The consumer theory: fundamentals and applications

Munich Personal RePEc Archive

The consumer theory: fundamentals and applications

Galvis Ciro, Juan Camilo and Henao Atehortúa, Edison Fred

Universidad Pontificia Bolivariana, Universidad Nacional de Colombia

10 January 2018

Online at https://mpra.ub.uni-muenchen.de/84507/

MPRA Paper No. 84507, posted 13 Feb 2018 09:21 UTC

La teoría del consumidor:

fundamentales y aplicaciones

Enero de 2018

Documento de Trabajo

Autor(es):

Juan Camilo Galvis Ciro

, Edison Fred Henao Atehortúa

1 PhD. Facultad de Economía. Universidad Pontificia Bolivariana. E-mail: jcgalvisciro@gmail.com

2 MSc. Facultad de Ciencias Humanas y Económicas. Universidad Nacional. E-mail: efhenao@unal.edu.co

❮♥❞✐❝❡ ❣❡♥❡'❛❧

■♥"#♦❞✉❝❝✐)♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸

▼♦❞❡❧♦ ❜'(✐❝♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳✺

❊❝♦♥♦♠/❛( ❆22♦✇✲❉❡❜2❡✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻

❈❛♣."✉❧♦ ✶

❈♦♥1✉♠✐❞♦#❡1 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼

✶✳✶ ■♥:2♦❞✉❝❝✐;♥✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳✼

✶✳✷ ❊❧ ❝♦♥❥✉♥:♦ ❞❡ ❝♦♥(✉♠♦ ♦ ❞❡ ❡❧❡❝❝✐;♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽

✶✳✸ ❈❛2❛❝:❡2/(:✐❝❛( ❞❡❧ ♠❡2❝❛❞♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵

✶✳✹ ❈♦♥❥✉♥:♦ ❞❡ 2❡(:2✐❝❝✐;♥ ♣2❡(✉♣✉❡(:❛2✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶

✶✳✺ ❊❧ ❝♦♥(✉♠✐❞♦2 ② ❧♦( ❝2✐:❡2✐♦( ❞❡ ❡❧❡❝❝✐;♥✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳✶✷

✶✳✻ ❘❡❧❛❝✐;♥ ❞❡ ♣2❡❢❡2❡♥❝✐❛(✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳✶✸

✶✳✻✳✶✳ ❆①✐♦♠❛( ❞❡ ♦2❞❡♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸

✶✳✻✳✷✳ ❆①✐♦♠❛ ❞❡ 2❛❝✐♦♥❛❧✐❞❛❞✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳✶✺

✶✳✻✳✸✳ ❆①✐♦♠❛( ❛♥❛❧/:✐❝♦( ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺

✶✳✻✳✶✳ ❆①✐♦♠❛( ❞❡ ♥♦ (❛❝✐❡❞❛❞ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼

✶✳✼ ❘❡♣2❡(❡♥:❛❝✐;♥ ❞❡ ❧❛( ♣2❡❢❡2❡♥❝✐❛(✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳✶✽

✶✳✽ I2♦♣✐❡❞❛❞❡( ❞❡ ❧❛ ❢✉♥❝✐;♥ ❞❡ ✉:✐❧✐❞❛❞ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶

✶✳✾ ❊❧ ♣2♦❜❧❡♠❛ ❞❡❧ ❝♦♥(✉♠✐❞♦2✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳✷✹

✶✳✶✵ ▲❛ ❢✉♥❝✐;♥ ❞❡ ✉:✐❧✐❞❛❞ ✐♥❞✐2❡❝:❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸

✶✳✶✶ ❊♥❢♦L✉❡ ❞❡(❞❡ ❡❧ ❛♥'❧✐(✐( ❞✐❢❡2❡♥❝✐❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼

✶✳✶✷ ❊❧ ❡❢❡❝:♦ (✉(:✐:✉❝✐;♥ ② ❡❧ ❛♣♦2:❡ ❞❡ ❙❧✉:(❦②✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳✹✹

✶✳✶✸ ❊❧ ♣2♦❜❧❡♠❛ ❛✉①✐❧✐❛2 ❞❡❧ ❝♦♥(✉♠✐❞♦2 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶

✶✳✶✹ ▲❛ ❢✉♥❝✐;♥ ❞❡ ❣❛(:♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✾

✶✳✶✺ ■♠♣♦2:❛♥❝✐❛ ❞❡ ❧♦( ♣2❡❝✐♦( ♠❛②♦2❡( ❛ ❝❡2♦ ② ❧❛ ♥♦ (❛❝✐❡❞❛❞ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✺

✶✳✶✻ ❊♥❢♦L✉❡ ❞❡(❞❡ ❡❧ ❛♥'❧✐(✐( ❞✐❢❡2❡♥❝✐❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✽

✶

✷

✶✳✶✼ ▲❛ ❡'✉✐✈❛❧❡♥❝✐❛ ❞❡ ❧♦0 ❡♥❢♦'✉❡0 ❞❡ ❍✐❝❦0 ② ❙❧✉60❦② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✸

✶✳✶✾ ❉❡ ❧♦ ♦❜0❡;✈❛❜❧❡ ❛ ❧♦ ♥♦ ♦❜0❡;✈❛❜❧❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✺

❈❛♣#$✉❧♦ ✷

❊❧ ♣*♦❜❧❡♠❛ ❞❡ ❧❛ ❛❣*❡❣❛❝✐2♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳✽✼

✷✳✶ ▲❛ ❢✉♥❝✐=♥ ❞❡ ❞❡♠❛♥❞❛ ❛❣;❡❣❛❞❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✼

