x) ✈s minx∈Rn ϕmin/max(x) s✳t✳ x ∈ X ■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❄
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
❚❤✐s ✐s t❤❡ ♣♦✐♥t ♦❢ ✈✐❡✇ ♣r❡s❡♥t❡❞ ✐♥ ❙t❡♣❤❛♥ ❉❡♠♣❡✬s ❜♦♦❦✿ minx∈Rn min / maxy∈Rm F(x, y) s✳t✳
y ∈ S(x) ✈s minx∈Rn ϕmin/max(x) s✳t✳ x ∈ X ■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❄
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
❚❤✐s ✐s t❤❡ ♣♦✐♥t ♦❢ ✈✐❡✇ ♣r❡s❡♥t❡❞ ✐♥ ❙t❡♣❤❛♥ ❉❡♠♣❡✬s ❜♦♦❦✿ minx∈Rn min / maxy∈Rm F(x, y) s✳t✳
y ∈ S(x) ✈s minx∈Rn ϕmin/max(x) s✳t✳ x ∈ 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❄
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
❆❝t✉❛❧❧② ✉s✉❛❧❧② ✇❤❡♥ ❝♦♥s✐❞❡r✐♥❣ ❇▲ minx∈Rn miny∈Rm F(x, y) s✳t✳ x ∈ X y ∈ S(x) ♣❡♦♣❧❡ s❛② ❙t❡♣ ❆✿ t❤❡ ❧❡❛❞❡r ♣❧❛②s ✜rst ❙t❡♣ ❇✿ t❤❡ ❢♦❧❧♦✇❡r r❡❛❝ts ❇✉t ✐♥ r❡❛❧ ❧✐❢❡ ✐t✬s ❛ ❧✐tt❧❡ ❜✐t ♠♦r❡ ❝♦♠♣❧❡①✳✳✳✳
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
❆❝t✉❛❧❧② ✐♥ r❡❛❧ ❧✐❢❡✱ ✇❤❡♥ ❝♦♥s✐❞❡r✐♥❣ ❇▲ minx∈Rn miny∈Rm F(x, y) s✳t✳ x ∈ X 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✐♦♥ ♦r ♦❢ t❤❡ ❇▲ ♠♦❞❡❧ ❙t❡♣ ✷✿ t❤❡ ❧❡❛❞❡r ♣❧❛②s ❙t❡♣ ✸✿ t❤❡ ❢♦❧❧♦✇❡r ❞❡❝✐❞❡s t♦ ♣❧❛②✳✳✳✇❤❛t❡✈❡r ❤❡ ✇❛♥ts✦✦✦
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
❆❝t✉❛❧❧② ✐♥ r❡❛❧ ❧✐❢❡✱ ✇❤❡♥ ❝♦♥s✐❞❡r✐♥❣ ❇▲ minx∈Rn miny∈Rm F(x, y) s✳t✳ x ∈ X 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✦✦✦
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
❉❡✜♥✐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❡♠ ∇ygi(x, y)d < 0 ∀ i ∈ I(x, y) := {j : gj(x, y) = 0} ❤❛s ❛ s♦❧✉t✐♦♥✳
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
❆ss✉♠❡ t❤❛t X = {x ∈ Rn : 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) ∈ gphS ✇✐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✐♦♥ ϕo(·) ✐s ❝♦♥t✐♥✉♦✉s ❛t x✳
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
❚❤❡♦r❡♠ ❆ss✉♠❡ t❤❛t t❤❡ s❡t {(x, y) : g(x, y) ≤ 0} ✐s ♥♦t ❡♠♣t② ❛♥❞ ❝♦♠♣❛❝t❀ ❛t ❡❛❝❤ ♣♦✐♥t (x, y) ∈ gphS ✇✐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✐♦♥✳
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
❲❡ ❝♦♥s✐❞❡r ❛ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠ ❝♦♥s✐st✐♥❣ ✐♥ ❛♥ ✉♣♣❡r✲❧❡✈❡❧ ✴ ❧❡❛❞❡r✬s ♣r♦❜❧❡♠ ✏ min
x∈Rn✑ F(x, y)
s✳t✳ y ∈ S(x), x ∈ X ✇❤❡r❡ ∅ = X ⊂ Rn✱ ❛♥❞ S(x) st❛♥❞s ❢♦r t❤❡ s♦❧✉t✐♦♥ ♦❢ ✐ts ❧♦✇❡r✲❧❡✈❡❧ ✴ ❢♦❧❧♦✇❡r✬s ♣r♦❜❧❡♠ min
y∈Rm f(x, y)
s✳t g(x, y) ≤ 0 ✇❤✐❝❤ ✇❡ ❛ss✉♠❡ t♦ ❜❡ ❝♦♥✈❡① ❛♥❞ s♠♦♦t❤✱ ✐✳❡✳ ∀x ∈ X, t❤❡ ❢✉♥❝t✐♦♥s f(x, ·) ❛♥❞ gi(x, ·) ❛r❡ s♠♦♦t❤ ❝♦♥✈❡① ❢✉♥❝t✐♦♥s✱ ❛♥❞ t❤❡ ❣r❛❞✐❡♥ts ∇ygi, ∇yf ❛r❡ ❝♦♥t✐♥✉♦✉s✱ i = 1, ..., p✳
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
❘❡♣❧❛❝✐♥❣ t❤❡ ❧♦✇❡r✲❧❡✈❡❧ ♣r♦❜❧❡♠ ❜② ✐ts ❑❑❚ ❝♦♥❞✐t✐♦♥s✱ ❣✐✈❡s ♣❧❛❝❡ t♦ ❛ ▼❛t❤❡♠❛t✐❝❛❧ Pr♦❣r❛♠ ✇✐t❤ ❈♦♠♣❧❡♠❡♥t❛r✐t② ❈♦♥str❛✐♥ts✳ ❇✐❧❡✈❡❧ ✏min
x∈X✑F(x, y)
s✳t✳ y ∈ S(x)
✇✐t❤ S(x) = ✏y s♦❧✈✐♥❣ min
y∈Rm f(x, y)
s✳t g(x, y) ≤ 0 ✑
▼P❈❈ ✏min
x∈X✑F(x, y)
s✳t✳ (y, u) ∈ KKT(x)
✇✐t❤ KKT(x) = ✏(y, u) s♦❧✈✐♥❣
0 ≤ u ⊥ −g(x, y) ≥ 0 ✑
❲❡ ✇r✐t❡ Λ(x, y) ❢♦r t❤❡ s❡t ♦❢ u s❛t✐s❢②✐♥❣ (y, u) ∈ KKT(x)✳
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
❊①❛♠♣❧❡ ✶
❈♦♥s✐❞❡r t❤❡ ❢♦❧❧♦✇✐♥❣ ❇✐❧❡✈❡❧ ♣r♦❜❧❡♠ ❛♥❞ ✐ts ▼P❈❈ r❡❢♦r♠✉❧❛t✐♦♥ ❇✐❧❡✈❡❧ ✏ min
x∈[−1,1]✑ x
s✳t✳ y ∈ S(x)
✇✐t❤ S(x) = ✏y s♦❧✈✐♥❣ min
y∈R
xy s✳t x2(y2 − 1) ≤ 0 ✑
▼P❈❈ ✏ min
x∈[−1,1]✑ x
s✳t✳ (y, u) ∈ KKT(x)
✇✐t❤ KKT(x) = ✏(y, u) s♦❧✈✐♥❣ x + u · 2yx2 = 0 0 ≤ u ⊥ −x2(y2 − 1) ≥ 0 ✑
✶
(0, −1, u) ✐s ❛ ❧♦❝❛❧ s♦❧✉t✐♦♥ ♦❢ ✏▼P❈❈✑✱ ❢♦r ❛♥② u ∈ Λ(0, −1) = R+
✷
(0, −1) ✐s ◆❖❚ ❛ ❧♦❝❛❧ s♦❧✉t✐♦♥ ♦❢ ✏❇✐❧❡✈❡❧✑
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
① ② ✉
KKT(·)
−∇F
✭❛✮ (0, −1, u) ✐s ❛ ❧♦❝❛❧ s♦❧✉t✐♦♥ ♦❢ ▼P❈❈✱ ∀u ∈ R+✳
② ① −∇F −1 1
S(·)
✭❜✮ (0, −1) ✐s♥✬t ❛ ❧♦❝❛❧ s♦❧✉t✐♦♥ ♦❢ t❤❡ ❇✐❧❡✈❡❧ ♣r♦❜❧❡♠✳
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
❚❤❡ ♦♣t✐♠✐st✐❝ ❇✐❧❡✈❡❧ ✭❖❇✮ ✐s min
x min y
F(x, y) s✳t✳ y ∈ S(x), x ∈ X. ❚❤❡ ♣❡ss✐♠✐st✐❝ ❇✐❧❡✈❡❧ ✭P❇✮ ✐s min
x max y
F(x, y) s✳t✳ y ∈ S(x), x ∈ X. ❚❤❡ ♦♣t✐♠✐st✐❝ ▼P❈❈ ✭❖▼P❈❈✮✿ s✳t✳ ❚❤❡ ♣❡ss✐♠✐st✐❝ ▼P❈❈ ✭P▼P❈❈✮✿ s✳t✳
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
❚❤❡ ♦♣t✐♠✐st✐❝ ❇✐❧❡✈❡❧ ✭❖❇✮ ✐s min
x min y
F(x, y) s✳t✳ y ∈ S(x), x ∈ X. ❚❤❡ ♣❡ss✐♠✐st✐❝ ❇✐❧❡✈❡❧ ✭P❇✮ ✐s min
x max y
F(x, y) s✳t✳ y ∈ S(x), x ∈ X. ❚❤❡ ♦♣t✐♠✐st✐❝ ▼P❈❈ ✭❖▼P❈❈✮✿ min
x min y
F(x, y) s✳t✳ (y, u) ∈ KKT(x), x ∈ X. ❚❤❡ ♣❡ss✐♠✐st✐❝ ▼P❈❈ ✭P▼P❈❈✮✿ min
x max y
F(x, y) s✳t✳ (y, u) ∈ KKT(x), x ∈ X.
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
■s ❜✐❧❡✈❡❧ ♣r♦❣r❛♠♠✐♥❣ ❛ s♣❡❝✐❛❧ ❝❛s❡ ♦❢ ❛ ▼P❈❈❄ ❙✳ ❉❡♠♣❡ ✲❏✳ ❉✉tt❛ ✭✷✵✶✷ ▼❛t❤✳ Pr♦❣✳✮ min
x min y
F(x, y) s✳t✳ y ∈ S(x), x ∈ X.
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
❉❡✜♥✐t✐♦♥ ❆ ❧♦❝❛❧ ✭r❡s♣✳ ❣❧♦❜❛❧✮ s♦❧✉t✐♦♥ ♦❢ ✭❖❇✮ ✐s ❛ ♣♦✐♥t (¯ x, ¯ y) ∈ Gr(S) ✐❢ t❤❡r❡ ❡①✐sts U ∈ N(¯ x, ¯ y) ✭r❡s♣✳ U = Rn × Rm✮ 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 = Rn × Rm × Rp✮ ✇✐t❤ F(¯ x, ¯ y) ≤ F(x, y), ∀(x, y, u) ∈ U ∩ Gr(KKT).
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
■♥ ❉❡♠♣❡✲❉✉tt❛ ✐t ✇❛s ❝♦♥s✐❞❡r❡❞ t❤❡ ❙❧❛t❡r t②♣❡ ❝♦♥str❛✐♥t q✉❛❧✐✜❝❛t✐♦♥ ❢♦r ❛ ♣❛r❛♠❡t❡r x ∈ X✿ ❙❧❛t❡r✿ ∃y(x) ∈ Rm s✳t✳ gi(x, y(x)) < 0✱ ∀i = 1, .., p.
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡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 ♦♥ ❛
♥❡✐❣❤❜♦✉r❤♦♦❞ ♦❢ ¯ x✱ Λ(¯ x, ¯ y) = ∅✱ ❛♥❞ (¯ x, ¯ y, u) ✐s ❛ ❧♦❝❛❧ s♦❧✉t✐♦♥ ♦❢ ✭❖▼P❈❈✮ ❢♦r ❡✈❡r② u ∈ Λ(¯ x, ¯ y)✳ ❚❤❡♥ (¯ x, ¯ y) ✐s ❛ ❧♦❝❛❧ s♦❧✉t✐♦♥ ♦❢ ✭❖❇✮✳
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
❯♥❞❡r t❤❡ ❝♦♥✈❡①✐t② ❛ss✉♠♣t✐♦♥ ❛♥❞ s♦♠❡ ❈◗ ❡♥s✉r✐♥❣ KKT(x) = ∅, ∀x ∈ X✿
(¯ x, ¯ y, ¯ u) s♦❧ ♦❢ ✭❖▼P❈❈✮ (¯ x, ¯ y) s♦❧ ♦❢ ✭❖❇✮ ∀¯ u ∈ Λ(¯ x, ¯ y)
❋✐❣✉r❡✿ ●❧♦❜❛❧ s♦❧✉t✐♦♥ ❝♦♠♣❛r✐s♦♥ ✐♥ ♦♣t✐♠✐st✐❝ ❛♣♣r♦❛❝❤
(¯ x, ¯ y) ❧♦❝❛❧ s♦❧ ♦❢ ✭❖❇✮ ∀¯ u ∈ Λ(¯ x, ¯ y)✱ (¯ x, ¯ y, ¯ u) ❧♦❝❛❧ s♦❧ ♦❢ ✭❖▼P❈❈✮ ✐❢ ❙❧❛t❡r✬s ❈◗ ❤♦❧❞s ❛r♦✉♥❞ ¯ x
❋✐❣✉r❡✿ ▲♦❝❛❧ s♦❧✉t✐♦♥ ❝♦♠♣❛r✐s♦♥ ✐♥ ♦♣t✐♠✐st✐❝ ❛♣♣r♦❛❝❤
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
❊①❛♠♣❧❡ ✶ ✭♦♣t✐♠✐st✐❝✮ ❈♦♥s✐❞❡r t❤❡ ❢♦❧❧♦✇✐♥❣ ♦♣t✐♠✐st✐❝ ❇✐❧❡✈❡❧ ♣r♦❜❧❡♠ min
x∈[−1,1] min y
x s✳t✳ y ∈ S(x), x ∈ R ✇✐t❤ ❧♦✇❡r✲❧❡✈❡❧ min
y
− xy s✳t x2(y2 − 1) ≤ 0.
