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❛♥❞ ❆✉❣♠❡♥t❡❞ ❉✉❛❧ ❉②♥❛♠✐❝ Pr♦❣r❛♠♠✐♥❣ ❙t♦❝❤❛st✐❝ ▲✐♣s❝❤✐t③ ❉②♥❛♠✐❝ Pr♦❣r❛♠♠✐♥❣ ❇❡r♥❛r❞♦ ❋r❡✐t❛s P❛✉❧♦ ❞❛ ❈♦st❛ ✭❯❋❘❏✮ ❏♦✐♥t ✇♦r❦ ✇✐t❤ ❙❤❛❜❜✐r ❆❤♠❡❞ ✭●❛❚❡❝❤✮ ❛♥❞ ❋✐❧✐♣❡ ●♦✉❧❛rt ❈❛❜r❛❧ ✭❖◆❙✮ ❆✉❣✉st ✶st✱ ✷✵✶✾ ✕ ■❈❙P✱ ❚r♦♥❞❤❡✐♠ ❙✳ ❆❤♠❡❞ ❋✳ ❈❛❜r❛❧ ❇✳ ❋P❈ ❙▲❉P ■❈❙P✬✶✾ ✶ ✴ ✶✽

❙t♦❝❤❛st✐❝ ▲✐♣s❝❤✐t③ ❉②♥❛♠✐❝ Pr♦❣r❛♠♠✐♥❣ ❛♥❞ ❆✉❣♠❡♥t❡❞ ❉✉❛❧ ❉②♥❛♠✐❝ Pr♦❣r❛♠♠✐♥❣ ❇❡r♥❛r❞♦ ❋r❡✐t❛s P❛✉❧♦ ❞❛ ❈♦st❛ ✭❯❋❘❏✮ ❏♦✐♥t ✇♦r❦ ✇✐t❤ ❙❤❛❜❜✐r ❆❤♠❡❞ ✭●❛❚❡❝❤✮ ❛♥❞ ❋✐❧✐♣❡ ●♦✉❧❛rt ❈❛❜r❛❧ ✭❖◆❙✮ ❆✉❣✉st ✶st✱ ✷✵✶✾ ✕ ■❈❙P✱ ❚r♦♥❞❤❡✐♠ ❙✳ ❆❤♠❡❞ ❋✳ ❈❛❜r❛❧ ❇✳ ❋P❈ ❙▲❉P ■❈❙P✬✶✾ ✶ ✴ ✶✽

■♥ ▼❡♠♦r✐❛♠✿ ❙❤❛❜❜✐r ❆❤♠❡❞ ❙✳ ❆❤♠❡❞ ❋✳ ❈❛❜r❛❧ ❇✳ ❋P❈ ❙▲❉P ■❈❙P✬✶✾ ✷ ✴ ✶✽

■❢ t❤❡ s❡t ✐s ♥♦t ❝♦♥✈❡①✱ t❤❡ ❢✉♥❝t✐♦♥ ♠✐❣❤t ❛❧s♦ ❜❡ ♥♦♥✲❝♦♥✈❡① ❀ ▼■❉❆❙ ❬P❤✐❧♣♦tt ❡t ❛❧✳ ✕ ✷✵✶✻❪ ❛ss✉♠❡s ♠♦♥♦t♦♥✐❝✐t② ♦❢ t❤❡ ❝♦st✲t♦✲❣♦ ❢✉♥❝t✐♦♥❀ ❙❉❉✐P ❬❩♦✉ ❡t ❛❧✳ ✕ ✷✵✶✽❪ ❧✐❢ts t❤❡ st❛t❡ ✈❛r✐❛❜❧❡s t♦ ❛ ❜✐♥❛r② ❝✉❜❡✳ ❚❤✐s t❛❧❦✿ ❛ ♥❡✇ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ✈❛❧✉❡ ❢✉♥❝t✐♦♥s✱ ✉s✐♥❣ ♥♦♥✲❝♦♥✈❡① ❝✉ts ✳ ❱❛❧✉❡ ❢✉♥❝t✐♦♥s ♦❢ ♦♣t✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠s ❆ ✈❛❧✉❡ ❢✉♥❝t✐♦♥ ✭❢♦r st❛❣❡ t ✮✿ c J Q t p x t ´ 1 q “ min t y t ` Q t ` 1 p x t q s✳t✳ p x t ´ 1 , x t , y t q P X t ❙✳ ❆❤♠❡❞ ❋✳ ❈❛❜r❛❧ ❇✳ ❋P❈ ❙▲❉P ■❈❙P✬✶✾ ✸ ✴ ✶✽