✷✳✷ ▲❛ ❞❡♠❛♥❞❛ ❛❣;❡❣❛❞❛ ❞❡ ❛❣❡♥6❡ ;❡♣;❡0❡♥6❛6✐✈♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✾

✷✳✸ ▲❛ ❛❣;❡❣❛❝✐=♥ ② ❧❛ ♣;❡❢❡;❡♥❝✐❛ ;❡✈❡❧❛❞❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✶

❈❛♣#$✉❧♦ ✸

❉❡♠❛♥❞❛✱ ❜✐❡♥❡9 ② ❡❧❛9$✐❝✐❞❛❞❡9 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✹

✸✳✶ ❇✐❡♥❡0 ❝♦♠♣❧❡♠❡♥6❛;✐♦0 ② ❜✐❡♥❡0 0✉06✐6✉6♦0 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✽

✸✳✷ ❊❧❛06✐❝✐❞❛❞ ♣;❡❝✐♦ ❡ ✐♥❣;❡0♦ ❞❡ ❧❛ ❞❡♠❛♥❞❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✷

✸✳✸ ❊❧❛06✐❝✐❞❛❞❡0 ② ❝♦♥❞✐❝✐♦♥❡0 ❞❡ ❛❣;❡❣❛❝✐=♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✹

✸✳✹ ❈✉;✈❛0 ❞❡ ✐♥❣;❡0♦ ✲ ❝♦♥0✉♠♦ ② ❝✉;✈❛ ❞❡ ❊♥❣❡❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✻

❈❛♣#$✉❧♦ ✹

❙✐9$❡♠❛9 ❞❡ ❞❡♠❛♥❞❛✿ ❛♣❧✐❝❛❝✐♦♥❡9 ❞❡ ❧❛ $❡♦*#❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✷

✹✳✶ ❊❧❛06✐❝✐❞❛❞❡0 ② ❡06✐♠❛❝✐♦♥❡0 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✷

✹✳✷ ❊❧ 0✐06❡♠❛ ❲♦;❦✐♥❣ ② ▲❡0❡; ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✸

✹✳✸ ❊❧ 0✐06❡♠❛ ❧✐♥❡❛❧ ❞❡ ❣❛06♦0 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✾

✹✳✹ ❙✐06❡♠❛ ❝✉❛❞;J6✐❝♦ ❞❡ ❣❛06♦0 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷✾

✹✳✺ ❙✐06❡♠❛ ❝❛0✐ ✐❞❡❛❧ ❞❡ ❞❡♠❛♥❞❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸✾

✹✳✻ ❙✐06❡♠❛0 ❞❡ ❞❡♠❛♥❞❛ ❝♦♥ ❞❡♠♦❣;❛❢K❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹✼

✹✳✼ ❙✐06❡♠❛ ❲♦;❦✐♥❣ ② ▲❡0❡; ❝♦♥ ❞❡♠♦❣;❛❢K❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺✷

✹✳✽ ❊❝♦♥♦♠K❛0 ❛ ❡0❝❛❧❛ ❡♥ ❝♦♥0✉♠♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺✻

❇✐❜❧✐♦❣*❛❢#❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻✵

◆♦$❛❝✐2♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻✹

■◆❚❘❖❉❯❈❈■(◆

❙✉♣♦♥❡& &❛❝✐♦♥❛❧✐❞❛❞ ♣❛&❛ ❝❛❞❛ ❛❣❡♥-❡ ❡❝♦♥.♠✐❝♦ ✭❝♦♥1✉♠✐❞♦&❡1 ② ♣&♦❞✉❝✲

-♦&❡1✮ ❡♥ ✉♥❛ ❡❝♦♥♦♠5❛ ❞❡1❝❡♥-&❛❧✐③❛❞❛✱ ❞♦♥❞❡ ❝❛❞❛ ❝✉❛❧ ♣❡&1✐❣✉❡ ✉♥ ♦❜❥❡-✐✈♦

❣✉✐❛❞♦ ♣♦& ❝&✐-❡&✐♦1 ✐♥❞✐✈✐❞✉❛❧✐1-❛1✱ ❤❛ 1✐❞♦ ❝❡♥-&♦ ❞❡ ❛♥<❧✐1✐1 ❡♥ ❧❛ -❡♦&5❛ ❡❝♦✲

♥.♠✐❝❛✳

❊♥ ❡❢❡❝-♦✱ ❡❧ ❤❡❝❤♦ ❞❡ @✉❡ ✉♥ 1✐1-❡♠❛ ❡❝♦♥.♠✐❝♦ ❞❛❞♦ ❝❛❞❛ ❛❣❡♥-❡ ❛❝-✉A 1✐❣✉✐❡♥❞♦ ♦❜❥❡-✐✈♦1 ❡❣♦51-❛1 ♣❧❛♥-❡❛ ❡❧ ♥❡❝❡1❛&✐♦ &❡1✉❧-❛❞♦ ❞❡❧ ❝❛♦1 ❝♦♠♦ ❝♦♥✲