✶ (0, −1, u) ✐s ❛ ❧♦❝❛❧ s♦❧✉t✐♦♥ ♦❢ ✭❖▼P❈❈✮✱ ❢♦r ❛♥②
u ∈ Λ(0, −1) = R+
✷ (0, −1) ✐s ◆❖❚ ❛ ❧♦❝❛❧ s♦❧✉t✐♦♥ ♦❢ ✭❖❇✮✳ ❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
■s ❜✐❧❡✈❡❧ ♣r♦❣r❛♠♠✐♥❣ ❛ s♣❡❝✐❛❧ ❝❛s❡ ♦❢ ❛ ✭▼P❈❈✮❄ ❆✉ss❡❧ ✲ ❙✈❡♥ss♦♥ ✭✷✵✶✾ ✲ ❏✳ ❖♣t✐♠✳ ❚❤❡♦r② ❆♣♣❧✳✮ min
x max y
F(x, y) s✳t✳ y ∈ S(x), x ∈ X.
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
❉❡✜♥✐t✐♦♥ ❆ ♣❛✐r (¯ x, ¯ y) ✐s s❛✐❞ t♦ ❜❡ ❛ ❧♦❝❛❧ ✭r❡s♣✳ ❣❧♦❜❛❧✮ s♦❧✉t✐♦♥ ❢♦r ✭P❇✮✱ ✐❢ (¯ x, ¯ y) ∈ Gr(Sp) ❛♥❞ ∃U ∈ N(¯ x, ¯ y) s✉❝❤ t❤❛t F(¯ x, ¯ y) ≤ F(x, y), ∀(x, y) ∈ U ∩ Gr(Sp). ✭✸✮ ✇❤❡r❡ Sp(x) := argmaxy {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(KKTp) ❛♥❞ ∃U ∈ N(¯ x, ¯ y, ¯ u) s✉❝❤ t❤❛t F(¯ x, ¯ y) ≤ F(x, y), ∀(x, y, u) ∈ U ∩ Gr(KKTp). ✭✹✮ ✇❤❡r❡ KKTp(x) := argmaxy,u {F(x, y) | (y, u) ∈ KKT(x)} .
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡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✐♦♥ KKTp ✐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❇✮✳
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
❊①❛♠♣❧❡ ✶ ✭♣❡ss✐♠✐st✐❝✮ ❈♦♥s✐❞❡r t❤❡ ❢♦❧❧♦✇✐♥❣ ♣❡ss✐♠✐st✐❝ ❇✐❧❡✈❡❧ ♣r♦❜❧❡♠ min
x∈[−1,1] max y
x s✳t✳ y ∈ S(x), x ∈ R ✇✐t❤ ❧♦✇❡r✲❧❡✈❡❧ min
y
− xy s✳t x2(y2 − 1) ≤ 0.
✶ (0, −1, u) ✐s ❛ ❧♦❝❛❧ s♦❧✉t✐♦♥ ♦❢ ✭P▼P❈❈✮✱ ❢♦r ❛♥②
u ∈ Λ(0, −1) = R+
✷ (0, −1) ✐s ◆❖❚ ❛ ❧♦❝❛❧ s♦❧✉t✐♦♥ ♦❢ ✭P❇✮✳ ❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
❊①❛♠♣❧❡ ✷
❈♦♥s✐❞❡r t❤❡ ❢♦❧❧♦✇✐♥❣ ❇✐❧❡✈❡❧ ♣r♦❜❧❡♠ ✏ min
x ✑x
s.t. y ∈ S(x) ✇✐t❤ S(x) t❤❡ s♦❧✉t✐♦♥ ♦❢ t❤❡ ❧♦✇❡r✲❧❡✈❡❧ ♣r♦❜❧❡♠ min
y
{−y | x + y ≤ 0, y ≤ 0} ❊✈❡♥ t❤♦✉❣❤ ❙❧❛t❡r✬s ❈◗ ❤♦❧❞s✱ ✇❡ ❤❛✈❡
✶ (0, 0, u1, u2) ✇✐t❤ (u1, u2) ∈ Λ(0, 0) = {(λ, 1 − λ) | λ ∈ [0, 1]} ✐s ❛
❧♦❝❛❧ s♦❧✉t✐♦♥ ♦❢ ✏✭▼P❈❈✮✑✱ ✐✛ u1 = 0✱
✷ (0, 0) ✐s ◆❖❚ ❛ ❧♦❝❛❧ s♦❧✉t✐♦♥ ❢♦r ✏✭❇✮✑✳ ❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
❯♥❞❡r t❤❡ ❝♦♥✈❡①✐t② ❛ss✉♠♣t✐♦♥ ❛♥❞ s♦♠❡ ✭❈◗✮ ❡♥s✉r✐♥❣ KKT(x) = ∅, ∀x ∈ X✿
(¯ x, ¯ y, ¯ u) s♦❧ ♦❢ ✭P▼P❈❈✮ (¯ x, ¯ y) s♦❧ ♦❢ ✭P❇✮ ∀¯ u ∈ Λ(¯ x, ¯ y)
❋✐❣✉r❡✿ ●❧♦❜❛❧ s♦❧✉t✐♦♥ ❝♦♠♣❛r✐s♦♥ ✐♥ ♣❡ss✐♠✐st✐❝ ❛♣♣r♦❛❝❤
(¯ x, ¯ y) ❧♦❝❛❧ s♦❧ ♦❢ ✭P❇✮ ∀¯ u ∈ Λ(¯ x, ¯ y)✱ (¯ x, ¯ y, ¯ u) ❧♦❝❛❧ s♦❧ ♦❢ ✭P▼P❈❈✮ ❙❧❛t❡r✬s ❈◗ ❢♦r ❛❧❧ x ❛r♦✉♥❞ ¯ x
❋✐❣✉r❡✿ ▲♦❝❛❧ s♦❧✉t✐♦♥s ❝♦♠♣❛r✐s♦♥ ✐♥ ♣❡ss✐♠✐st✐❝ ❛♣♣r♦❛❝❤
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
◆♦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 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✳
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
◆♦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 xi ∈ Rni✳ ❚❤❡ ✏t♦t❛❧ str❛t❡❣② ✈❡❝t♦r✑ ✐s x ✇❤✐❝❤ ✇✐❧❧ ❜❡ ♦❢t❡♥ ❞❡♥♦t❡❞ ❜② x = (xi, x−i). ✇❤❡r❡ x−i ✐s t❤❡ str❛t❡❣② ✈❡❝t♦r ♦❢ t❤❡ ♦t❤❡r ♣❧❛②❡rs✳
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
◆❛s❤ ❊q✉✐❧✐❜r✐✉♠ Pr♦❜❧❡♠ ✭◆❊P✮ ❚❤❡ str❛t❡❣② ♦❢ ♣❧❛②❡r i ❜❡❧♦♥❣s t♦ ❛ str❛t❡❣② s❡t xi ∈ Xi
♦❢ 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
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
◆❛s❤ ❊q✉✐❧✐❜r✐✉♠ Pr♦❜❧❡♠ ✭◆❊P✮ ❚❤❡ str❛t❡❣② ♦❢ ♣❧❛②❡r i ❜❡❧♦♥❣s t♦ ❛ str❛t❡❣② s❡t xi ∈ Xi
✐s t♦ ❝❤♦♦s❡ ❛ str❛t❡❣② xi s♦❧✈✐♥❣ Pi(x−i) max θi(xi, x−i) s✳t✳ xi ∈ Xi ✇❤❡r❡ θi(·, x−i) : Rni → R ✐s t❤❡ ❞❡❝✐s✐♦♥ ❢✉♥❝t✐♦♥ ❢♦r ♣❧❛②❡r i✳ ❆ ✈❡❝t♦r ✐s ❛ ◆❛s❤ ❊q✉✐❧✐❜r✐✉♠ ✐❢ ❢♦r ❛♥② s♦❧✈❡s
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
◆❛s❤ ❊q✉✐❧✐❜r✐✉♠ Pr♦❜❧❡♠ ✭◆❊P✮ ❚❤❡ str❛t❡❣② ♦❢ ♣❧❛②❡r i ❜❡❧♦♥❣s t♦ ❛ str❛t❡❣② s❡t xi ∈ Xi
✐s t♦ ❝❤♦♦s❡ ❛ str❛t❡❣② xi s♦❧✈✐♥❣ Pi(x−i) max θi(xi, x−i) s✳t✳ xi ∈ Xi ✇❤❡r❡ θi(·, x−i) : Rni → R ✐s t❤❡ ❞❡❝✐s✐♦♥ ❢✉♥❝t✐♦♥ ❢♦r ♣❧❛②❡r i✳ ❆ ✈❡❝t♦r ¯ x ✐s ❛ ◆❛s❤ ❊q✉✐❧✐❜r✐✉♠ ✐❢ ❢♦r ❛♥② i, ¯ xi s♦❧✈❡s Pi(¯ x−i).
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
❆ ❣❡♥❡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 xi ∈ Xi(x−i) ✇❤✐❝❤ ❞❡♣❡♥❞s ♦♥ t❤❡ ❞❡❝✐s✐♦♥ ✈❛r✐❛❜❧❡s ♦❢ t❤❡ ♦t❤❡r ♣❧❛②❡rs✳
♦❢ 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 ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
❆ ❣❡♥❡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 xi ∈ Xi(x−i) ✇❤✐❝❤ ❞❡♣❡♥❞s ♦♥ t❤❡ ❞❡❝✐s✐♦♥ ✈❛r✐❛❜❧❡s ♦❢ t❤❡ ♦t❤❡r ♣❧❛②❡rs✳
xi s♦❧✈✐♥❣ Pi(x−i) max θi(xi, x−i) s✳t✳ xi ∈ Xi(x−i) ✇❤❡r❡ θi(·, x−i) : Rni → R ✐s t❤❡ ❞❡❝✐s✐♦♥ ❢✉♥❝t✐♦♥ ❢♦r ♣❧❛②❡r i✳ ❆ ✈❡❝t♦r ✐s ❛ ●❡♥❡r❛❧✐③❡❞ ◆❛s❤ ❊q✉✐❧✐❜r✐✉♠ ✐❢ ❢♦r ❛♥② s♦❧✈❡s ❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
❆ ❣❡♥❡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 xi ∈ Xi(x−i) ✇❤✐❝❤ ❞❡♣❡♥❞s ♦♥ t❤❡ ❞❡❝✐s✐♦♥ ✈❛r✐❛❜❧❡s ♦❢ t❤❡ ♦t❤❡r ♣❧❛②❡rs✳
xi s♦❧✈✐♥❣ Pi(x−i) max θi(xi, x−i) s✳t✳ xi ∈ Xi(x−i) ✇❤❡r❡ θi(·, x−i) : Rni → R ✐s t❤❡ ❞❡❝✐s✐♦♥ ❢✉♥❝t✐♦♥ ❢♦r ♣❧❛②❡r i✳ ❆ ✈❡❝t♦r ¯ x ✐s ❛ ●❡♥❡r❛❧✐③❡❞ ◆❛s❤ ❊q✉✐❧✐❜r✐✉♠ ✐❢ ❢♦r ❛♥② i, ¯ xi s♦❧✈❡s Pi(¯ x−i). ❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
min
x1
θ1(x) s✳t✳ x1 ∈ X1(x−1) . . . min
xn
θn(x) s✳t✳
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
❚❤❡♦r❡♠ ✭■❝❤✐✐s❤✐✲◗✉✐♥③✐✐ ✶✾✽✸✮ ▲❡t ❛ ●◆❊P ❜❡ ❣✐✈❡♥ ❛♥❞ s✉♣♣♦s❡ t❤❛t
✶ ❋♦r ❡❛❝❤ ν = 1, ..., N t❤❡r❡ ❡①✐st ❛ ♥♦♥❡♠♣t②✱ ❝♦♥✈❡① ❛♥❞
❝♦♠♣❛❝t s❡t Kν ⊂ Rnν s✉❝❤ t❤❛t t❤❡ ♣♦✐♥t✲t♦✲s❡t ♠❛♣ Xν : K−ν ⇒ Kν✱ ✐s ❜♦t❤ ✉♣♣❡r ❛♥❞ ❧♦✇❡r s❡♠✐❝♦♥t✐♥✉♦✉s ✇✐t❤ ♥♦♥❡♠♣t② ❝❧♦s❡❞ ❛♥❞ ❝♦♥✈❡① ✈❛❧✉❡s✱ ✇❤❡r❡ K−ν :=
ν′=ν Kν✳
✷ ❋♦r ❡✈❡r② ♣❧❛②❡r ν✱ t❤❡ ❢✉♥❝t✐♦♥ θν ✐s ❝♦♥t✐♥✉♦✉s ❛♥❞
θν(·, x−ν) ✐s q✉❛s✐✲❝♦♥✈❡① ♦♥ Xν(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✉❡✳
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
❊①❛♠♣❧❡
▲❡t x = (x1, x2) ∈ [0, 4]2 ❛♥❞ f ν(x) := dTν(x)2✱ ✇❤❡r❡ T1 ✐s t❤❡ tr✐❛♥❣❧❡ ✇✐t❤ ✈❡rt✐❝❡s (0, 0)✱ (0, 4) ❛♥❞ (1, 2)✱ ❛♥❞ T2 ✐s t❤❡ tr✐❛♥❣❧❡ ✇❤♦s❡ ✈❡rt✐❝❡s ❛r❡ (0, 0)✱ (4, 0) ❛♥❞ (2, 1)✳ ▲❡t Sν(x−ν) := ❛r❣♠✐♥xν
S1(x2) =
S1(x2) = {2} ❢♦r ❛❧❧ x2 ∈ (1, 4]) S2(x1) =
S2(x1) = {2} ❢♦r ❛❧❧ x1 ∈ (1, 4])✳
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
x1 x2 S1(·) S2(·)
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
❆ ♣❛rt✐❝✉❧❛r ❝❛s❡ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r✲●❛♠❡ ✭▼▲❋●✮✿ min
x1 y1,..,yp
θ1(x, y) s✳t✳ x1 ∈ X1(x−1) y ∈ Y (x) . . . min
xn y1,..,yp
θn(x, y) s✳t✳ xn ∈ Xn(x−n) y ∈ Y (x) ↓↑ ↓↑ miny1,..,yp φ1(x, y) s✳t✳ y ∈ Y (x) . . . miny1,..,yp φp(x, y) s✳t✳ y ∈ Y (x)
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
❛♥❞ ❛♥♦t❤❡r ♣r♦❜❧❡♠ ❙✐♥❣❧❡✲▲❡❛❞❡r✲▼✉❧t✐✲❋♦❧❧♦✇❡r✲●❛♠❡ ✭❙▲▼❋●✮✿ min
x y1,..,yp
θ1(x, y) s✳t✳
y ∈ Y (x) ↓↑ miny1,..,yp φ1(x, y) s✳t✳ y ∈ Y (x) . . . miny1,..,yp φp(x, y) s✳t✳ y ∈ Y (x)
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
❆ ♣❛rt✐❝✉❧❛r ❝❛s❡ ▼✉❧t✐✲▲❡❛❞❡r✲❙✐♥❣❧❡✲❋♦❧❧♦✇❡r✲●❛♠❡ ✭▼▲❙❋●✮✿ min
x1 y1,..,yp
θ1(x, y) s✳t✳ x1 ∈ X1(x−1) y ∈ Y (x) . . . min
xn y1,..,yp
θn(x, y) s✳t✳ xn ∈ Xn(x−n) y ∈ Y (x) ↓↑ miny1,..,yp φ1(x, y) s✳t✳ y ∈ Y (x)
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
minx1,y θ1(x1, x2, y) = x1.y s✳t✳
y ∈ S(x1, x2) minx2,y θ1(x1, x2, y) = −x2.y s✳t✳
y ∈ S(x1, x2) ✇✐t❤ miny f(x1, x2, y) = 1
3y3 − (x1 + x2)2y
s✳t✳ y ∈ R ❊①❡r❝✐s❡✿ P❧❡❛s❡ ❛♥❛❧②s❡ t❤✐s s♠❛❧❧ ❡①❛♠♣❧❡✳✳✳
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
minx1,y θ1(x1, x2, y) = x1.y s✳t✳
y ∈ S(x1, x2) minx2,y θ1(x1, x2, y) = −x2.y s✳t✳
y ∈ S(x1, x2) ✇✐t❤ miny f(x1, x2, y) = 1
3y3 − (x1 + x2)2y
s✳t✳ y ∈ R ❊①❡r❝✐s❡✿ P❧❡❛s❡ ❛♥❛❧②s❡ t❤✐s s♠❛❧❧ ❡①❛♠♣❧❡✳✳✳
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
❚❤❡ ❢♦❧❧♦✇❡r ♣r♦❜❧❡♠ ✜rst miny f(x1, x2, y) = 1
3y3 − (x1 + x2)2y
s✳t✳ y ∈ R ❚❤❡ s♦❧✉t✐♦♥ ♠❛♣ ♦❢ t❤✐s ❢♦❧❧♦✇❡r ♣r♦❜❧❡♠ ✐s
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
❚❤❡ ❢♦❧❧♦✇❡r ♣r♦❜❧❡♠ ✜rst miny f(x1, x2, y) = 1
3y3 − (x1 + x2)2y
s✳t✳ y ∈ R ❚❤❡ s♦❧✉t✐♦♥ ♠❛♣ ♦❢ t❤✐s ❢♦❧❧♦✇❡r ♣r♦❜❧❡♠ ✐s S(x1, x2) = {y1 = x1 + x2, y2 = −x1 − x2}.