▼■❉❆❙ ❬P❤✐❧♣♦tt ❡t ❛❧✳ ✕ ✷✵✶✻❪ ❛ss✉♠❡s ♠♦♥♦t♦♥✐❝✐t② ♦❢ t❤❡ ❝♦st✲t♦✲❣♦ ❢✉♥❝t✐♦♥❀ ❙❉❉✐P ❬❩♦✉ ❡t ❛❧✳ ✕ ✷✵✶✽❪ ❧✐❢ts t❤❡ st❛t❡ ✈❛r✐❛❜❧❡s t♦ ❛ ❜✐♥❛r② ❝✉❜❡✳ ❚❤✐s t❛❧❦✿ ❛ ♥❡✇ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ✈❛❧✉❡ ❢✉♥❝t✐♦♥s✱ ✉s✐♥❣ ♥♦♥✲❝♦♥✈❡① ❝✉ts ✳ ❱❛❧✉❡ ❢✉♥❝t✐♦♥s ♦❢ ♦♣t✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠s ❆ ✈❛❧✉❡ ❢✉♥❝t✐♦♥ ✭❢♦r st❛❣❡ t ✮✿ c J Q t p x t ´ 1 q “ min t y t ` Q t ` 1 p x t q s✳t✳ p x t ´ 1 , x t , y t q P X t ■❢ t❤❡ s❡t X t ✐s ♥♦t ❝♦♥✈❡①✱ t❤❡ ❢✉♥❝t✐♦♥ Q t ♠✐❣❤t ❛❧s♦ ❜❡ ♥♦♥✲❝♦♥✈❡① ❀ ❙✳ ❆❤♠❡❞ ❋✳ ❈❛❜r❛❧ ❇✳ ❋P❈ ❙▲❉P ■❈❙P✬✶✾ ✸ ✴ ✶✽

❚❤✐s t❛❧❦✿ ❛ ♥❡✇ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ✈❛❧✉❡ ❢✉♥❝t✐♦♥s✱ ✉s✐♥❣ ♥♦♥✲❝♦♥✈❡① ❝✉ts ✳ ❱❛❧✉❡ ❢✉♥❝t✐♦♥s ♦❢ ♦♣t✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠s ❆ ✈❛❧✉❡ ❢✉♥❝t✐♦♥ ✭❢♦r st❛❣❡ t ✮✿ c J Q t p x t ´ 1 q “ min t y t ` Q t ` 1 p x t q s✳t✳ p x t ´ 1 , x t , y t q P X t ■❢ t❤❡ s❡t X t ✐s ♥♦t ❝♦♥✈❡①✱ t❤❡ ❢✉♥❝t✐♦♥ Q t ♠✐❣❤t ❛❧s♦ ❜❡ ♥♦♥✲❝♦♥✈❡① ❀ ▼■❉❆❙ ❬P❤✐❧♣♦tt ❡t ❛❧✳ ✕ ✷✵✶✻❪ ❛ss✉♠❡s ♠♦♥♦t♦♥✐❝✐t② ♦❢ t❤❡ ❝♦st✲t♦✲❣♦ ❢✉♥❝t✐♦♥❀ ❙❉❉✐P ❬❩♦✉ ❡t ❛❧✳ ✕ ✷✵✶✽❪ ❧✐❢ts t❤❡ st❛t❡ ✈❛r✐❛❜❧❡s t♦ ❛ ❜✐♥❛r② ❝✉❜❡✳ ❙✳ ❆❤♠❡❞ ❋✳ ❈❛❜r❛❧ ❇✳ ❋P❈ ❙▲❉P ■❈❙P✬✶✾ ✸ ✴ ✶✽