1❡❝✉❡♥❝✐❛ ❞❡ -❛❧ ❝♦♥❞✉❝-❛✳ ❊1-❛ ✐♥@✉✐❡-✉❞ ❤❛❜5❛ 1✐❞♦ ❛♥❛❧✐③❛❞❛ ❞❡1❞❡ ❡❧ ♠✐1♠♦

❆❞❛♠ ❙♠✐-❤✱ @✉✐A♥ ♣♦1-✉❧. ❧❛ ✜❣✉&❛ ❞❡ ❧❛ ♠❛♥♦ ✐♥✈✐1✐❜❧❡ ❝♦♠♦ ❛@✉❡❧❧❛ @✉❡ ❣✉5❛

❧❛1 ❛❝❝✐♦♥❡1 ✐♥❞✐✈✐❞✉❛❧❡1 ❤❛❝✐❛ ❡❧ ❜✐❡♥ ❝♦♠D♥ ✭♥♦ ❝❛♦1✮ ❞❡ ✉♥❛ ♠❛♥❡&❛ @✉❡ ♥♦

❡1-❛❜❛ ❡♥ ❧❛ -♦♠❛ ❞❡ ❞❡❝✐1✐♦♥❡1 ❞❡ ❝❛❞❛ ❛❣❡♥-❡✳ ❊♥ ❡❢❡❝-♦✱ ❞✐❝❡ ❙♠✐-❤ ✭✶✼✼✻✮✿

▲✐❜&♦ ■❱ ✭❝❛♣✳ ■■✱ ✹✵✷✮✱

◆✐♥❣✉♥♦ &❡ ♣)♦♣♦♥❡✱ ♣♦) ❧♦ ❣❡♥❡)❛❧✱ ♣)♦♠♦✈❡) ❡❧ ✐♥/❡)0& ♣1❜❧✐❝♦✱ ♥✐ &❛❜❡ ❤❛&✲

/❛ 6✉0 ♣✉♥/♦ ❧♦ ♣)♦♠✉❡✈❡✳ ❈✉❛♥❞♦ ♣)❡✜❡)❡ ❧❛ ❛❝/✐✈✐❞❛❞ ❡❝♦♥;♠✐❝❛ ❞❡ &✉ ♣❛<& ❛

❧❛ ❡①/)❛♥❥❡)❛✱ 1♥✐❝❛♠❡♥/❡ ❝♦♥&✐❞❡)❛ &✉ &❡❣✉)✐❞❛❞✱ ② ❝✉❛♥❞♦ ❞✐)✐❣❡ ❧❛ ♣)✐♠❡)❛ ❞❡

/❛❧ ❢♦)♠❛ 6✉❡ &✉ ♣)♦❞✉❝/♦ )❡♣)❡&❡♥/❡ ❡❧ ♠❛②♦) ✈❛❧♦)✱ &;❧♦ ♣✐❡♥&❛ ❡♥ &✉ ❣❛♥❛♥❝✐❛

♣)♦♣✐❛❀ ♣❡)♦ ❡♥ 0&/❡ ❝♦♠♦ ❡♥ ♦/)♦& ♠✉❝❤♦& ❝❛&♦&✱ ❡& ❝♦♥❞✉❝✐❞♦ ♣♦) ✉♥❛ ♠❛♥♦

✐♥✈✐&✐❜❧❡ ❛ ♣)♦♠♦✈❡) ✉♥ ✜♥ 6✉❡ ♥♦ ❡♥/)❛❜❛ ❡♥ &✉& ✐♥/❡♥❝✐♦♥❡& ❬✳✳✳❪ ♣✉❡& ❛❧ ♣❡)✲

&❡❣✉✐) &✉ ♣)♦♣✐♦ ✐♥/❡)0&✱ ♣)♦♠✉❡✈❡ ❡❧ ❞❡ ❧❛ &♦❝✐❡❞❛❞ ❞❡ ✉♥❛ ♠❛♥❡)❛ ♠D& ❡❢❡❝/✐✈❛

6✉❡ &✐ ❡&/♦ ❡♥/)❛)❛ ❡♥ &✉& ❞❡&✐❣♥✐♦&✳

O❛&❛ ❙♠✐-❤✱ ❡&❛ ❡❧ ♠❡&❝❛❞♦ @✉✐❡♥ -❡♥5❛ ❧❛ ❝❛♣❛❝✐❞❛❞ ❞❡ ❝♦♦&❞✐♥❛& ❛❞❡❝✉❛❞❛✲

♠❡♥-❡ ❧❛ ❛❝-✐✈✐❞❛❞ ❡❝♦♥.♠✐❝❛ ② 1❡ &❡✜&✐. ❛ ❡1❛ ❝❛♣❛❝✐❞❛❞ ❜❛❥♦ ❧❛ ❞❡♥♦♠✐♥❛❝✐.♥

♣♦A-✐❝❛ ❞❡ ♠❛♥♦ ✐♥✈✐1✐❜❧❡ ✭❆&&♦✇ ② ❍❛❤♥✱ ✶✾✼✼✮✳ ◆♦ ♦❜1-❛♥-❡✱ ❧❛ ♣&❡❣✉♥-❛ ✐♥-❡✲

❧❡❝-✉❛❧ ❝♦♥ &❡1♣❡❝-♦ ❛ ❧❛ ♣♦1✐❜✐❧✐❞❛❞ ② ❧❛ ✈✐❛❜✐❧✐❞❛❞ ❞❡ ✉♥❛ 1♦❝✐❡❞❛❞ ♠❡&❝❛♥-✐❧✱