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
❚❤❡ s♦❧✉t✐♦♥ ♠❛♣ ♦❢ t❤✐s ❢♦❧❧♦✇❡r ♣r♦❜❧❡♠ ✐s S(x1, x2) = {y1 = x1 + x2, y2 = −x1 − x2}.
❚❤❡ ❧❡❛❞❡r ✶ ♣r♦❜❧❡♠ θ1(x, y) = x1.y = x2
1 + x1.x2
✐❢ y = y1 −x2
1 − x1.x2
✐❢ y = y2 ❚❤✉s t❤❡ r❡s♣♦♥s❡ ❢✉♥❝t✐♦♥ ♦❢ ♣❧❛②❡r ✶ ✐s ✐❢ ✇✐t❤ ❛ ♣❛②♦✛ ✐❢ ✇✐t❤ ❛ ♣❛②♦✛
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
❚❤❡ s♦❧✉t✐♦♥ ♠❛♣ ♦❢ t❤✐s ❢♦❧❧♦✇❡r ♣r♦❜❧❡♠ ✐s S(x1, x2) = {y1 = x1 + x2, y2 = −x1 − x2}.
❚❤❡ ❧❡❛❞❡r ✶ ♣r♦❜❧❡♠ θ1(x, y) = x1.y = x2
1 + x1.x2
✐❢ y = y1 −x2
1 − x1.x2
✐❢ y = y2 ❚❤✉s t❤❡ r❡s♣♦♥s❡ ❢✉♥❝t✐♦♥ ♦❢ ♣❧❛②❡r ✶ ✐s R1(x2) = {0} ✐❢ y = y1 ✇✐t❤ ❛ ♣❛②♦✛ = 0 {1} ✐❢ y = y2 ✇✐t❤ ❛ ♣❛②♦✛ = −1 − x2
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
❚❤❡ s♦❧✉t✐♦♥ ♠❛♣ ♦❢ t❤✐s ❢♦❧❧♦✇❡r ♣r♦❜❧❡♠ ✐s S(x1, x2) = {y1 = x1 + x2, y2 = −x1 − x2}.
❚❤❡ ❧❡❛❞❡r ✷ ♣r♦❜❧❡♠ θ1(x, y) = −x2.y = −x2
1 − x1.x2
✐❢ y = y1 x2
1 + x1.x2
✐❢ y = y2 ❚❤✉s t❤❡ r❡s♣♦♥s❡ ❢✉♥❝t✐♦♥ ♦❢ ♣❧❛②❡r ✶ ✐s ✐❢ ✇✐t❤ ❛ ♣❛②♦✛ ✐❢ ✇✐t❤ ❛ ♣❛②♦✛
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
❚❤❡ s♦❧✉t✐♦♥ ♠❛♣ ♦❢ t❤✐s ❢♦❧❧♦✇❡r ♣r♦❜❧❡♠ ✐s S(x1, x2) = {y1 = x1 + x2, y2 = −x1 − x2}.
❚❤❡ ❧❡❛❞❡r ✷ ♣r♦❜❧❡♠ θ1(x, y) = −x2.y = −x2
1 − x1.x2
✐❢ y = y1 x2
1 + x1.x2
✐❢ y = y2 ❚❤✉s t❤❡ r❡s♣♦♥s❡ ❢✉♥❝t✐♦♥ ♦❢ ♣❧❛②❡r ✶ ✐s R2(x1) = {1} ✐❢ y = y1 ✇✐t❤ ❛ ♣❛②♦✛ = −1 − x1 {0} ✐❢ y = y2 ✇✐t❤ ❛ ♣❛②♦✛ = 0
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
R1(x2) =
✇✐t❤ ❛ ♣❛②♦✛ = 0 {(1, y = y2)} ✇✐t❤ ❛ ♣❛②♦✛ = −1 − x2 R2(x1) = {(1, y = y1)} ✇✐t❤ ❛ ♣❛②♦✛ = −1 − x1 {(0, y = y2)} ✇✐t❤ ❛ ♣❛②♦✛ = 0 ❙♦ t❤❡ ◆❛s❤ ❡q✉✐❧✐❜r✐✉♠ ✇✐❧❧ ❜❡ ❜✉t✳✳✳✳
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
R1(x2) =
✇✐t❤ ❛ ♣❛②♦✛ = 0 {(1, y = y2)} ✇✐t❤ ❛ ♣❛②♦✛ = −1 − x2 R2(x1) = {(1, y = y1)} ✇✐t❤ ❛ ♣❛②♦✛ = −1 − x1 {(0, y = y2)} ✇✐t❤ ❛ ♣❛②♦✛ = 0 ❙♦ t❤❡ ◆❛s❤ ❡q✉✐❧✐❜r✐✉♠ ✇✐❧❧ ❜❡ (x1, x2) = (1, 1) ❜✉t✳✳✳✳
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
❋♦r t❤❡ ❉❡♠❛♥❞✲s✐❞❡ ♠❛♥❛❣❡♠❡♥t✱ ✇❡ r❡❝❡♥t❧② ✐♥tr♦❞✉❝❡❞ t❤❡ ▼✉❧t✐✲▲❡❛❞❡r✲❉✐s❥♦✐♥t✲❋♦❧❧♦✇❡r ❣❛♠❡
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
▲❡t ✉s ❝♦♥s✐❞❡r ❛ ✷✲❧❡❛❞❡r✲s✐♥❣❧❡✲❢♦❧❧♦✇❡r ❣❛♠❡✿
♠✐♥x1,y
1 2 x1 + y
♠✐♥x2,y − 1
2 x2 − y
y ∈ S(x1, x2)
y ∈ S(x1, x2)
✇❤❡r❡ S(x1, x2) ✐s t❤❡ s♦❧✉t✐♦♥ ♠❛♣ ♦❢ ♠✐♥y≥0 y(−1 + x1 + x2) + 1 2y2
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
▲❡t ✉s ❝♦♥s✐❞❡r ❛ ✷✲❧❡❛❞❡r✲s✐♥❣❧❡✲❢♦❧❧♦✇❡r ❣❛♠❡✿
♠✐♥x1,y1
1 2 x1 + y1
♠✐♥x2,y2 − 1
2 x2 − y2
y1 ∈ S(x1, x2)
y2 ∈ S(x1, x2)
✇❤❡r❡ S(x1, x2) ✐s t❤❡ s♦❧✉t✐♦♥ ♠❛♣ ♦❢ ♠✐♥y≥0 y(−1 + x1 + x2) + 1 2y2
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
❆❝t✉❛❧❧② S(x1, x2) = max{0, 1 − x1 − x2} t❤✉s t❤❡ ♣r♦❜❧❡♠ ❜❡❝♦♠❡s
♠✐♥x1,y1
1 2 x1 + y1
♠✐♥x2,y2 − 1
2 x2 − y2
y1 = max{0, 1 − x1 − x2}
y2 = max{0, 1 − x1 − x2}
❚❤❡♥ t❤❡ ❘❡s♣♦♥s❡ ♠❛♣s ❛r❡ ❛♥❞ ❛♥❞ t❤✉s t❤❡r❡ ✐s ♥♦ ◆❛s❤ ❡q✉✐❧✐❜r✐✉♠✳✳✳✳✳✳✳
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
❆❝t✉❛❧❧② S(x1, x2) = max{0, 1 − x1 − x2} t❤✉s t❤❡ ♣r♦❜❧❡♠ ❜❡❝♦♠❡s
♠✐♥x1,y1
1 2 x1 + y1
♠✐♥x2,y2 − 1
2 x2 − y2
y1 = max{0, 1 − x1 − x2}
y2 = max{0, 1 − x1 − x2}
❚❤❡♥ t❤❡ ❘❡s♣♦♥s❡ ♠❛♣s ❛r❡ R1(x2) = {1 − x2} ❛♥❞ R2(x1) = {0} x1 ∈ [0, 1
2[
{0, 1} x1 = 1
2
{1} x1 ∈] 1
2, 1]
❛♥❞ t❤✉s t❤❡r❡ ✐s ♥♦ ◆❛s❤ ❡q✉✐❧✐❜r✐✉♠✳✳✳✳✳✳✳
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
❇✉t ❧❡t ✉s ❝♦♥s✐❞❡r t❤❡ s❧✐❣❤t❧② ♠♦❞✐✜❡❞ ♣r♦❜❧❡♠✳✳✳✳✳✳✳
♠✐♥x1,y1
1 2 x1 + y1
♠✐♥x2,y2 − 1
2 x2 − y2
x1 ∈ [0, 1] y1 = max{0, 1 − x1 − x2} y2 = max{0, 1 − x1 − x2} x2 ∈ [0, 1] y1 = max{0, 1 − x1 − x2} y2 = max{0, 1 − x1 − x2}
t❤❛t ❝❛♥ ❜❡ ♣r♦✈❡❞ t♦ ❤❛✈❡ ❛ ✭✉♥✐q✉❡✮ ◆❛s❤ ❡q✉✐❧✐❜r✐✉♠ ♥❛♠❡❧② ✇✐t❤ ✦✦✦✦
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
❇✉t ❧❡t ✉s ❝♦♥s✐❞❡r t❤❡ s❧✐❣❤t❧② ♠♦❞✐✜❡❞ ♣r♦❜❧❡♠✳✳✳✳✳✳✳
♠✐♥x1,y1
1 2 x1 + y1
♠✐♥x2,y2 − 1
2 x2 − y2
x1 ∈ [0, 1] y1 = max{0, 1 − x1 − x2} y2 = max{0, 1 − x1 − x2} x2 ∈ [0, 1] y1 = max{0, 1 − x1 − x2} y2 = max{0, 1 − x1 − x2}
t❤❛t ❝❛♥ ❜❡ ♣r♦✈❡❞ t♦ ❤❛✈❡ ❛ ✭✉♥✐q✉❡✮ ◆❛s❤ ❡q✉✐❧✐❜r✐✉♠ ♥❛♠❡❧② (x1, x2) = (0, 1) ✇✐t❤ y1 = y2 = 0✦✦✦✦
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
❚❤❡ ❦✐♥❞ ♦❢ ✏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✳✳✳✳
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
❊❧❡❝tr✐❝✐t② ♠❛r❦❡ts
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
❱♦❧✉♠❡ ♦❢ ❡①❝❤❛♥❣❡s ❇✐❞ s❝❤❡❞✉❧❡ ♦❢ t❤❡ s♣♦t ♠❛r❦❡t
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
❱♦❧✉♠❡ ♦❢ ❡①❝❤❛♥❣❡s ❇✐❞ s❝❤❡❞✉❧❡ ♦❢ t❤❡ s♣♦t ♠❛r❦❡t
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
❡❧❡❝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 qi ♦❢ ❣❡♥❡r❛t❡❞✴❝♦♥s✉♠❡❞ ❡❧❡❝tr✐❝✐t② ✐✐✐✮ ❡❧❡❝tr✐❝✐t② ❞✐s♣❛t❝❤ te ✇✐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 ❣❛♠❡
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
❡❧❡❝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 qi ♦❢ ❣❡♥❡r❛t❡❞✴❝♦♥s✉♠❡❞ ❡❧❡❝tr✐❝✐t② ✐✐✐✮ ❡❧❡❝tr✐❝✐t② ❞✐s♣❛t❝❤ te ✇✐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 ❣❛♠❡