❱❛❧✉❡ ❢✉♥❝t✐♦♥s ♦❢ ♦♣t✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠s ❆ ✈❛❧✉❡ ❢✉♥❝t✐♦♥ ✭❢♦r st❛❣❡ t ✮✿ c J Q t p x t ´ 1 q “ min t y t ` Q t ` 1 p x t q s✳t✳ p x t ´ 1 , x t , y t q P X t ■❢ t❤❡ s❡t X t ✐s ♥♦t ❝♦♥✈❡①✱ t❤❡ ❢✉♥❝t✐♦♥ Q t ♠✐❣❤t ❛❧s♦ ❜❡ ♥♦♥✲❝♦♥✈❡① ❀ ▼■❉❆❙ ❬P❤✐❧♣♦tt ❡t ❛❧✳ ✕ ✷✵✶✻❪ ❛ss✉♠❡s ♠♦♥♦t♦♥✐❝✐t② ♦❢ t❤❡ ❝♦st✲t♦✲❣♦ ❢✉♥❝t✐♦♥❀ ❙❉❉✐P ❬❩♦✉ ❡t ❛❧✳ ✕ ✷✵✶✽❪ ❧✐❢ts t❤❡ st❛t❡ ✈❛r✐❛❜❧❡s t♦ ❛ ❜✐♥❛r② ❝✉❜❡✳ ❚❤✐s t❛❧❦✿ ❛ ♥❡✇ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ✈❛❧✉❡ ❢✉♥❝t✐♦♥s✱ ✉s✐♥❣ ♥♦♥✲❝♦♥✈❡① ❝✉ts ✳ ❙✳ ❆❤♠❡❞ ❋✳ ❈❛❜r❛❧ ❇✳ ❋P❈ ❙▲❉P ■❈❙P✬✶✾ ✸ ✴ ✶✽

▲✐♥❡❛r ❝✉ts ❲❡ ✇✐❧❧ ✉s❡ t❤❡ ✏❲✑ ❢✉♥❝t✐♦♥ ❜❡❧♦✇ ❛s ❛ r✉♥♥✐♥❣ ❡①❛♠♣❧❡✳ 2.5 Opt. Value 2.0 1.5 1.0 0.5 0.0 0.5 3 2 1 0 1 2 3 ❙✳ ❆❤♠❡❞ ❋✳ ❈❛❜r❛❧ ❇✳ ❋P❈ ❙▲❉P ■❈❙P✬✶✾ ✹ ✴ ✶✽

❚❤✐s ❝♦rr❡s♣♦♥❞s t♦ ❛ ❙tr❡♥❣t❤❡♥❡❞ ❇❡♥❞❡rs ❝✉t✳ ❱❛r②✐♥❣ ❣✐✈❡s ♥❡✇ ❝✉ts✳ ▲✐♥❡❛r ❝✉ts ❲❡ ✇✐❧❧ ✉s❡ t❤❡ ✏❲✑ ❢✉♥❝t✐♦♥ ❜❡❧♦✇ ❛s ❛ r✉♥♥✐♥❣ ❡①❛♠♣❧❡✳ 2.5 1 Linear cut Opt. Value 2.0 1.5 1.0 0.5 0.0 0.5 3 2 1 0 1 2 3 ▲✐♥❡❛r ❝✉ts ❝❛♥ ❜❡ ❝❛❧❝✉❧❛t❡❞ ❜② ✜①✐♥❣ λ ❛♥❞ ❞❡t❡r♠✐♥✐♥❣ t❤❡ ✐♥t❡r❝❡♣t α s✉❝❤ t❤❛t α ´ λ J x t ´ 1 ď Q t p x t ´ 1 q ❢♦r ❛❧❧ x t ´ 1 . ❙✳ ❆❤♠❡❞ ❋✳ ❈❛❜r❛❧ ❇✳ ❋P❈ ❙▲❉P ■❈❙P✬✶✾ ✹ ✴ ✶✽