❡1 ❞❡❝✐&❀ ✉♥❛ 1♦❝✐❡❞❛❞ ❞♦♥❞❡ ❧❛1 ❛❝❝✐♦♥❡1 1❡ -♦♠❛♥ ❞❡ ♠❛♥❡&❛ ❞❡1❝❡♥-&❛❧✐③❛❞❛✱

-❛&❞❛&5❛ -✐❡♠♣♦ ♣❛&❛ ♣&✐♠❡&♦ ❡①♣&❡1❛&1❡ ❡♥ -A&♠✐♥♦1 ❞❡ ✉♥ ♠♦❞❡❧♦ ❡❝♦♥.♠✐❝♦

②✱ ❛D♥ ❤♦②✱ ♥♦ 1❡ ❤❛ &❡1✉❡❧-♦✳

❙❡ ❛-&✐❜✉②❡ ❛ ▲❡.♥ ❲❛❧&❛1 ✭✶✽✼✹✮ ❧❛ ♣&✐♠❡&❛ ❝♦♥1-&✉❝❝✐.♥ &✐❣✉&♦1❛ ❞❡ ✉♥

✸

✹

♠♦❞❡❧♦ ❞❡ ❡&✉✐❧✐❜*✐♦ ❣❡♥❡*❛❧ ❞♦♥❞❡ ❧❛. ♦❢❡*0❛. ② ❧❛. ❞❡♠❛♥❞❛. *❡.✉❧0❛❜❛♥ ❢✉♥✲

❝✐♦♥❡. ❞❡ ❧❛ ♣*✐♥❝✐♣❛❧ ✈❛*✐❛❜❧❡ ❞❡❧ ♠❡*❝❛❞♦✿ ❡❧ ♣*❡❝✐♦✳ ❲❛❧*❛. ❜✉.❝❛❜❛ ❧❛ ♠❛♥❡*❛

❡♥ ❝9♠♦ ❧♦. ♠❡*❝❛❞♦. .❡ ✈❛❝✐❛❜❛♥ ❛❧ ✜♥❛❧ ❞❡ ❝❛❞❛ ❥♦*♥❛❞❛ ② ❢✉♥❝✐♦♥❛❜❛♥ ❞❡

♠❛♥❡*❛ ❝♦❤❡*❡♥0❡ ❝♦♥ ❡❧ ✉.♦ ❞❡ ❧♦. *❡❝✉*.♦.✳ ◆♦ ♦❜.0❛♥0❡✱ ❡❧ ♦❜❥❡0✐✈♦ ❞❡ ❙♠✐0❤

❜✉.❝❛❞♦ ♣♦* ❲❛❧*❛. ❛ ♣❛*0✐* ❞❡ ✉♥ ♠♦❞❡❧♦ ❡❝♦♥9♠✐❝♦ ♥♦ 0❡♥@❛ ✉♥❛ .♦❧✉❝✐9♥

❝♦♥✈✐♥❝❡♥0❡ ②❛ &✉❡ ❡❧ .9❧♦ ❤❡❝❤♦ ❞❡ &✉❡ ❧❛. ♦❢❡*0❛. ② ❞❡♠❛♥❞❛. ❞❡❧ .✐.0❡♠❛ ✇❛❧✲

*❛.✐❛♥♦✱ ❞✐❡*❛♥ ❧✉❣❛* ❛ &✉❡ ❧❛. ❡❝✉❛❝✐♦♥❡. ❢✉❡*❛♥ ✐❣✉❛❧❡. ❛ ❧❛. ✐♥❝9❣♥✐0❛.✱ ♥♦ ❡*❛

✉♥❛ ♣*✉❡❜❛ *✐❣✉*♦.❛✳

▲♦. ✐♥0❡♥0♦. .✐❣✉✐❡*♦♥ ♣♦* ❡❧ ❧❛❞♦ ❈❛..❡❧ ✱ ❲❛❧❞ ② ❑♦♦♣♠❛♥.^✶♣❡*♦ .♦❧♦ ❤❛.0❛

❡❧ ❛E♦ ✶✾✺✹ ❝♦♥ ❑❡♥♥❡0❤ ❆**♦✇ ② ●❡*❛*❞ ❉❡❜*❡✉✱ ▼❝❦❡♥③✐❡✱ ❡♥0*❡ ♦0*♦.✱ .❡ ❧♦❣*9

❝♦♥.0*✉✐* ✉♥ ♠♦❞❡❧♦ 0❡9*✐❝♦ ❜❛❥♦ ♣*❡♠✐.❛. 0❡9*✐❝❛. ❞❡ *❛❝✐♦♥❛❧✐❞❛❞ ✐♥❞✐✈✐❞✉❛❧

&✉❡ ♣*♦❜❛❜❛ ❧❛ ❡①✐.0❡♥❝✐❛ ❞❡ ✉♥ ❡&✉✐❧✐❜*✐♦ ♣❛*❛ ✉♥❛ ❡❝♦♥♦♠@❛ ❤✐♣♦0P0✐❝❛✳

❊. ❞❡❝✐*✱ ❆**♦✇ ② ❉❡❜*❡✉ ♠✉❡.0*❛♥ ❧❛. ❝♦♥❞✐❝✐♦♥❡. ❧9❣✐❝❛. ♣❛*❛ ❧❛ ❡①✐.0❡♥✲

❝✐❛ ❞❡ ✉♥❛ .✐0✉❛❝✐9♥ ❞❡ ❡&✉✐❧✐❜*✐♦ ❞❡ ♣*❡❝✐♦. &✉❡ ❤❛❝❡♥ ❝♦♠♣❛0✐❜❧❡ ❧❛. ❛❝❝✐♦♥❡.