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
❆ ❝❧❛ss✐❝❛❧ ♣r♦❜❧❡♠ ✭♦❢ ❛ ♣r♦❞✉❝❡r✮ ✐s t❤❡ ❜❡st r❡s♣♦♥s❡ s❡❛r❝❤
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
❊❧❡❝tr✐❝✐t② ♠❛r❦❡ts ✇✐t❤♦✉t tr❛♥s♠✐ss✐♦♥ ❧♦ss❡s✿ ❳✳ ❍✉ ✫ ❉✳ ❘❛❧♣❤✱ ❯s✐♥❣ ❊P❊❈s t♦ ▼♦❞❡❧ ❇✐❧❡✈❡❧ ●❛♠❡s ✐♥ ❘❡str✉❝t✉r❡❞ ❊❧❡❝tr✐❝✐t② ▼❛r❦❡ts ✇✐t❤ ▲♦❝❛t✐♦♥❛❧ Pr✐❝❡s✱ ❖♣❡r❛t✐♦♥s ❘❡s❡❛r❝❤ ✭✷✵✵✼✮✳ ❜✐❞✲♦♥✲a✲♦♥❧② ❊❧❡❝tr✐❝✐t② ♠❛r❦❡ts ✇✐t❤ tr❛♥s♠✐ss✐♦♥ ❧♦ss❡s✿
❍❡♥r✐♦♥✱ ❘✳✱ ❖✉tr❛t❛✱ ❏✳ ✫ ❙✉r♦✇✐❡❝✱ ❚✳✱ ❆♥❛❧②s✐s ♦❢ ▼✲st❛t✐♦♥❛r② ♣♦✐♥ts t♦ ❛♥ ❊P❊❈ ♠♦❞❡❧✐♥❣ ♦❧✐❣♦♣♦❧✐st✐❝ ❝♦♠♣❡t✐t✐♦♥ ✐♥ ❛♥ ❡❧❡❝tr✐❝✐t② s♣♦t ♠❛r❦❡t✱ ❊❙❆■▼✿ ❈❖❈❱ ✭✷✵✶✷✮✳ ▼✲st❛t✐♦♥❛r② ♣♦✐♥ts ❉✳ ❆✳✱ ❘✳ ❈♦rr❡❛ ✫ ▼✳ ▼❛r❡❝❤❛❧ ❙♣♦t ❡❧❡❝tr✐❝✐t② ♠❛r❦❡t ✇✐t❤ tr❛♥s♠✐ss✐♦♥ ❧♦ss❡s✱ ❏✳ ■♥❞✉str✐❛❧ ▼❛♥❛❣✳ ❖♣t✐♠ ✭✷✵✶✸✮✳ ❡①✐st❡♥❝❡ ♦❢ ◆❛s❤ ❡q✉✐❧✳✱ ❝❛s❡ ♦❢ ❛ t✇♦ ✐s❧❛♥❞ ♠♦❞❡❧ ❉✳❆✳✱ ▼✳ ❈❡r✈✐♥❦❛ ✫ ▼✳ ▼❛r❡❝❤❛❧✱ ❉❡r❡❣✉❧❛t❡❞ ❡❧❡❝tr✐❝✐t② ♠❛r❦❡ts ✇✐t❤ t❤❡r♠❛❧ ❧♦ss❡s ❛♥❞ ♣r♦❞✉❝t✐♦♥ ❜♦✉♥❞s✱ ❘❆■❘❖ ✭✷✵✶✻✮ ♣r♦❞✉❝t✐♦♥ ❜♦✉♥❞s✱ ✇❡❧❧✲♣♦s❡❞♥❡ss ♦❢ ♠♦❞❡❧
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
❊❧❡❝tr✐❝✐t② ♠❛r❦❡ts ✇✐t❤♦✉t tr❛♥s♠✐ss✐♦♥ ❧♦ss❡s✿ ❳✳ ❍✉ ✫ ❉✳ ❘❛❧♣❤✱ ❯s✐♥❣ ❊P❊❈s t♦ ▼♦❞❡❧ ❇✐❧❡✈❡❧ ●❛♠❡s ✐♥ ❘❡str✉❝t✉r❡❞ ❊❧❡❝tr✐❝✐t② ▼❛r❦❡ts ✇✐t❤ ▲♦❝❛t✐♦♥❛❧ Pr✐❝❡s✱ ❖♣❡r❛t✐♦♥s ❘❡s❡❛r❝❤ ✭✷✵✵✼✮✳ ❜✐❞✲♦♥✲a✲♦♥❧② ❊❧❡❝tr✐❝✐t② ♠❛r❦❡ts ✇✐t❤ tr❛♥s♠✐ss✐♦♥ ❧♦ss❡s✿
❍❡♥r✐♦♥✱ ❘✳✱ ❖✉tr❛t❛✱ ❏✳ ✫ ❙✉r♦✇✐❡❝✱ ❚✳✱ ❆♥❛❧②s✐s ♦❢ ▼✲st❛t✐♦♥❛r② ♣♦✐♥ts t♦ ❛♥ ❊P❊❈ ♠♦❞❡❧✐♥❣ ♦❧✐❣♦♣♦❧✐st✐❝ ❝♦♠♣❡t✐t✐♦♥ ✐♥ ❛♥ ❡❧❡❝tr✐❝✐t② s♣♦t ♠❛r❦❡t✱ ❊❙❆■▼✿ ❈❖❈❱ ✭✷✵✶✷✮✳ ▼✲st❛t✐♦♥❛r② ♣♦✐♥ts ❉✳ ❆✳✱ ❘✳ ❈♦rr❡❛ ✫ ▼✳ ▼❛r❡❝❤❛❧ ❙♣♦t ❡❧❡❝tr✐❝✐t② ♠❛r❦❡t ✇✐t❤ tr❛♥s♠✐ss✐♦♥ ❧♦ss❡s✱ ❏✳ ■♥❞✉str✐❛❧ ▼❛♥❛❣✳ ❖♣t✐♠ ✭✷✵✶✸✮✳ ❡①✐st❡♥❝❡ ♦❢ ◆❛s❤ ❡q✉✐❧✳✱ ❝❛s❡ ♦❢ ❛ t✇♦ ✐s❧❛♥❞ ♠♦❞❡❧ ❉✳❆✳✱ ▼✳ ❈❡r✈✐♥❦❛ ✫ ▼✳ ▼❛r❡❝❤❛❧✱ ❉❡r❡❣✉❧❛t❡❞ ❡❧❡❝tr✐❝✐t② ♠❛r❦❡ts ✇✐t❤ t❤❡r♠❛❧ ❧♦ss❡s ❛♥❞ ♣r♦❞✉❝t✐♦♥ ❜♦✉♥❞s✱ ❘❆■❘❖ ✭✷✵✶✻✮ ♣r♦❞✉❝t✐♦♥ ❜♦✉♥❞s✱ ✇❡❧❧✲♣♦s❡❞♥❡ss ♦❢ ♠♦❞❡❧
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
❇❡st r❡s♣♦♥s❡ ✐♥ ❡❧❡❝tr✐❝✐t② ♠❛r❦❡ts✿
❊✳ ❆♥❞❡rs♦♥ ❛♥❞ ❆✳ P❤✐❧♣♦tt✱ ❖♣t✐♠❛❧ ❖✛❡r ❈♦♥str✉❝t✐♦♥ ✐♥ ❊❧❡❝tr✐❝✐t② ▼❛r❦❡ts✱ ▼❛t❤❡♠❛t✐❝s ♦❢ ❖♣❡r❛t✐♦♥s ❘❡s❡❛r❝❤ ✭✷✵✵✷✮✳ ▲✐♥❡❛r ❜✐❞ ❢✉♥❝t✐♦♥ ✲ ♥❡❝❡ss❛r② ♦♣t✐♠❛❧✐t② ❝♦♥❞✳ ❢♦r ❧♦❝❛❧ ❜❡st r❡s♣♦♥s❡ ✐♥ t✐♠❡ ❞❡♣❡♥❞❡♥t ❝❛s❡ ❉✳ ❆✉ss❡❧✱ P✳ ❇❡♥❞♦tt✐ ❛♥❞ ▼✳ P✐➨t➙❦✱ ◆❛s❤ ❊q✉✐❧✐❜r✐✉♠ ✐♥ P❛②✲❛s✲❜✐❞ ❊❧❡❝tr✐❝✐t② ▼❛r❦❡t ✿ P❛rt ✷ ✲ ❇❡st ❘❡s♣♦♥s❡ ♦❢ Pr♦❞✉❝❡r✱ ❖♣t✐♠✐③❛t✐♦♥ ✭✷✵✶✼✮ ❧✐♥❡❛r ✉♥✐t ❜✐❞ ❢✉♥❝t✐♦♥✱ ❡①♣❧✐❝✐t ❢♦r♠✉❧❛ ❢♦r ❜❡st r❡s♣♦♥s❡
❊①♣❧✐❝✐t ❢♦r♠✉❧❛ ❢♦r ❡q✉✐❧✐❜r✐❛ ❉✳ ❆✉ss❡❧✱ P✳ ❇❡♥❞♦tt✐ ❛♥❞ ▼✳ P✐➨t➙❦✱ ◆❛s❤ ❊q✉✐❧✐❜r✐✉♠ ✐♥ P❛②✲❛s✲❜✐❞ ❊❧❡❝tr✐❝✐t② ▼❛r❦❡t ✿ P❛rt ✶ ✲ ❊①✐st❡♥❝❡ ❛♥❞ ❈❤❛r❛❝t❡r✐s❛t✐♦♥✱ ❖♣t✐♠✐③❛t✐♦♥ ✭✷✵✶✼✮ ❡①♣❧✐❝✐t ❢♦r♠✉❧❛ ❢♦r ❡q✉✐❧✐❜r✐❛
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
❇❡st r❡s♣♦♥s❡ ✐♥ ❡❧❡❝tr✐❝✐t② ♠❛r❦❡ts✿
❊✳ ❆♥❞❡rs♦♥ ❛♥❞ ❆✳ P❤✐❧♣♦tt✱ ❖♣t✐♠❛❧ ❖✛❡r ❈♦♥str✉❝t✐♦♥ ✐♥ ❊❧❡❝tr✐❝✐t② ▼❛r❦❡ts✱ ▼❛t❤❡♠❛t✐❝s ♦❢ ❖♣❡r❛t✐♦♥s ❘❡s❡❛r❝❤ ✭✷✵✵✷✮✳ ▲✐♥❡❛r ❜✐❞ ❢✉♥❝t✐♦♥ ✲ ♥❡❝❡ss❛r② ♦♣t✐♠❛❧✐t② ❝♦♥❞✳ ❢♦r ❧♦❝❛❧ ❜❡st r❡s♣♦♥s❡ ✐♥ t✐♠❡ ❞❡♣❡♥❞❡♥t ❝❛s❡ ❉✳ ❆✉ss❡❧✱ P✳ ❇❡♥❞♦tt✐ ❛♥❞ ▼✳ P✐➨t➙❦✱ ◆❛s❤ ❊q✉✐❧✐❜r✐✉♠ ✐♥ P❛②✲❛s✲❜✐❞ ❊❧❡❝tr✐❝✐t② ▼❛r❦❡t ✿ P❛rt ✷ ✲ ❇❡st ❘❡s♣♦♥s❡ ♦❢ Pr♦❞✉❝❡r✱ ❖♣t✐♠✐③❛t✐♦♥ ✭✷✵✶✼✮ ❧✐♥❡❛r ✉♥✐t ❜✐❞ ❢✉♥❝t✐♦♥✱ ❡①♣❧✐❝✐t ❢♦r♠✉❧❛ ❢♦r ❜❡st r❡s♣♦♥s❡
❊①♣❧✐❝✐t ❢♦r♠✉❧❛ ❢♦r ❡q✉✐❧✐❜r✐❛ ❉✳ ❆✉ss❡❧✱ P✳ ❇❡♥❞♦tt✐ ❛♥❞ ▼✳ P✐➨t➙❦✱ ◆❛s❤ ❊q✉✐❧✐❜r✐✉♠ ✐♥ P❛②✲❛s✲❜✐❞ ❊❧❡❝tr✐❝✐t② ▼❛r❦❡t ✿ P❛rt ✶ ✲ ❊①✐st❡♥❝❡ ❛♥❞ ❈❤❛r❛❝t❡r✐s❛t✐♦♥✱ ❖♣t✐♠✐③❛t✐♦♥ ✭✷✵✶✼✮ ❡①♣❧✐❝✐t ❢♦r♠✉❧❛ ❢♦r ❡q✉✐❧✐❜r✐❛
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
◆♦♥ ❛ ♣r✐♦r✐ str✉❝t✉r❡❞ ❜✐❞ ❢✉♥❝t✐♦♥s
❊s❝♦❜❛r✱ ❏✳❋✳ ❛♥❞ ❏♦❢ré✱ ❆✳✱ ▼♦♥♦♣♦❧✐st✐❝ ❝♦♠♣❡t✐t✐♦♥ ✐♥ ❡❧❡❝tr✐❝✐t② ♥❡t✇♦r❦s ✇✐t❤ r❡s✐st❛♥❝❡ ❧♦ss❡s✱ ❊❝♦♥♦♠✳ ❚❤❡♦r② ✹✹ ✭✷✵✶✵✮✳ ❊s❝♦❜❛r✱ ❏✳❋✳ ❛♥❞ ❏♦❢ré✱ ❆✳✱ ❊q✉✐❧✐❜r✐✉♠ ❛♥❛❧②s✐s ♦❢ ❡❧❡❝tr✐❝✐t② ❛✉❝t✐♦♥s✱ ♣r❡♣r✐♥t ✭✷✵✶✹✮✳ ❊✳ ❆♥❞❡rs♦♥✱ P✳ ❍♦❧♠❜❡r❣ ❛♥❞ ❆✳ P❤✐❧♣♦tt✱ ▼✐①❡❞ str❛t❡❣✐❡s ✐♥ ❞✐s❝r✐♠✐♥❛t♦r② ❞✐✈✐s✐❜❧❡✲❣♦♦❞ ❛✉❝t✐♦♥s✱ ❚❤❡ ❘❆◆❉ ❏♦✉r♥❛❧ ♦❢ ❊❝♦♥♦♠✐❝s ✭✷✵✶✸✮✳ ♥❡❝❡ss❛r② ♦♣t✐♠❛❧✐t② ❝♦♥❞✳ ❢♦r ❧♦❝❛❧ ❜❡st r❡s♣♦♥s❡
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
❉✳ ❆✉ss❡❧✱ P✳ ❇❡♥❞♦tt✐ ❛♥❞ ▼✳ P✐➨t➙❦✱ ◆❛s❤ ❊q✉✐❧✐❜r✐✉♠ ✐♥ P❛②✲❛s✲❜✐❞ ❊❧❡❝tr✐❝✐t② ▼❛r❦❡t ✿ ❖♣t✐♠✐③❛t✐♦♥ P❛rt ✶ ✲ ❊①✐st❡♥❝❡ ❛♥❞ ❈❤❛r❛❝t❡r✐s❛t✐♦♥✱ ✻✻✿✻ ✭✷✵✶✼✮ P❛rt ✷ ✲ ❇❡st ❘❡s♣♦♥s❡ ♦❢ Pr♦❞✉❝❡r✱ ✻✻✿✻ ✭✷✵✶✼✮✳
▲❡t ❝♦♥s✐❞❡r ❛ ✜①❡❞ t✐♠❡ ✐♥st❛♥t ❛♥❞ ❞❡♥♦t❡ D > 0 ❜❡ t❤❡ ♦✈❡r❛❧❧ ❡♥❡r❣② ❞❡♠❛♥❞ ♦❢ ❛❧❧ ❝♦♥s✉♠❡rs N ❜❡ t❤❡ s❡t ♦❢ ♣r♦❞✉❝❡rs qi ≥ 0 ❜❡ t❤❡ ♣r♦❞✉❝t✐♦♥ ♦❢ i✲t❤ ♣r♦❞✉❝❡r✱ i ∈ N ❲❡ ❛ss✉♠❡ t❤❛t ♣r♦❞✉❝❡r ♣r♦✈✐❞❡s t♦ t❤❡ ■❙❖ ❛ q✉❛❞r❛t✐❝ ❜✐❞ ❢✉♥❝t✐♦♥ ❣✐✈❡♥ ❜② ✳ ❙✐♠✐❧❛r❧②✱ ❧❡t ❜❡ t❤❡ tr✉❡ ♣r♦❞✉❝t✐♦♥ ❝♦st ♦❢ ✲t❤ ♣r♦❞✉❝❡r ✇✐t❤ ❛♥❞ r❡✢❡❝t✐♥❣ t❤❡ ✐♥❝r❡❛s✐♥❣ ♠❛r❣✐♥❛❧ ❝♦st ♦❢ ♣r♦❞✉❝t✐♦♥✳
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
❉✳ ❆✉ss❡❧✱ P✳ ❇❡♥❞♦tt✐ ❛♥❞ ▼✳ P✐➨t➙❦✱ ◆❛s❤ ❊q✉✐❧✐❜r✐✉♠ ✐♥ P❛②✲❛s✲❜✐❞ ❊❧❡❝tr✐❝✐t② ▼❛r❦❡t ✿ ❖♣t✐♠✐③❛t✐♦♥ P❛rt ✶ ✲ ❊①✐st❡♥❝❡ ❛♥❞ ❈❤❛r❛❝t❡r✐s❛t✐♦♥✱ ✻✻✿✻ ✭✷✵✶✼✮ P❛rt ✷ ✲ ❇❡st ❘❡s♣♦♥s❡ ♦❢ Pr♦❞✉❝❡r✱ ✻✻✿✻ ✭✷✵✶✼✮✳