❚❤✐s ❝♦rr❡s♣♦♥❞s t♦ ❛ ❙tr❡♥❣t❤❡♥❡❞ ❇❡♥❞❡rs ❝✉t✳ ❱❛r②✐♥❣ ❣✐✈❡s ♥❡✇ ❝✉ts✳ ▲✐♥❡❛r ❝✉ts ❲❡ ✇✐❧❧ ✉s❡ t❤❡ ✏❲✑ ❢✉♥❝t✐♦♥ ❜❡❧♦✇ ❛s ❛ r✉♥♥✐♥❣ ❡①❛♠♣❧❡✳ 2.5 1 Linear cut Opt. Value 2.0 1.5 1.0 0.5 0.0 0.5 3 2 1 0 1 2 3 ▲✐♥❡❛r ❝✉ts ❝❛♥ ❜❡ ❝❛❧❝✉❧❛t❡❞ ❜② ✜①✐♥❣ λ ❛♥❞ ❞❡t❡r♠✐♥✐♥❣ t❤❡ ✐♥t❡r❝❡♣t α s✉❝❤ t❤❛t α ď Q t p x t ´ 1 q ` λ J x t ´ 1 ❢♦r ❛❧❧ x t ´ 1 . ❙✳ ❆❤♠❡❞ ❋✳ ❈❛❜r❛❧ ❇✳ ❋P❈ ❙▲❉P ■❈❙P✬✶✾ ✹ ✴ ✶✽

❱❛r②✐♥❣ ❣✐✈❡s ♥❡✇ ❝✉ts✳ ▲✐♥❡❛r ❝✉ts ❲❡ ✇✐❧❧ ✉s❡ t❤❡ ✏❲✑ ❢✉♥❝t✐♦♥ ❜❡❧♦✇ ❛s ❛ r✉♥♥✐♥❣ ❡①❛♠♣❧❡✳ 2.5 1 Linear cut Opt. Value 2.0 1.5 1.0 0.5 0.0 0.5 3 2 1 0 1 2 3 ▲✐♥❡❛r ❝✉ts ❝❛♥ ❜❡ ❝❛❧❝✉❧❛t❡❞ ❜② ✜①✐♥❣ λ ❛♥❞ ❞❡t❡r♠✐♥✐♥❣ t❤❡ ✐♥t❡r❝❡♣t α s✉❝❤ t❤❛t x t ´ 1 Q t p x t ´ 1 q ` λ J x t ´ 1 . α : “ min ❚❤✐s ❝♦rr❡s♣♦♥❞s t♦ ❛ ❙tr❡♥❣t❤❡♥❡❞ ❇❡♥❞❡rs ❝✉t✳ ❙✳ ❆❤♠❡❞ ❋✳ ❈❛❜r❛❧ ❇✳ ❋P❈ ❙▲❉P ■❈❙P✬✶✾ ✹ ✴ ✶✽

▲✐♥❡❛r ❝✉ts ❲❡ ✇✐❧❧ ✉s❡ t❤❡ ✏❲✑ ❢✉♥❝t✐♦♥ ❜❡❧♦✇ ❛s ❛ r✉♥♥✐♥❣ ❡①❛♠♣❧❡✳ 2.5 2 Linear cuts Opt. Value 2.0 1.5 1.0 0.5 0.0 0.5 3 2 1 0 1 2 3 ▲✐♥❡❛r ❝✉ts ❝❛♥ ❜❡ ❝❛❧❝✉❧❛t❡❞ ❜② ✜①✐♥❣ λ ❛♥❞ ❞❡t❡r♠✐♥✐♥❣ t❤❡ ✐♥t❡r❝❡♣t α s✉❝❤ t❤❛t x t ´ 1 Q t p x t ´ 1 q ` λ J x t ´ 1 . α : “ min ❚❤✐s ❝♦rr❡s♣♦♥❞s t♦ ❛ ❙tr❡♥❣t❤❡♥❡❞ ❇❡♥❞❡rs ❝✉t✳ ❱❛r②✐♥❣ λ ❣✐✈❡s ♥❡✇ ❝✉ts✳ ❙✳ ❆❤♠❡❞ ❋✳ ❈❛❜r❛❧ ❇✳ ❋P❈ ❙▲❉P ■❈❙P✬✶✾ ✹ ✴ ✶✽