❞❡❝✐❞✐❞❛. ♣♦* ❧♦. ❛❣❡♥0❡.✳ ▲❛ ♣*❡❣✉♥0❛ &✉❡ .✉*❣❡ ♥❡❝❡.❛*✐❛♠❡♥0❡ ❡.✿ ➽▲♦❣*❛♥

*❡.♣♦♥❞❡* ❧❛ ♣*❡❣✉♥0❛ .♦❜*❡ ❡❧ P①✐0♦ ❞❡❧ ♠❡❝❛♥✐.♠♦ ❞❡ ♠❡*❝❛❞♦❄ ❊. ❡①0❡♥.❛ ❧❛

❝*@0✐❝❛ &✉❡ *❡.♣♦♥❞❡ ♥❡❣❛0✐✈❛♠❡♥0❡ ❛ ❡.❛ ♣*❡❣✉♥0❛ ② ❝♦♠♣❛*0✐♠♦. ♣❛*0❡ ❞❡ ❡❧❧❛✱

♣❡*♦ ❝*❡❡♠♦. &✉❡ ❡.0♦ ♥♦ ✐♠♣♦.✐❜✐❧✐0❛ ❛❧ ❧❧❛♠❛❞♦ ♠♦❞❡❧♦ ❆**♦✇✲❉❡❜*❡✉ ♣❛*❛

❝♦♥.0✐0✉✐*.❡ ❡♥ ✉♥❛ ♣♦❞❡*♦.❛ ❝❛❥❛ ❞❡ ❤❡**❛♠✐❡♥0❛. ♣❛*❛ ❛❝❡*❝❛*.❡ ❛ ❧❛ *❡❛❧✐❞❛❞✱

❛❞❡♠T. ❞❡ &✉❡ .❡ ❤❛ ❝♦♥✈❡*0✐❞♦ ❡♥ ❡❧ ❧❧❛♠❛❞♦ ♠♦❞❡❧♦ ❜❛.❡ ♣♦* .❡* ❝❛♣❛③ ❞❡

❛❜.♦*❜❡* ② ❞✐.❝✉0✐* ❝♦♥ ❝✉❛❧&✉✐❡* ♦0*❛ 0❡♦*@❛^✷✳ ❊♥ ❡❢❡❝0♦✱ ❧❛ 0❡♦*@❛ ❡❝♦♥9♠✐❝❛

❝♦♥0❡♠♣♦*T♥❡❛ ❝♦♥0✐♥✉❛ ❜✉.❝❛♥❞♦ *❡.♣✉❡.0❛ ❛ ❧❛ ♣*❡❣✉♥0❛ ❞❡ ❧❛ ✈✐❛❜✐❧✐❞❛❞ ❞❡

✉♥❛ ❡❝♦♥♦♠@❛ ❞❡ ♠❡*❝❛❞♦ ❝♦♠♣❡0✐0✐✈❛ ♣❡*♦ ♣❛*0✐❡♥❞♦ ❞❡❧ ♠♦❞❡❧♦ ❜❛.❡ ❡.0❛✲

❜❧❡❝✐❞♦ ♣♦* ❆**♦✇ ② ❉❡❜*❡✉ ②❛ &✉❡ ♠✉❝❤❛. ♣*♦♣♦.✐❝✐♦♥❡. ❢✉♥❞❛♠❡♥0❛❧❡.✱ ♣❛*❛

*❡.♦❧✈❡* ❧❛ ♣*❡❣✉♥0❛ ❛❜.0*❛❝0❛ ❞❡ ❙♠✐0❤✱ .9❧♦ .❡ ❤❛♥ *❡✈❡❧❛❞♦ ❛ ♣❛*0✐* ❞❡❧ ♠♦❞❡❧♦

&✉❡ ❡❧❧♦. ♣*♦♣♦♥❡♥✳

❈*❡❡♠♦. ❛❞❡♠T. &✉❡ ❧♦. ♣*♦❜❧❡♠❛. ❞❡❧ ♠♦❞❡❧♦ ❜❛.❡ ❡.0T♥ ♠T. ❞❡❧ ❧❛❞♦ ❞❡

❧❛ ❜U.&✉❡❞❛ ❞❡❧ ❡&✉✐❧✐❜*✐♦ ♦ ❡0❛♣❛ ♣♦.0❡*✐♦* ❛ ❧❛ ♠♦❞❡❧✐③❛❝✐9♥ ❞❡ ❧♦. ❛❣❡♥0❡. &✉❡

❤❛❝❡♥ ❛♣❛*❡❝❡* ❧❛ ❧❧❛♠❛❞❛ ✜❣✉*❛ ❞❡❧ .✉❜❛.0❛❞♦* ♥❡❝❡.❛*✐❛♠❡♥0❡ ❝♦♠♦ ✐♥❝♦♠♦❞❛

❛ ❧❛ 0❡♦*@❛ ② ❡❧ ♦❜❥❡0✐✈♦ ❜✉.❝❛❞♦✳ ▲♦ ❛♥0❡*✐♦* ♠✉② ❛.♦❝✐❛❞♦✱ ❞❡.❞❡ ❧✉❡❣♦✱ ❛❧

✶▼♦♥#❛❧✈❡ ✭✷✵✶✵✮ ❤❛❝❡ /❡❢❡/❡♥❝✐❛ ❛ ✉♥❛ 3/❛❞✐❝✐5♥ ❛❧❡♠❛♥❛ 7✉❡ #❡ ❞✐❢❡/❡♥❝✐5 ❞❡ ❧❛ ❝♦//✐❡♥3❡

♠8# ❛❢9♥ ❛ ❧❛ 3/❛❞✐❝✐5♥ ♣❛/❡3✐❛♥❛ ❝/✐#3❛❧✐③❛❞❛ ❡♥ ❍✐❝❦#✱ ❡♥ ❡❧ #❡♥3✐❞♦ ❞❡ 7✉❡ ❧❛ ♣/✐♠❡/❛ ❜✉#❝5