▲❡t ❝♦♥s✐❞❡r ❛ ✜①❡❞ t✐♠❡ ✐♥st❛♥t ❛♥❞ ❞❡♥♦t❡ D > 0 ❜❡ t❤❡ ♦✈❡r❛❧❧ ❡♥❡r❣② ❞❡♠❛♥❞ ♦❢ ❛❧❧ ❝♦♥s✉♠❡rs N ❜❡ t❤❡ s❡t ♦❢ ♣r♦❞✉❝❡rs qi ≥ 0 ❜❡ t❤❡ ♣r♦❞✉❝t✐♦♥ ♦❢ i✲t❤ ♣r♦❞✉❝❡r✱ i ∈ N ❲❡ ❛ss✉♠❡ t❤❛t ♣r♦❞✉❝❡r i ∈ N ♣r♦✈✐❞❡s t♦ t❤❡ ■❙❖ ❛ q✉❛❞r❛t✐❝ ❜✐❞ ❢✉♥❝t✐♦♥ aiqi + biq2
i ❣✐✈❡♥ ❜② ai, bi ≥ 0✳
❙✐♠✐❧❛r❧②✱ ❧❡t ❜❡ t❤❡ tr✉❡ ♣r♦❞✉❝t✐♦♥ ❝♦st ♦❢ ✲t❤ ♣r♦❞✉❝❡r ✇✐t❤ ❛♥❞ r❡✢❡❝t✐♥❣ t❤❡ ✐♥❝r❡❛s✐♥❣ ♠❛r❣✐♥❛❧ ❝♦st ♦❢ ♣r♦❞✉❝t✐♦♥✳
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
❉✳ ❆✉ss❡❧✱ P✳ ❇❡♥❞♦tt✐ ❛♥❞ ▼✳ P✐➨t➙❦✱ ◆❛s❤ ❊q✉✐❧✐❜r✐✉♠ ✐♥ P❛②✲❛s✲❜✐❞ ❊❧❡❝tr✐❝✐t② ▼❛r❦❡t ✿ ❖♣t✐♠✐③❛t✐♦♥ P❛rt ✶ ✲ ❊①✐st❡♥❝❡ ❛♥❞ ❈❤❛r❛❝t❡r✐s❛t✐♦♥✱ ✻✻✿✻ ✭✷✵✶✼✮ P❛rt ✷ ✲ ❇❡st ❘❡s♣♦♥s❡ ♦❢ Pr♦❞✉❝❡r✱ ✻✻✿✻ ✭✷✵✶✼✮✳
▲❡t ❝♦♥s✐❞❡r ❛ ✜①❡❞ t✐♠❡ ✐♥st❛♥t ❛♥❞ ❞❡♥♦t❡ D > 0 ❜❡ t❤❡ ♦✈❡r❛❧❧ ❡♥❡r❣② ❞❡♠❛♥❞ ♦❢ ❛❧❧ ❝♦♥s✉♠❡rs N ❜❡ t❤❡ s❡t ♦❢ ♣r♦❞✉❝❡rs qi ≥ 0 ❜❡ t❤❡ ♣r♦❞✉❝t✐♦♥ ♦❢ i✲t❤ ♣r♦❞✉❝❡r✱ i ∈ N ❲❡ ❛ss✉♠❡ t❤❛t ♣r♦❞✉❝❡r i ∈ N ♣r♦✈✐❞❡s t♦ t❤❡ ■❙❖ ❛ q✉❛❞r❛t✐❝ ❜✐❞ ❢✉♥❝t✐♦♥ aiqi + biq2
i ❣✐✈❡♥ ❜② ai, bi ≥ 0✳
❙✐♠✐❧❛r❧②✱ ❧❡t Aiqi + Biq2
i ❜❡ t❤❡ tr✉❡ ♣r♦❞✉❝t✐♦♥ ❝♦st ♦❢ i✲t❤ ♣r♦❞✉❝❡r
✇✐t❤ Ai ≥ 0 ❛♥❞ Bi > 0 r❡✢❡❝t✐♥❣ t❤❡ ✐♥❝r❡❛s✐♥❣ ♠❛r❣✐♥❛❧ ❝♦st ♦❢ ♣r♦❞✉❝t✐♦♥✳
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
P❡❝✉❧✐❛r✐t② ♦❢ ❡❧❡❝tr✐❝✐t② ♠❛r❦❡ts ✐s t❤❡✐r ❜✐✲❧❡✈❡❧ str✉❝t✉r❡✿ Pi(a−i, b−i, D) max
ai,bi max qi
aiqi + biq2
i − (Aiqi + Biq2 i )
s✉❝❤ t❤❛t
(qj)j∈N ∈ Q(a, b) ✇❤❡r❡ s❡t✲✈❛❧✉❡❞ ♠❛♣♣✐♥❣ Q(a, b) ❞❡♥♦t❡s s♦❧✉t✐♦♥ s❡t ♦❢ ISO(a, b, D) Q(a, b) = ❛r❣♠✐♥
q
i )
s✉❝❤ t❤❛t qi ≥ 0 , ∀i ∈ N
qi = D
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
■♥❞✉str✐❛❧ ❊❝♦✲P❛r❦s
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
❊①❛♠♣❧❡ ♦❢ ✇❛t❡r ♠❛♥❛❣❡♠❡♥t ■♥ ❛ ❣❡♦❣r❛♣❤✐❝❛❧ ❛r❡❛✱ t❤❡r❡ ❛r❡ ❞✐✛❡r❡♥t ❝♦♠♣❛♥✐❡s 1, . . . , n ❊❛❝❤ ♦❢ t❤❡♠ ✐s ❜✉②✐♥❣ ❢r❡s❤ ✇❛t❡r ✭❤✐❣❤ ♣r✐❝❡✮ ❢♦r t❤❡✐r ♣r♦❞✉❝t✐♦♥ ♣r♦❝❡ss❡s ❊❛❝❤ ❝♦♠♣❛♥② ❣❡♥❡r❛t❡s s♦♠❡ ✧❞✐rt② ✇❛t❡r✧ ❛♥❞ ❤❛✈❡ t♦ ♣❛② ❢♦r ❞✐s❝❤❛r❣❡ ❙t❛♥❞ ❛❧♦♥❡ s✐t✉❛t✐♦♥
Company 2 Company 3 Fresh water Discharge water Company 1 Company 1
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
❚❤❡ ❛✐♠s ✐♥ ❞❡s✐❣♥✐♥❣ ■♥❞✉str✐❛❧ ❊❝♦✲♣❛r❦ ✭■❊P✮ ❛r❡ ❛✮ ❘❡❞✉❝❡ ❝♦st ♦❢ ♣r♦❞✉❝t✐♦♥ ♦❢ ❡❛❝❤ ❝♦♠♣❛♥② ❜✮ ❘❡❞✉❝❡ t❤❡ ❡♥✈✐r♦♥♠❡♥t❛❧ ✐♠♣❛❝t ♦❢ t❤❡ ✇❤♦❧❡ ♣r♦❞✉❝t✐♦♥ ❚❤✉s ✧❊❝♦✧ ♦❢ ■❊P ✐s ❛t t❤❡ s❛♠❡ t✐♠❡ ❊❝♦♥♦♠✐❝❛❧ ❛♥❞ ❡❝♦❧♦❣✐❝❛❧
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
❊①❛♠♣❧❡ ♦❢ ✇❛t❡r ♠❛♥❛❣❡♠❡♥t ❍♦✇ t♦ r❡❛❝❤ t❤❡s❡ ❛✐♠s❄ ❛✮ ❝r❡❛t❡ ❛ ♥❡t✇♦r❦ ✭✇❛t❡r t✉❜❡s✮ ❜❡t✇❡❡♥ t❤❡ ❝♦♠♣❛♥✐❡s ❜✮ ❊✈❡♥t✉❛❧❧② ✐♥st❛❧❧ s♦♠❡ r❡❣❡♥❡r❛t✐♦♥ ✉♥✐t ✭❝❧❡❛♥✐♥❣ ♦❢ t❤❡ ✇❛t❡r✮ ■t ✐s ✐♠♣♦rt❛♥t t♦ ✉♥❞❡rst❛♥❞ t❤❛t t❤✐s ❛♣♣r♦❛❝❤ ✐s ♥♦t ❧✐♠✐t❡❞ t♦ ✇❛t❡r✳ ■t ❝❛♥ ❜❡ ❛♣♣❧✐❡❞ t♦ ✈❛♣♦r✱ ❣❛s✱ ❝♦❛❧✐♥❣ ✢✉✐❞s✱ ❤✉♠❛♥ r❡s♦✉r❝❡s✳✳✳
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
❆♥ s②♠❜♦❧✐❝ ❡①❛♠♣❧❡ ♦❢ ■♥❞✉str✐❛❧ ❡❝♦✲♣❛r❦ ✐s ❑❛❧✉♥❞❜♦r❣ ✭❉❛♥❡♠❛r❦✮
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
❲❤❛t ✐s ❛♥ ✓ ❊❝♦✲♣❛r❦ ✔ ❄ ■♥ ♦r❞❡r t♦ ❝♦♥✈✐♥❝❡ ❝♦♠♣❛♥✐❡s t♦ ♣❛rt✐❝✐♣❛t❡ t♦ t❤❡ ❊❝♦♣❛r❦✱ ♦✉r ♠♦❞❡❧ s❤♦✉❧❞ ❣✉❛r❛♥t❡❡ t❤❛t✿ ❛✮ ❡❛❝❤ ❝♦♠♣❛♥② ✇✐❧❧ ❤❛✈❡ ❛ ❧♦✇❡r ❝♦st ♦❢ ♣r♦❞✉❝t✐♦♥ ✐♥ ❊❝♦✲♣❛r❦ ♦r❣❛♥✐③❛t✐♦♥ t❤❛♥ ✐♥ st❛♥❞✲❛❧♦♥❡ ♦r❣❛♥✐③❛t✐♦♥ ❜✮ t❤❡ ❡❝♦✲♣❛r❦ ♦r❣❛♥✐③❛t✐♦♥ ♠✉st ❣❡♥❡r❛t❡ ❛ ❧♦✇❡r ❢r❡s❤✇❛t❡r ❝♦♥s✉♠♣t✐♦♥ t❤❛♥ ✇✐t❤ ❛ st❛♥❞✲❛❧♦♥❡ ♦r❣❛♥✐③❛t✐♦♥
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
❚❤❡ ❊❝♦✲♣❛r❦ ❞❡s✐❣♥ ✇❛s ❞♦♥❡ t❤r♦✉❣❤ ▼✉❧t✐✲♦❜❥❡❝t✐✈❡ ❖♣t✐♠✐③❛t✐♦♥ ❜② t❤❡ ❡✈❛❧✉❛t✐♦♥ ♦❢ P❛r❡t♦ ❢r♦♥ts ✭●♦❧❞ ♣r♦❣r❛♠♠✐♥❣ ❛❧❣♦r✐t❤♠s✱ s❝❛❧❛r✐③❛t✐♦♥✳✳✳✮✳ min ❋r❡s❤ ✇❛t❡r ❝♦♥s✉♠♣t✐♦♥ ■♥❞✐✈✐❞✉❛❧ ❝♦sts ♦❢ ♣r♦❞✉❝❡r 1 ✳ ✳ ✳ ■♥❞✐✈✐❞✉❛❧ ❝♦sts ♦❢ ♣r♦❞✉❝❡r n s✳t✳ ❲❛t❡r ❜❛❧❛♥❝❡s ❚♦♣♦❧♦❣✐❝❛❧ ❝♦♥str❛✐♥ts ❲❛t❡r q✉❛❧✐t② ❝r✐t❡r✐❛
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
❚❤❡ ♥❡❡❞❡❞ ❝❤❛♥❣❡ ✿ . . . t♦ ❤❛✈❡ ❛♥ ✐♥❞❡♣❡♥❞❛♥t ❞❡s✐❣♥❡r✴r❡❣✉❧❛t♦r . . . t♦ ❤❛✈❡ ❢❛✐r s♦❧✉t✐♦♥s ❢♦r t❤❡ ❝♦♠♣❛♥✐❡s ❚❤✉s ✇❡ ♣r♦♣♦s❡ t♦ ✉s❡ t✇♦ ❞✐✛❡r❡♥t ♣♦ss✐❜❧❡ ♠♦❞❡❧s✿ ❍✐❡r❛r❝❤✐❝❛❧ ♦♣t✐♠✐s❛t✐♦♥ ✭❜✐✲❧❡✈❡❧ ♦♣t✐♠✳✮ ◆❛s❤ ❣❛♠❡ ❝♦♥❝❡♣t ❜❡t✇❡❡♥ t❤❡ ❝♦♠♣❛♥✐❡s
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
❚❤✐s ✈❡r② ❞✐✣❝✉❧t ♣r♦❜❧❡♠ ✐s tr❡❛t❡❞ ❛s ❢♦❧❧♦✇s✿ ✜rst ✇❡ r❡♣❧❛❝❡ t❤❡ ❧♦✇❡r✲❧❡✈❡❧ ✭❝♦♥✈❡①✮ ♦♣t✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠ ❜② t❤❡✐r ❑❑❚ s②st❡♠s❀ t❤❡ r❡s✉❧t✐♥❣ ♣r♦❜❧❡♠ ✐s ❛♥ ▼❛t❤❡♠❛t✐❝❛❧ Pr♦❣r❛♠♠✐♥❣ ✇✐t❤ ❈♦♠♣❧❡♠❡♥t❛r✐t② ❈♦♥str❛✐♥ts ✭▼P❈❈✮❀ s❡❝♦♥❞ t❤❡ ▼P❈❈ ♣r♦❜❧❡♠ ✐s s♦❧✈❡❞ ❜② ♣❡♥❛❧✐③❛t✐♦♥ ♠❡t❤♦❞s ◆✉♠❡r✐❝❛❧ r❡s✉❧ts ❤❛✈❡ ❜❡❡♥ ♦❜t❛✐♥❡❞ ✇✐t❤ ❏✉❧✐❛ ♠❡t❛✲s♦❧✈❡r ❝♦✉♣❧❡❞ ✇✐t❤ ●✉r♦❜✐✱ ■P❖P❚ ❛♥❞ ❇❛r♦♥✳
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
❉✳❆ ✫ ❆✳ ❙✈❡♥ss♦♥ ✭❏✳ ❖♣t✐♠✳ ❚❤❡♦r② ❆♣♣❧✳ ✶✽✷ ✭✷✵✶✾✮✮
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
❚❤❡♦r❡♠ ❆ss✉♠❡ t❤❛t F ✐s ❧♦✇❡r s❡♠✐✲❝♦♥t✐♥✉♦✉s✱ ❛♥❞ ❢♦r ❡❛❝❤ ❢♦❧❧♦✇❡r i = 1, ..., M t❤❡ ♦❜❥❡❝t✐✈❡ fi ✐s ❝♦♥t✐♥✉♦✉s ❛♥❞ (x, y−i) → Ci(x, y−i) := {yi | gi(x, y) ≤ 0} ✐s ❛ ❧♦✇❡r s❡♠✐✲❝♦♥t✐♥✉♦✉s s❡t✲✈❛❧✉❡❞ ♠❛♣ ✇❤✐❝❤ ❤❛s ♥♦♥❡♠♣t② ❝♦♠♣❛❝t ❣r❛♣❤✳ ■❢ t❤❡ ❣r❛♣❤ ♦❢ GNEP ✐s ♥♦♥❡♠♣t②✱ t❤❡♥ t❤❡ ❙▲▼❋ ❣❛♠❡ ❛❞♠✐ts ❛♥ ♦♣t✐♠✐st✐❝ s♦❧✉t✐♦♥✳
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
▲❡t ✉s ❝♦♥s✐❞❡r t❤❡ ❙▲▼❋● ✇✐t❤ t✇♦ ❢♦❧❧♦✇❡rs min
x∈[0,4]
max
y∈●◆❊P(x) −x + (y1 + y2).