❡#3❛❜❧❡❝❡/ ❜❛❥♦ 7✉A ❝♦♥❞✐❝✐♦♥❡# ❡①✐#3❡ ❡❧ ❡7✉✐❧✐❜/✐♦ ♠✐❡♥3/❛# ❧❛ #❡❣✉♥❞❛ ❧♦ ❞❛❜❛ ♣♦/ ❤❡❝❤♦✳

E❛/❛ ▼♦♥#❛❧✈❡✱ ❧❛ 3/❛❞✐❝✐5♥ ♣❛/❡3✐❛♥❛ ❡#38 ✐♠♣/❡❣♥❛❞❛ ❡♥ ❆//♦✇ ② ❉❡❜/❡✉ ② ❣/❛♥ ♣❛/3❡ ❞❡

❧❛# ♣/❡♦❝✉♣❛❝✐♦♥❡# ❞❡ ❲❛❧/❛# ❢✉❡/♦♥ ✈✐❝✐❛❞❛#✳

✷❊❧ ❧❧❛♠❛❞♦ ♠♦❞❡❧♦ ❜❛#❡ ❝♦♥#3✐3✉②❡ ❤♦② ❡❧ ♥L❝❧❡♦ ❞❡ ❧❛ 3❡♦/9❛ ❡❝♦♥5♠✐❝❛ ❝♦♥3❡♠♣♦/8♥❡❛ ②

❧♦ 7✉❡ ❡# ♠8# ✐♠♣♦/3❛♥3❡✿ #❡ ❝♦♥✈✐❡/3❡ ❡♥ ❧❛ ♥♦/♠❛ ❞❡ ❧❛ ❡❝♦♥♦♠9❛ ♣♦#✐3✐✈❛✱ ❞♦♥❞❡ ❧♦ ♥♦/♠❛3✐✈♦

♥♦ ❡# #5❧♦ ✉♥❛ /❡❣❧❛ ♣❛/❛ ❤❛❝❡/ /❡❝♦♠❡♥❞❛❝✐♦♥❡# ❞❡ ♣♦❧93✐❝❛ #✐♥♦ 3❛♠❜✐A♥ ✉♥❛ ♠❡3♦❞♦❧♦❣9❛

♣❛/❛ ❧❛ ❝♦♥#3/✉❝❝✐5♥ ❞❡❧ #❛❜❡/ ❝✐❡♥39✜❝♦ ✭❇❡♥♥❡3✐ ✫ ❈❛/3❡❧✐❡/✱ ✶✾✾✽✮✳

✺

♣"♦❜❧❡♠❛ ❞❡ ❧❛ ✐♥❝♦"♣♦"❛❝✐-♥ ❞❡❧ ❞✐♥❡"♦ ♣❛"❛ .❡♥❡" ✉♥❛ ❡❝♦♥♦♠0❛ ♠♦♥❡.❛"✐❛ ②

❧♦2 ♣♦2✐❜❧❡2 ♣"♦❜❧❡♠❛2 ② ♣❡"2♣❡❝.✐✈❛2 4✉❡ ❡❧❧♦ ."❛❡✳ ❆ ♣❡2❛" ❞❡ ❡❧❧♦✱ ❝"❡❡♠♦2 4✉❡

❡2.♦ ♥♦ ✐♠♣♦2✐❜✐❧✐.❛ ❧❛ ✈❛❧✐❞❡③ ❞❡ ✉♥ ♠❛"❝♦ .❡-"✐❝♦ 2♦❜"❡ ❧❛ .♦♠❛ ❞❡ ❞❡❝✐2✐♦♥❡2 ❞❡

❧♦2 ❛❣❡♥.❡2✱ ❝♦♥ ❜❛2❡ ❛ ♥♦❝✐♦♥❡2 ❞❡ "❛❝✐♦♥❛❧✐❞❛❞ ♠0♥✐♠❛♠❡♥.❡ ❛❝❡♣.❛❜❧❡2 ❝♦♠♦

❧❛2 4✉❡ ♣♦2.✉❧❛ ❧♦2 ❛①✐♦♠❛2 ✐♠♣✉❡2.♦2 ❛ ❧❛ ❝♦♥❞✉❝.❛ ❞❡ ❧♦2 ❛❣❡♥.❡2 ❡♥ ❡❧ ♠♦❞❡❧♦

❆""♦✇✕❉❡❜"❡✉✳ ▲❛ ❞✐2❝✉2✐-♥ ♣♦❞"0❛ ❡①.❡♥❞❡"2❡ ❞❡♠❛2✐❛❞♦ ♣♦" ❧♦ 4✉❡ ❛2♣✐"❛♠♦2

❛ 4✉❡ ❧❛ ❤✐2.♦"✐❛ 2❡❛ ❡❧ ❥✉❡③ ② ❞❡ 2✉ ✈❡"❡❞✐❝.♦ 2♦❜"❡ ❧❛ ♣♦2.✉"❛ 4✉❡ ❞❡❢❡♥❞❡♠♦2✳

▲♦2 ❝♦♠❡♥.❛"✐♦2 ❛♥.❡"✐♦"❡2 2❡ ❤❛♥ ❤❡❝❤♦ ❞❡❜✐❞♦ ❛ 4✉❡ ❡❧ ♦❜❥❡.✐✈♦✱ ♣♦" ❧♦ ♠❡✲

♥♦2 ❡❧ ♣"✐♥❝✐♣❛❧✱ ❜✉2❝❛❞♦ ❡♥ ❡2.❡ .❡①.♦ ❡2 ❞❛" ❧♦2 ❧✐♥❡❛♠✐❡♥.♦2 ❣❡♥❡"❛❧❡2 2♦❜"❡ ❧❛