✇✐t❤ miny1 y1 s✳t✳ y1 ≥ 0 2y2 − y1 ≤ 2 y1 + y2 ≥ x miny2 y2 s✳t✳ y2 ≥ 0 2y1 − y2 ≤ 2 y1 + y2 ≥ x
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
❚❤❡ s♦❧✉t✐♦♥ ♦❢ t❤❡ ♣❛r❛♠❡tr✐❝ ●◆❊P ♦❢ t❤❡ ❢♦❧❧♦✇❡rs ✐s ❣✐✈❡♥ ❜②
{(0, 0), (2, 2)} ✐❢ x ≥ 4, {(0, 0)} ✐❢ x ∈ [0, 4[ ∅ ♦t❤❡r✇✐s❡✳ ✭✺✮ ◆♦t✐❝❡ t❤❛t t❤❡ ❢✉♥❝t✐♦♥ ϕmax(x) := max
y∈●◆❊P(x) −x + (y1 + y2)
=
−x ✐❢ x ∈ [0, 4[ ✐s ♥♦t ❧♦✇❡r s❡♠✐✲❝♦♥t✐♥✉♦✉s✱ s♦ t❤❛t ❲❡✐❡rstr❛ss t❤❡♦r❡♠ ❛r❣✉♠❡♥t ❝❛♥♥♦t ❜❡ ❛♣♣❧✐❡❞✳ ❆♥❞ ✐♥ ❢❛❝t✱ t❤❡ ✈❛❧✉❡ ♦❢ t❤❡ ♣r♦❜❧❡♠ ♦❢ t❤❡ ❧❡❛❞❡r ✐s −4✱ ✇❤✐❧❡ t❤❡r❡ ❞♦❡s ♥♦t ❡①✐st ❛ ♣♦✐♥t x ∈ [0, 4] ✇✐t❤ t❤❛t ✈❛❧✉❡✳ ❚❤❡ ♣❡ss✐♠✐st✐❝ ❧✐♥❡❛r s✐♥❣❧❡✲❧❡❛❞❡r✲t✇♦✲❢♦❧❧♦✇❡r ♣r♦❜❧❡♠ ❤❛s ♥♦ ♦♣t✐♠❛❧ s♦❧✉t✐♦♥✳
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
■♥ t✇♦ ✈❡r② r❡❝❡♥t ✇♦r❦s ✇❡ s✉❣❣❡st❡❞ s♦♠❡ r❡❢♦r♠✉❧❛t✐♦♥s ♦❢ t❤❡ ♦♣t✐♠❛❧ ❞❡s✐❣♥ ♣r♦❜❧❡♠✿ ✉♥❞❡r s♦♠❡ ❤②♣♦t❤❡s✐s ✭✉♥✐q✉❡ ♣r♦❝❡ss ❢♦r ❡❛❝❤ ❝♦♠♣❛♥②✱ ❧✐♥❡❛r✐③❛t✐♦♥ ✐♥ t❤❡ ❝❛s❡ ♦❢ r❡❣❡♥❡r❛t✐♦♥ ✉♥✐ts✮✱ ✇❡ s❤♦✇♥ t❤❛t t❤❡ ♦♣t✐♠❛❧ ❞❡s✐❣♥ ♣r♦❜❧❡♠ ❝❛♥ ❜❡ r❡❢♦r♠✉❧❛t❡❞ ❛s ❛ ❝❧❛ss✐❝❛❧ ▼✐①❡❞ ■♥t❡❣❡r ▲✐♥❡❛r Pr♦❣r❛♠♠✐♥❣ ♣r♦❜❧❡♠ ✭▼■▲P✮❀ t❤✐s ♣r♦❜❧❡♠ ❝❛♥ ❜❡ tr❡❛t❡❞ ✇✐t❤ ❝❧❛ss✐❝❛❧ t♦♦❧s ✭❈P▲❊❳✮❀ ▼♦r❡♦✈❡r ✇❡ ✐♥s❡rt❡❞ ❛ ✧♠✐♥✐♠❛❧ ❣❛✐♥✧ ❝♦♥❞✐t✐♦♥ Costi(xi, xP
−i, xR, E) ≤ αi · STCi,
∀i ∈ IP . ❡♥s✉r✐♥❣ t❤❛t ❡❛❝❤ ♣❛rt✐❝✐♣❛t✐♥❣ ❝♦♠♣❛♥② ✇✐❧❧ ❣❛✐♥ ❛t ♠✐♥✐♠✉♠ α% ♦♥ ✐ts ♣r♦❞✉❝t✐♦♥ ❝♦st✳
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
❚❤❡♦r❡♠ ❋♦r E ∈ E ❛♥❞ xR ≥ 0 ✜①❡❞✱ t❤❡ ❡q✉✐❧✐❜r✐✉♠ s❡t Eq(xR, E) ✐s ❣✐✈❡♥ ❜② Eq(xR, E) = xP : ∀i ∈ IP , zi(x−i) +
xk,i =
xi,j gi(x−i) ≤ 0 zi(x−i) ≥ 0 xi
i,act
= 0 xi ≥ 0 ✭✻✮ ❚❤✉s✱ t❤❡ ♦♣t✐♠❛❧ ❞❡s✐❣♥ ♣r♦❜❧❡♠ ✐s ❡q✉✐✈❛❧❡♥t t♦ min
E∈E,x∈R|Emax| Z(x)
s.t. x ∈ X, zi(x−i) +
xk,i = +
xi,j, ∀i ∈ I xi
i,act
= 0, ∀i ∈ I gi(x−i) ≤ 0, ∀i ∈ I zi(x−i) ≥ 0, ∀i ∈ I Costi(xi, xP
−i, xR, E) ≤ αi · ST Ci,
∀i ∈ IP x ≥ 0. ✭✼✮ ❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
11 12 13 3 6 4 8 15 14 7 9 10 5 2 1
❋✐❣✉r❡✿ ❚❤❡ ❝♦♥✜❣✉r❛t✐♦♥ ✐♥ t❤❡ ❝❛s❡ ✇✐t❤♦✉t r❡❣❡♥❡r❛t✐♦♥ ✉♥✐ts✱ αi = 0.95 ❛♥❞ Coef = 1✳ ●r❛② ♥♦❞❡s ❛r❡ ❝♦♥s✉♠✐♥❣ str✐❝t❧② ♣♦s✐t✐✈❡ ❢r❡s❤ ✇❛t❡r✳
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
0.75 0.8 0.85 0.9 0.95 1
α
1 2 3 4 5 6 7
The number of enterprises operating stand-alone
360 370 380 390 400 410 420 430
Global freshwater consumption
The number of enterprises operating STC Global Freshwater consumption
❋✐❣✉r❡✿ ❚❤❡ ♥✉♠❜❡r ♦❢ ❡♥t❡r♣r✐s❡s ♦♣❡r❛t✐♥❣ st❛♥❞✲❛❧♦♥❡ ❛♥❞ t❤❡ ❣❧♦❜❛❧ ❢r❡s❤✇❛t❡r ❝♦♥s✉♠♣t✐♦♥ ✇✐t❤ Coef = 1✳
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
2 4 6 8 10 12 14 16 18 20
coef
1 2 3 4 5 6 7
The number of enterprises operating stand-alone
365 366 367 368 369 370 371 372 373 374 375
Global freshwater consumption
The number of enterprises operating STC Global Freshwater consumption
❋✐❣✉r❡✿ ❚❤❡ ♥✉♠❜❡r ♦❢ ❡♥t❡r♣r✐s❡s ♦♣❡r❛t✐♥❣ st❛♥❞✲❛❧♦♥❡ ❛♥❞ t❤❡ ❣❧♦❜❛❧ ❢r❡s❤✇❛t❡r ❝♦♥s✉♠♣t✐♦♥ ✇✐t❤ α = 0.99✳
❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
❉✳ ❆✉ss❡❧✱ ❆✳ ❙✈❡♥ss♦♥✱ ❚♦✇❛r❞s ❚r❛❝t❛❜❧❡ ❈♦♥str❛✐♥t ◗✉❛❧✐✜❝❛t✐♦♥s ❢♦r P❛r❛♠❡tr✐❝ ❖♣t✐♠✐s❛t✐♦♥ Pr♦❜❧❡♠s ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s t♦ ●❡♥❡r❛❧✐s❡❞ ◆❛s❤ ●❛♠❡s✱ ❏✳ ❖♣t✐♠✳ ❚❤❡♦r② ❆♣♣❧✳ ✶✽✷ ✭✷✵✶✾✮✱ ✹✵✹✲✹✶✻✳ ❉✳ ❆✉ss❡❧✱ ▲✳ ❇r♦t❝♦r♥❡✱ ❙✳ ▲❡♣❛✉❧✱ ▲✳ ✈♦♥ ◆✐❡❞❡r❤ä✉s❡r♥✱ ❆ ❚r✐❧❡✈❡❧ ▼♦❞❡❧ ❢♦r ❇❡st ❘❡s♣♦♥s❡ ✐♥ ❊♥❡r❣② ❉❡♠❛♥❞ ❙✐❞❡ ▼❛♥❛❣❡♠❡♥t✱ ❊✉r✳ ❏✳ ❖♣❡r❛t✐♦♥s ❘❡s❡❛r❝❤ ✷✽✶ ✭✷✵✷✵✮✱ ✷✾✾✲✸✶✺✳ ❉✳ ❆✉ss❡❧✱ ❑✳ ❈❛♦ ❱❛♥✱ ❉✳ ❙❛❧❛s✱ ◗✉❛s✐✲✈❛r✐❛t✐♦♥❛❧ ■♥❡q✉❛❧✐t② Pr♦❜❧❡♠s ♦✈❡r Pr♦❞✉❝t s❡ts ✇✐t❤ ◗✉❛s✐♠♦♥♦t♦♥❡ ❖♣❡r❛t♦rs✱ ❙■❖P❚ ✷✾ ✭✷✵✶✾✮✱ ✶✺✺✽✲✶✺✼✼✳ ❉✳ ❆✉ss❡❧ ✫ ❆✳ ❙✈❡♥ss♦♥✱ ■s P❡ss✐♠✐st✐❝ ❇✐❧❡✈❡❧ Pr♦❣r❛♠♠✐♥❣ ❛ ❙♣❡❝✐❛❧ ❈❛s❡ ♦❢ ❛ ▼❛t❤❡♠❛t✐❝❛❧ Pr♦❣r❛♠ ✇✐t❤ ❈♦♠♣❧❡♠❡♥t❛r✐t② ❈♦♥str❛✐♥ts❄✱ ❏✳ ❖♣t✐♠✳ ❚❤❡♦r② ❆♣♣❧✳ ✶✽✶✭✷✮ ✭✷✵✶✾✮✱ ✺✵✹✲✺✷✵✳ ❉✳ ❆✉ss❡❧ ✫ ❆✳ ❙✈❡♥ss♦♥✱ ❙♦♠❡ r❡♠❛r❦s ♦♥ ❡①✐st❡♥❝❡ ♦❢ ❡q✉✐❧✐❜r✐❛✱ ❛♥❞ t❤❡ ✈❛❧✐❞✐t② ♦❢ t❤❡ ❊P❈❈ r❡❢♦r♠✉❧❛t✐♦♥ ❢♦r ♠✉❧t✐✲❧❡❛❞❡r✲❢♦❧❧♦✇❡r ❣❛♠❡s✱ ❏✳ ◆♦♥❧✐♥❡❛r ❈♦♥✈❡① ❆♥❛❧✳ ✶✾ ✭✷✵✶✽✮✱ ✶✶✹✶✲✶✶✻✷✳ ❊✳ ❆❧❧❡✈✐✱ ❉✳ ❆✉ss❡❧ ✫ ❘✳ ❘✐❝❝❛r❞✐✱ ❖♥ ❛ ❡q✉✐❧✐❜r✐✉♠ ♣r♦❜❧❡♠ ✇✐t❤ ❝♦♠♣❧❡♠❡♥t❛r✐t② ❝♦♥str❛✐♥ts ❢♦r♠✉❧❛t✐♦♥ ♦❢ ♣❛②✲❛s✲❝❧❡❛r ❡❧❡❝tr✐❝✐t② ♠❛r❦❡t ✇✐t❤ ❞❡♠❛♥❞ ❡❧❛st✐❝✐t②✱ ❏✳ ●❧♦❜❛❧ ❖♣t✐♠✳ ✼✵ ✭✷✵✶✽✮✱ ✸✷✾✲✸✹✻✳ ❉✳ ❆✉ss❡❧✱ P✳ ❇❡♥❞♦tt✐ ❛♥❞ ▼✳ P✐➨t➙❦✱ ◆❛s❤ ❊q✉✐❧✐❜r✐✉♠ ✐♥ P❛②✲❛s✲❜✐❞ ❊❧❡❝tr✐❝✐t② ▼❛r❦❡t ✿ P❛rt ✶ ✲ ❊①✐st❡♥❝❡ ❛♥❞ ❈❤❛r❛❝t❡r✐s❛t✐♦♥✱ ❖♣t✐♠✐③❛t✐♦♥ ✻✻✿✻ ✭✷✵✶✼✮✱ ✶✵✶✸✲✶✵✷✺✳ ❉✳ ❆✉ss❡❧✱ P✳ ❇❡♥❞♦tt✐ ❛♥❞ ▼✳ P✐➨t➙❦✱ ◆❛s❤ ❊q✉✐❧✐❜r✐✉♠ ✐♥ P❛②✲❛s✲❜✐❞ ❊❧❡❝tr✐❝✐t② ▼❛r❦❡t ✿ P❛rt ✷ ✲ ❇❡st ❘❡s♣♦♥s❡ ♦❢ Pr♦❞✉❝❡r✱ ❖♣t✐♠✐③❛t✐♦♥ ✻✻✿✻ ✭✷✵✶✼✮✱ ✶✵✷✼✲✶✵✺✸✳ ❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
▼✳ ❘❛♠♦s✱ ▼✳ ❇♦✐①✱ ❉✳ ❆✉ss❡❧✱ ▲✳ ▼♦♥t❛str✉❝✱ ❙✳ ❉♦♠❡♥❡❝❤✱ ❲❛t❡r ✐♥t❡❣r❛t✐♦♥ ✐♥ ❊❝♦✲■♥❞✉str✐❛❧ P❛r❦s ❯s✐♥❣ ❛ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ❆♣♣r♦❛❝❤✱ ❈♦♠♣✉t❡rs ✫ ❈❤❡♠✐❝❛❧ ❊♥❣✐♥❡❡r✐♥❣ ✽✼ ✭✷✵✶✻✮ ✶✾✵✲✷✵✼✳ ▼✳ ❘❛♠♦s✱ ▼✳ ❇♦✐①✱ ❉✳ ❆✉ss❡❧✱ ▲✳ ▼♦♥t❛str✉❝✱ ❙✳ ❉♦♠❡♥❡❝❤✱ ❖♣t✐♠❛❧ ❉❡s✐❣♥ ♦❢ ❲❛t❡r ❊①❝❤❛♥❣❡s ✐♥ ❊❝♦✲■♥❞✉❝tr✐❛❧ P❛r❦s ❚❤r♦✉❣❤ ❛ ●❛♠❡ ❚❤❡♦r② ❆♣♣r♦❛❝❤✱ ❈♦♠♣✉t❡rs ❆✐❞❡❞ ❈❤❡♠✐❝❛❧ ❊♥❣✐♥❡❡r✐♥❣ ✸✽ ✭✷✵✶✻✮ ✶✶✼✼✲✶✶✽✸✳ ▼✳ ❘❛♠♦s✱ ▼✳ ❘♦❝❛❢✉❧❧✱ ▼✳ ❇♦✐①✱ ❉✳ ❆✉ss❡❧✱ ▲✳ ▼♦♥t❛str✉❝ ✫ ❙✳ ❉♦♠❡♥❡❝❤✱ ❯t✐❧✐t② ◆❡t✇♦r❦ ❖♣t✐♠✐③❛t✐♦♥ ✐♥ ❊❝♦✲■♥❞✉str✐❛❧ P❛r❦s ❜② ❛ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ❣❛♠❡ ▼❡t❤♦❞♦❧♦❣②✱ ❈♦♠♣✉t❡rs ✫ ❈❤❡♠✐❝❛❧ ❊♥❣✐♥❡❡r✐♥❣ ✶✶✷ ✭✷✵✶✽✮✱ ✶✸✷✲✶✺✸✳ ❉✳ ❙❛❧❛s✱ ❈❛♦ ❱❛♥ ❑✐❡♥✱ ❉✳ ❆✉ss❡❧✱ ▲✳ ▼♦♥t❛str✉❝✱ ❖♣t✐♠❛❧ ❞❡s✐❣♥ ♦❢ ❡①❝❤❛♥❣❡ ♥❡t✇♦r❦s ✇✐t❤ ❜❧✐♥❞ ✐♥♣✉ts ❛♥❞ ✐ts ❛♣♣❧✐❝❛t✐♦♥ t♦ ❊❝♦✲■♥❞✉str✐❛❧ ♣❛r❦s✱ ❈♦♠♣✉t❡rs ✫ ❈❤❡♠✐❝❛❧ ❊♥❣✐♥❡❡r✐♥❣ ✶✹✸ ✭✷✵✷✵✮✱ ✶✽ ♣♣✱ ♣✉❜❧✐s❤❡❞ ♦♥❧✐♥❡✳ ❆✉ss❡❧✱ ❑✳ ❈❛♦ ❱❛♥✱ ❈♦♥tr♦❧✲✐♥♣✉t ❛♣♣r♦❛❝❤ ♦❢ ♦❢ ✇❛t❡r ❡①❝❤❛♥❣❡ ✐♥ ❊❝♦✲■♥❞✉str✐❛❧ P❛r❦s✱ s✉❜♠✐tt❡❞ ✭✷✵✷✵✮✳ ❉✐❞✐❡r ❆✉ss❡❧ ❇✐❧❡✈❡❧ Pr♦❜❧❡♠s✱ ▼P❈❈s✱ ❛♥❞ ▼✉❧t✐✲▲❡❛❞❡r✲❋♦❧❧♦✇❡r ●❛♠❡s
Quasiconvex optimization Now the case of GNEP...
Didier Aussel
ALOP autumn school - October 14th, 2020
Coauthors: N. Hadjisavvas (Greece and Saudia), M. Pistek (Czech Republic), Jane Ye (Canada) Didier Aussel Univ. de Perpignan, France MLFG in the setting of quasiconvex optimization
Quasiconvex optimization Now the case of GNEP...
Didier Aussel Univ. de Perpignan, France MLFG in the setting of quasiconvex optimization
Quasiconvex optimization Now the case of GNEP...
Quasiconvexity A function f : X → I R ∪ {+∞} is said to be quasiconvex on K if, for all x, y ∈ K and all t ∈ [0, 1], f (tx + (1 − t)y) ≤ max{f (x), f (y)}.
Didier Aussel Univ. de Perpignan, France MLFG in the setting of quasiconvex optimization
Quasiconvex optimization Now the case of GNEP...
Quasiconvexity A function f : X → I R ∪ {+∞} is said to be quasiconvex on K if, for all x, y ∈ K and all t ∈ [0, 1], f (tx + (1 − t)y) ≤ max{f (x), f (y)}.
for all λ ∈ I R, the sublevel set Sλ = {x ∈ X : f (x) ≤ λ} is convex.
Didier Aussel Univ. de Perpignan, France MLFG in the setting of quasiconvex optimization
Quasiconvex optimization Now the case of GNEP...
Quasiconvexity A function f : X → I R ∪ {+∞} is said to be quasiconvex on K if, for all x, y ∈ K and all t ∈ [0, 1], f (tx + (1 − t)y) ≤ max{f (x), f (y)}.
for all λ ∈ I R, the sublevel set Sλ = {x ∈ X : f (x) ≤ λ} is convex.
f differentiable f is quasiconvex ⇐ ⇒ df is quasimonotone
Didier Aussel Univ. de Perpignan, France MLFG in the setting of quasiconvex optimization
Quasiconvex optimization Now the case of GNEP...
Quasiconvexity A function f : X → I R ∪ {+∞} is said to be quasiconvex on K if, for all x, y ∈ K and all t ∈ [0, 1], f (tx + (1 − t)y) ≤ max{f (x), f (y)}.
for all λ ∈ I R, the sublevel set Sλ = {x ∈ X : f (x) ≤ λ} is convex.
f differentiable f is quasiconvex ⇐ ⇒ df is quasimonotone
f is quasiconvex ⇐ ⇒ ∂f is quasimonotone
Didier Aussel Univ. de Perpignan, France MLFG in the setting of quasiconvex optimization
Quasiconvex optimization Now the case of GNEP...
Quasiconvexity A function f : X → I R ∪ {+∞} is said to be quasiconvex on K if, for all λ ∈ I R, the sublevel set Sλ = {x ∈ X : f (x) ≤ λ} is convex. A function f : X → I R ∪ {+∞} is said to be semistrictly quasiconvex
f (x) < f (y) ⇒ f (z) < f (y), ∀ z ∈ [x, y[.
Didier Aussel Univ. de Perpignan, France MLFG in the setting of quasiconvex optimization
Quasiconvex optimization Now the case of GNEP...
Why not a subdifferential for quasiconvex programming?
Didier Aussel Univ. de Perpignan, France MLFG in the setting of quasiconvex optimization
Quasiconvex optimization Now the case of GNEP...
Why not a subdifferential for quasiconvex programming? No (upper) semicontinuity of ∂f if f is not supposed to be Lipschitz
Didier Aussel Univ. de Perpignan, France MLFG in the setting of quasiconvex optimization
Quasiconvex optimization Now the case of GNEP...
Why not a subdifferential for quasiconvex programming? No (upper) semicontinuity of ∂f if f is not supposed to be Lipschitz No sufficient optimality condition ¯ x ∈ Sstr(∂f , C) = ⇒ ¯ x ∈ arg min
C f
❍ ❍ ❍ ✟ ✟ ✟
Didier Aussel Univ. de Perpignan, France MLFG in the setting of quasiconvex optimization
Quasiconvex optimization Now the case of GNEP...
Didier Aussel Univ. de Perpignan, France MLFG in the setting of quasiconvex optimization
Quasiconvex optimization Now the case of GNEP...
A first approach Sublevel set: Sλ = {x ∈ X : f (x) ≤ λ} S>
λ = {x ∈ X : f (x) < λ}
Normal operator: Define Nf (x) : X → 2X ∗ by Nf (x) = N(Sf (x), x) = {x∗ ∈ X ∗ : x∗, y − x ≤ 0, ∀ y ∈ Sf (x)}. With the corresponding definition for N>
f (x)
Didier Aussel Univ. de Perpignan, France MLFG in the setting of quasiconvex optimization
Quasiconvex optimization Now the case of GNEP...
But ... Nf (x) = N(Sf (x), x) has no upper-semicontinuity properties N>
f (x) = N(S> f (x), x) has no quasimonotonicity properties
Didier Aussel Univ. de Perpignan, France MLFG in the setting of quasiconvex optimization
Quasiconvex optimization Now the case of GNEP...
But ... Nf (x) = N(Sf (x), x) has no upper-semicontinuity properties N>
f (x) = N(S> f (x), x) has no quasimonotonicity properties
Example Define f : R2 → R by f (a, b) =
if |a| + |b| ≤ 1 1, if |a| + |b| > 1 . Then f is quasiconvex. Didier Aussel Univ. de Perpignan, France MLFG in the setting of quasiconvex optimization
Quasiconvex optimization Now the case of GNEP...
But ... Nf (x) = N(Sf (x), x) has no upper-semicontinuity properties N>
f (x) = N(S> f (x), x) has no quasimonotonicity properties
Example Define f : R2 → R by f (a, b) =
if |a| + |b| ≤ 1 1, if |a| + |b| > 1 . Then f is quasiconvex. Consider x = (10, 0), x∗ = (1, 2), y = (0, 10) and y∗ = (2, 1). We see that x∗ ∈ N<(x) and y∗ ∈ N< (y) (since |a| + |b| < 1 implies (1, 2) · (a − 10, b) ≤ 0 and (2, 1) · (a, b − 10) ≤ 0) while
Didier Aussel Univ. de Perpignan, France MLFG in the setting of quasiconvex optimization
Quasiconvex optimization Now the case of GNEP...
But ...another example Nf (x) = N(Sf (x), x) has no upper-semicontinuity properties N>
f (x) = N(S> f (x), x) has no quasimonotonicity properties
Example
Then f is quasiconvex. Didier Aussel Univ. de Perpignan, France MLFG in the setting of quasiconvex optimization
Quasiconvex optimization Now the case of GNEP...
But ...another example Nf (x) = N(Sf (x), x) has no upper-semicontinuity properties N>
f (x) = N(S> f (x), x) has no quasimonotonicity properties
Example
Then f is quasiconvex. We easily see that N(x) is not upper semicontinuous.... Didier Aussel Univ. de Perpignan, France MLFG in the setting of quasiconvex optimization
Quasiconvex optimization Now the case of GNEP...
But ...another example Nf (x) = N(Sf (x), x) has no upper-semicontinuity properties N>
f (x) = N(S> f (x), x) has no quasimonotonicity properties
Example
Then f is quasiconvex. We easily see that N(x) is not upper semicontinuous....
These two operators are essentially adapted to the class of semi-strictly quasiconvex functions. Indeed in this case, for each x ∈ dom f \ arg min f , the sets Sf (x) and S<
f (x) have the same closure and Nf (x) = N< f (x).
Didier Aussel Univ. de Perpignan, France MLFG in the setting of quasiconvex optimization
Quasiconvex optimization Now the case of GNEP...
Didier Aussel Univ. de Perpignan, France MLFG in the setting of quasiconvex optimization
Quasiconvex optimization Now the case of GNEP...
Definition Adjusted sublevel set For any x ∈ dom f , we define Sa
f (x) = Sf (x) ∩ B(S< f (x), ρx)
where ρx = dist(x, S<
f (x)), if S< f (x) = ∅
and Sa
f (x) = Sf (x) if S< f (x) = ∅.
Didier Aussel Univ. de Perpignan, France MLFG in the setting of quasiconvex optimization
Quasiconvex optimization Now the case of GNEP...
Definition Adjusted sublevel set For any x ∈ dom f , we define Sa
f (x) = Sf (x) ∩ B(S< f (x), ρx)
where ρx = dist(x, S<
f (x)), if S< f (x) = ∅
and Sa
f (x) = Sf (x) if S< f (x) = ∅.