❢♦"♠❛ ❡♥ 4✉❡ 2❡ ♠♦❞❡❧❛ ❛ ❧♦2 ❝♦♥2✉♠✐❞♦"❡2 ❡♥ ✉♥❛ ❡❝♦♥♦♠0❛ .✐♣♦ ❆""♦✇✕❉❡❜"❡✉✳

C❛"❛ ❡❧❧♦ ❞❡2❛""♦❧❧❛"❡♠♦2 ✉♥❛ ❝✉✐❞❛❞♦2❛ ❢♦"♠❛❧✐③❛❝✐-♥ ❞❡❧ ❛❣❡♥.❡ ❝♦♥2✉♠✐❞♦" ②

❛♥❛❧✐③❛"❡♠♦2 ❞❡.❛❧❧❛❞❛♠❡♥.❡ ❧♦ 4✉❡ 2❡ ❝♦♥2✐❞❡"❛ "❛❝✐♦♥❛❧✐❞❛❞ ✐♥❞✐✈✐❞✉❛❧✳ C❛"❛

✐♥✐❝✐❛" ❡2.❛ .❛"❡❛✱ ❡"❛ ♥❡❝❡2❛"✐❛ ❧❛ ❛♥.❡"✐♦" ❝♦♥.❡①.✉❛❧✐③❛❝✐-♥ ❤✐2.-"✐❝❛ ❞❡❧ ♠♦✲

❞❡❧♦ 4✉❡ 2❡ ✈❛ ❛ ❞❡2❛""♦❧❧❛" ♣❛"❛ ♣❛2❛" ❧✉❡❣♦✱ 2✐♥ ❣"❛♥❞❡2 2❛❧.♦2✱ ❛ ❧♦ 4✉❡ 2❡

❝♦♥2✐❞❡"❛ ❡❧ ♠♦❞❡❧♦ ❜D2✐❝♦ ♦ ❡❝♦♥♦♠0❛ .✐♣♦ ❆""♦✇ ✲ ❉❡❜"❡✉✳

▼♦❞❡❧♦ ❇&'✐❝♦

❙❡ ❞❡♥♦♠✐♥❛"D ♠♦❞❡❧♦ ❜D2✐❝♦ ❛❧ ♠♦❞❡❧♦ ❞❡ ❡4✉✐❧✐❜"✐♦ ❣❡♥❡"❛❧ ❝♦♠♣❡.✐.✐✈♦

❞❡2❛""♦❧❧❛❞♦ ♣♦" ❑❡♥♥❡.❤ ❆""♦✇ ② ●❡"❛"❞ ❉❡❜"❡✉ ✭✶✾✺✹✮✱ ❡♥."❡ ♠✉❝❤♦2 ♦."♦2 ②

❝"✐2.❛❧✐③❛❞♦ ❡♥ ❉❡❜"❡✉ ✭✶✾✼✸✮ ② ❆""♦✇ ② ❍❛❤♥ ✭✶✾✼✼✮✳

❈♦♠♦ ♠❡♥❝✐♦♥❛♠♦2 ❛♥.❡"✐♦"♠❡♥.❡✱ ❆""♦✇✲❉❡❜"❡✉ 2❡ ✐♥2❝"✐❜❡♥ ❞❡♥."♦ ❞❡

✉♥❛ ."❛❞✐❝✐-♥ ❞❡ ❡❝♦♥♦♠✐2.❛2 4✉❡ ❞❡2❞❡ ❆❞❛♠ ❙♠✐.❤ ❤❛2.❛ ❡❧ ♣"❡2❡♥.❡ ❤❛♥

."❛.❛❞♦ ❞❡ ❞❡♠♦2."❛" ❝-♠♦ ✉♥❛ ❡❝♦♥♦♠0❛ ❞❡2❝❡♥."❛❧✐③❛❞❛ ② ♠♦.✐✈❛❞❛ ♣♦" ❡❧

✐♥.❡"Q2 ✐♥❞✐✈✐❞✉❛❧✱ ② ❣✉✐❛❞❛ ♣♦" ❧❛2 2❡R❛❧❡2 ❞❡ ❧♦2 ♣"❡❝✐♦2✱ ❡2 ❝♦♠♣❛.✐❜❧❡ ❝♦♥

✉♥❛ ❞✐2♣♦2✐❝✐-♥ ❝♦❤❡"❡♥.❡ ❞❡ ❧♦2 "❡❝✉"2♦2 ❡❝♦♥-♠✐❝♦2✱ 4✉❡ ♣♦❞"0❛ ❝♦♥2✐❞❡"❛"2❡

♠❡❥♦" 4✉❡ ✉♥ ❣"❛♥ ♥S♠❡"♦ ❞❡ ❞✐2♣♦2✐❝✐♦♥❡2 ❛❧.❡"♥❛.✐✈❛2 ♣♦2✐❜❧❡2 ✭❆""♦✇ ② ❍❛❤♥✱

✶✾✼✼✿✾✮✱ ✭❈❛.❛R♦✿ ✶✾✾✼✿✶✶✻✮✳

❊❧ ♦❜❥❡.✐✈♦✱ ❡♥.♦♥❝❡2✱ 4✉❡ .✐❡♥❡ ❧❛ .❡♦"0❛ ❞❡ ❡4✉✐❧✐❜"✐♦ ❣❡♥❡"❛❧ ♥❡♦✇❛❧"❛2✐❛♥❛

❡2 "❡2♣♦♥❞❡" ✉♥❛ ♣"❡❣✉♥.❛ ✐♥.❡❧❡❝.✉❛❧ ✭❛❜2."❛❝.❛✮ ❝♦♥ "❡2♣❡❝.♦ ❛ ❧❛ ♣♦2✐❜✐❧✐❞❛❞

② ❧❛ ✈✐❛❜✐❧✐❞❛❞ ❞❡ ✉♥❛ 2♦❝✐❡❞❛❞ ♠❡"❝❛♥.✐❧ ✭❈❛.❛R♦✿ ✶✾✾✼✿✶✶✼✮✳ C❛"❛ ❡♠♣❡③❛"