Sa
f (x) coincides with Sf (x) if cl(S> f (x)) = Sf (x)
Didier Aussel Univ. de Perpignan, France MLFG in the setting of quasiconvex optimization
Quasiconvex optimization Now the case of GNEP...
Definition Adjusted sublevel set For any x ∈ dom f , we define Sa
f (x) = Sf (x) ∩ B(S< f (x), ρx)
where ρx = dist(x, S<
f (x)), if S< f (x) = ∅
and Sa
f (x) = Sf (x) if S< f (x) = ∅.
Sa
f (x) coincides with Sf (x) if cl(S> f (x)) = Sf (x)
e.g. f is semistrictly quasiconvex Didier Aussel Univ. de Perpignan, France MLFG in the setting of quasiconvex optimization
Quasiconvex optimization Now the case of GNEP...
Definition Adjusted sublevel set For any x ∈ dom f , we define Sa
f (x) = Sf (x) ∩ B(S< f (x), ρx)
where ρx = dist(x, S<
f (x)), if S< f (x) = ∅
and Sa
f (x) = Sf (x) if S< f (x) = ∅.
Sa
f (x) coincides with Sf (x) if cl(S> f (x)) = Sf (x)
e.g. f is semistrictly quasiconvex
Proposition Let f : X → I R ∪ {+∞} be any function, with domain dom f . Then f is quasiconvex ⇐ ⇒ Sa
f (x) is convex , ∀ x ∈ dom f .
Didier Aussel Univ. de Perpignan, France MLFG in the setting of quasiconvex optimization
Quasiconvex optimization Now the case of GNEP...
Adjusted normal operator Adjusted sublevel set: For any x ∈ dom f , we define Sa
f (x) = Sf (x) ∩ B(S< f (x), ρx)
where ρx = dist(x, S<
f (x)), if S< f (x) = ∅.
Ajusted normal operator: Na
f (x) = {x∗ ∈ X ∗ : x∗, y − x ≤ 0,
∀ y ∈ Sa
f (x)}
Didier Aussel Univ. de Perpignan, France MLFG in the setting of quasiconvex optimization
Quasiconvex optimization Now the case of GNEP...
Example
x
Didier Aussel Univ. de Perpignan, France MLFG in the setting of quasiconvex optimization
Quasiconvex optimization Now the case of GNEP...
Example
x
B(S<
f (x), ρx)
Sa
f (x) = Sf (x) ∩ B(S< f (x), ρx)
Didier Aussel Univ. de Perpignan, France MLFG in the setting of quasiconvex optimization
Quasiconvex optimization Now the case of GNEP...
Example
Sa
f (x) = Sf (x) ∩ B(S< f (x), ρx)
Na
f (x) = {x∗ ∈ X ∗ : x∗, y − x ≤ 0,
∀ y ∈ Sa
f (x)}
Didier Aussel Univ. de Perpignan, France MLFG in the setting of quasiconvex optimization
Quasiconvex optimization Now the case of GNEP...
An exercice......... Let us draw the normal operator value Na
f (x, y) at the points
(x, y) = (0.5, 0.5), (x, y) = (0, 1), (x, y) = (1, 0), (x, y) = (1, 2), (x, y) = (1.5, 0) and (x, y) = (0.5, 2).
Didier Aussel Univ. de Perpignan, France MLFG in the setting of quasiconvex optimization
Quasiconvex optimization Now the case of GNEP...
An exercice......... Let us draw the normal operator value Na
f (x, y) at the points
(x, y) = (0.5, 0.5), (x, y) = (0, 1), (x, y) = (1, 0), (x, y) = (1, 2), (x, y) = (1.5, 0) and (x, y) = (0.5, 2). Operator Na
f provide information at any point!!!
Didier Aussel Univ. de Perpignan, France MLFG in the setting of quasiconvex optimization
Quasiconvex optimization Now the case of GNEP...
Basic properties of Na
f
Nonemptyness: Proposition Let f : X → I R ∪ {+∞} be lsc. Assume that rad. continuous on dom f
f is quasiconvex ⇔ Na
f (x) \ {0} = ∅,
∀ x ∈ dom f \ arg min f . Quasimonotonicity: The normal operator Na
f is always quasimonotone
Didier Aussel Univ. de Perpignan, France MLFG in the setting of quasiconvex optimization
Quasiconvex optimization Now the case of GNEP...
Upper sign-continuity
x, y ∈ K, one have : ∀ t ∈ ]0, 1[, inf
x∗∈T(xt)x∗, y − x ≥ 0
= ⇒ sup
x∗∈T(x)
x∗, y − x ≥ 0 where xt = (1 − t)x + ty. upper semi-continuous ⇓ upper hemicontinuous ⇓ upper sign-continuous
Didier Aussel Univ. de Perpignan, France MLFG in the setting of quasiconvex optimization
Quasiconvex optimization Now the case of GNEP...
locally upper sign continuity Definition Let T : K → 2X ∗ be a set-valued map. T is called locally upper sign-continuous on K if, for any x ∈ K there exist a neigh. Vx of x and a upper sign-continuous set-valued map Φx(·) : Vx → 2X ∗ with nonempty convex w ∗-compact values such that Φx(y) ⊆ T(y) \ {0}, ∀ y ∈ Vx
Didier Aussel Univ. de Perpignan, France MLFG in the setting of quasiconvex optimization
Quasiconvex optimization Now the case of GNEP...
locally upper sign continuity Definition Let T : K → 2X ∗ be a set-valued map. T is called locally upper sign-continuous on K if, for any x ∈ K there exist a neigh. Vx of x and a upper sign-continuous set-valued map Φx(·) : Vx → 2X ∗ with nonempty convex w ∗-compact values such that Φx(y) ⊆ T(y) \ {0}, ∀ y ∈ Vx Continuity of normal operator Proposition Let f be lsc quasiconvex function such that int(Sλ) = ∅, ∀ λ > inf f . Then Na
f is locally upper sign-continuous on dom f \ arg min f .
Didier Aussel Univ. de Perpignan, France MLFG in the setting of quasiconvex optimization
Quasiconvex optimization Now the case of GNEP...
Didier Aussel Univ. de Perpignan, France MLFG in the setting of quasiconvex optimization
Quasiconvex optimization Now the case of GNEP...
Quasiconvex programming Let f : X → I R ∪ {+∞} and K ⊆ dom f be a convex subset. (P) find ¯ x ∈ K : f (¯ x) = inf
x∈K f (x)
Didier Aussel Univ. de Perpignan, France MLFG in the setting of quasiconvex optimization
Quasiconvex optimization Now the case of GNEP...
Quasiconvex programming Let f : X → I R ∪ {+∞} and K ⊆ dom f be a convex subset. (P) find ¯ x ∈ K : f (¯ x) = inf
x∈K f (x)
Perfect case: f convex f : X → I R ∪ {+∞} a proper convex function K a nonempty convex subset of X, ¯ x ∈ K + C.Q. Then f (¯ x) = inf
x∈K f (x)
⇐ ⇒ ¯ x ∈ Sstr(∂f , K)
Didier Aussel Univ. de Perpignan, France MLFG in the setting of quasiconvex optimization
Quasiconvex optimization Now the case of GNEP...
Quasiconvex programming Let f : X → I R ∪ {+∞} and K ⊆ dom f be a convex subset. (P) find ¯ x ∈ K : f (¯ x) = inf
x∈K f (x)
Perfect case: f convex f : X → I R ∪ {+∞} a proper convex function K a nonempty convex subset of X, ¯ x ∈ K + C.Q. Then f (¯ x) = inf
x∈K f (x)
⇐ ⇒ ¯ x ∈ Sstr(∂f , K) What about f quasiconvex case? ¯ x ∈ Sstr(∂f (¯ x), K) = ⇒ ¯ x ∈ arg min
K f
❍ ❍ ❍ ✟ ✟ ✟
Didier Aussel Univ. de Perpignan, France MLFG in the setting of quasiconvex optimization
Quasiconvex optimization Now the case of GNEP...
Sufficient optimality condition Theorem f : X → I R ∪ {+∞} quasiconvex, radially cont. on dom f C ⊆ X such that conv(C) ⊂ dom f . Suppose that C ⊂ int(dom f ) or AffC = X. Then ¯ x ∈ S(Na
f \ {0}, C)
= ⇒ ∀ x ∈ C, f (¯ x) ≤ f (x).
where ¯ x ∈ S(Na
f \ {0}, K) means that there exists ¯
x∗ ∈ Na
f (¯
x) \ {0} such that ¯ x∗, c − x ≥ 0, ∀ c ∈ C.
Didier Aussel Univ. de Perpignan, France MLFG in the setting of quasiconvex optimization
Quasiconvex optimization Now the case of GNEP...
Necessary and Sufficient conditions Proposition Let C be a closed convex subset of X, ¯ x ∈ C and f : X → I R be continuous semistrictly quasiconvex such that int(Sa
f (¯
x)) = ∅ and f (¯ x) > infX f . Then the following assertions are equivalent: i) f (¯ x) = minC f ii) ¯ x ∈ Sstr(Na
f \ {0}, C)
iii) 0 ∈ Na
f (¯
x) \ {0} + NK(C, ¯ x).
Didier Aussel Univ. de Perpignan, France MLFG in the setting of quasiconvex optimization
Quasiconvex optimization Now the case of GNEP...
GNEP reformulation in quasiconvex case
To simplify the notations, we will denote, for any i and any x ∈ Rn, by Si(x) and Ai(x−i) the subsets of Rni Si(x) = Sa
θi (·,x−i )(xi)
and Ai(x−i) = arg min
Rni θi(·, x−i).
In order to construct the variational inequality problem we define the following set-valued map Na
θ : Rn → 2Rn which is described,
for any x = (x1, . . . , xp) ∈ Rn1 × . . . × Rnp, by Na
θ(x) = F1(x) × . . . × Fp(x),
where Fi(x) =
if xi ∈ Ai(x−i) co(Na
θi (xi) ∩ Si(0, 1))
The set-valued map Na
θ has nonempty convex compact values. Didier Aussel Univ. de Perpignan, France MLFG in the setting of quasiconvex optimization
Quasiconvex optimization Now the case of GNEP...
Sufficient condition
In the following we assume that X is a given nonempty subset X of Rn, such that for any i, the set Xi (x−i ) is given as Xi (x−i ) = {xi ∈ Rni : (xi , x−i ) ∈ X}.
Theorem Let us assume that, for any i, the function θi is continuous and quasiconvex with respect to the i-th variable. Then every solution of S(Na
θ, X) is a solution of the GNEP.
Didier Aussel Univ. de Perpignan, France MLFG in the setting of quasiconvex optimization
Quasiconvex optimization Now the case of GNEP...
Sufficient condition
In the following we assume that X is a given nonempty subset X of Rn, such that for any i, the set Xi (x−i ) is given as Xi (x−i ) = {xi ∈ Rni : (xi , x−i ) ∈ X}.
Theorem Let us assume that, for any i, the function θi is continuous and quasiconvex with respect to the i-th variable. Then every solution of S(Na
θ, X) is a solution of the GNEP.
Note that the link between GNEP and variational inequality is valid even if the constraint set X is neither convex nor compact.
Didier Aussel Univ. de Perpignan, France MLFG in the setting of quasiconvex optimization
Quasiconvex optimization Now the case of GNEP...
Lemma Let i ∈ {1, . . . , p}. If the function θi is continuous quasiconvex with respect to the i-th variable, then, 0 ∈ Fi(¯ x) ⇐ ⇒ ¯ xi ∈ Ai(¯ x−i).
x such that ¯ xi ∈ Ai (¯ x−i ). Since θi (·, ¯ x−i ) is continuous at ¯ xi , the interior of Si (¯ x) is nonempty. Let us denote by Ki the convex cone Ki = Na
θi (¯
xi ) = (Si (¯ x) − ¯ xi )◦. By quasiconvexity of θi , Ki is not reduced to {0}. Let us first observe that, since Si (¯ x) has a nonempty interior, Ki is a pointed cone, that is Ki ∩ (−Ki ) = {0}. Now let us suppose that 0 ∈ Fi (¯ x). By Caratheodory theorem, there exist vectors vi ∈ [Ki ∩ Si (0, 1)], i = 1, . . . , n + 1 and scalars λi ≥ 0, i = 1, . . . , n + 1 with
n+1
λi = 1 and 0 =
n+1
λi vi . Didier Aussel Univ. de Perpignan, France MLFG in the setting of quasiconvex optimization
Quasiconvex optimization Now the case of GNEP... Since there exists at least one r ∈ {1, . . . , n + 1} such that λr > 0 we have vr = −
n+1
λi λr vi which clearly shows that vr is an element of the convex cone −Ki . But vr ∈ Si (0, 1) and thus vr = 0. This contradicts the fact that Ki is pointed and the proof is complete. Didier Aussel Univ. de Perpignan, France MLFG in the setting of quasiconvex optimization
Quasiconvex optimization Now the case of GNEP...
Proof of necessary condition
x to be a solution of S(Na
θ, X). There exists v ∈ Na θ(¯
x) such that v, y − ¯ x ≥ 0, ∀ y ∈ X. (∗) Let i ∈ {1, . . . , p}. If ¯ xi ∈ Ai (¯ x−i ) then obviously ¯ xi ∈ Soli (¯ x−i ). Otherwise vi ∈ Fi (¯ x) = co(Na
θi (¯
xi ) ∩ Si (0, 1)). Thus, according to Lemma 2, there exist λ > 0 and ui ∈ Na
θi (¯
xi ) \ {0} satisfying vi = λui . Now for any xi ∈ Xi (¯ x−i ), consider y =
x1, . . . , ¯ xi−1, xi , ¯ xi+1, . . . , ¯ xp . From (∗) one immediately obtains that ui , xi − ¯ xi ≥ 0. Since xi is an arbitrary element of Xi (¯ x−i ), we have that ¯ xi is a solution of S(Na
θi \ {0}, Xi (¯
x−i )) and therefore, according to Prop. 4, ¯ xi ∈ Soli (¯ x−i ) Since i was arbitrarily chosen we conclude that ¯ x solves the GNEP. Didier Aussel Univ. de Perpignan, France MLFG in the setting of quasiconvex optimization
Quasiconvex optimization Now the case of GNEP...
Necessary and sufficient condition Theorem Let us suppose that, for any i, the loss function θi is continuous and semistrictly quasiconvex with respect to the i-th variable. Further assume that the set X is a nonempty convex subset of RN. Then any solution of the variational inequality S(Na
θ, X) is a solution
and any solution of the GNEP is a solution of the quasi-variational inequality QVI(Na
θ, X)
where X stands for the set-valued map defined on R2 by X(x) =
p
Xi(x−i) .
D.A. & J. Dutta, Oper. Res. Letters, 2008. Didier Aussel Univ. de Perpignan, France MLFG in the setting of quasiconvex optimization