❛ ❝♦♥2."✉✐" ❛♥❛❧0.✐❝❛♠❡♥.❡ ✉♥ ♠♦❞❡❧♦ 4✉❡ ♣"❡.❡♥❞❛ ❞❛" "❡2♣✉❡2.❛ ❛❧ ♦❜❥❡.✐✈♦

❛♥.❡"✐♦"✱ ❡2 ♥❡❝❡2❛"✐♦ ♣"✐♠❡"♦ ❝♦♥2."✉✐" ✉♥ ♠♦❞❡❧♦ 4✉❡ ✐♥.❡♥.❡ "❡♣"❡2❡♥.❛" ❧❛

"❡❛❧✐❞❛❞ ❡♥ 2✉2 ♣"✐♥❝✐♣❛❧❡2 ❡❧❡♠❡♥.♦2 ② ♣❛"❛ ❡❧❧♦✱ ✈❛♠♦2 ❡♥.♦♥❝❡2 ❛ ❞❡✜♥✐"✱ ❡♥

❢♦"♠❛ ❣❡♥❡"❛❧ ② "D♣✐❞❛✱ ❧❛ ♠❛♥❡"❛ ❡♥ 4✉❡ 2❡ ❞❡✜♥❡ ❧❛ ❡❝♦♥♦♠0❛ ❡♥ ❡❧ ♠♦❞❡❧♦

❆""♦✇✲❉❡❜"❡✉ ♣❛"❛ ❞❡2♣✉Q2 ❡♠♣❡③❛" ❝♦♥ ❡❧ ."❛.❛♠✐❡♥.♦ ♠❡.♦❞♦❧-❣✐❝♦ ❞❡❧ ❛❣❡♥.❡

❝♦♥2✉♠✐❞♦"✳

✻

!❛#❛ ❝♦♥'(#✉✐# ✉♥ ♠♦❞❡❧♦ ❞❡ ❡/✉✐❧✐❜#✐♦ ❣❡♥❡#❛❧✱ ❝♦♠♦ #❡♣#❡'❡♥(❛❝✐4♥ ❛❜'✲

(#❛❝(❛ ❞❡ ✉♥❛ ❡❝♦♥♦♠6❛ ❝♦♠♣❡(✐(✐✈❛✱ '❡ #❡/✉✐❡#❡ ❧❛ ❡'♣❡❝✐✜❝❛❝✐4♥ ❞❡ ❛❧ ♠❡♥♦' (#❡' ❡❧❡♠❡♥(♦'✿ ❧❛' ♠❡#❝❛♥❝6❛' ② ❧♦' ♣#❡❝✐♦' /✉❡ '♦♥ ❧❛' ✈❛#✐❛❜❧❡' ❞❡❧ ♠♦❞❡❧♦✱ ❧♦'

❛❣❡♥(❡' /✉❡ '♦♥ ❧❛' ✉♥✐❞❛❞❡' ❞❡ ❞❡❝✐'✐4♥ ❞❡❧ ♠♦❞❡❧♦ ② ❧❛ ♥❛(✉#❛❧❡③❛ ❞❡ ❧❛ ❢♦#♠❛

❡♥ /✉❡ '❡ (♦♠❛♥ ❧❛' ❞❡❝✐'✐♦♥❡' /✉❡ ❞❛ ❡❧ ♠❛#❝♦ ❡♥ /✉❡ '❡ ♠✉❡✈❡♥ ❧♦' ❛❣❡♥(❡'

✭❱✐❧❧❛#✱ ✶✾✾✻✿✶✷✮✳ ❊❧ ♦❜❥❡(✐✈♦ ❞❡ ❡'(❡ ❧✐❜#♦ '❡ ❡♥♠❛#❝❛ '4❧♦ ❡♥ ❧❛ ❡'♣❡❝✐✜❝❛❝✐4♥

❞❡ ❧♦' ❞♦' ♣#✐♠❡#♦' ❡❧❡♠❡♥(♦' ❞❡❥❛♥❞♦ ❛❢✉❡#❛ ❧❛ ♥❛(✉#❛❧❡③❛ ❞❡❧ ♣#♦❝❡'♦ ❞❡ (♦♠❛

❞❡❝✐'✐♦♥❡'✳

❊❝♦♥♦♠%❛' ❆))♦✇ ✲ ❉❡❜)❡✉

❱❛♠♦' ❛ ❞❡✜♥✐# ♥✉❡'(#❛ ❡❝♦♥♦♠6❛ ❝♦♠♦ ✉♥❛ (#✐❛❞❛ ❢♦#♠❛❞❛ ♣♦#✿

ǫ= [(Xi,%_i, βi)^m_i=1,(Yj, πj)ⁿ_j=1, ω]

❉♦♥❞❡ Xi ❡' ❡❧ ❝♦♥❥✉♥(♦ ❞❡ ❝♦♥'✉♠♦ ❞❡❧ ❝♦♥'✉♠✐❞♦# i✱ %i ❡' ❧❛ #❡❧❛❝✐4♥

❜✐♥❛#✐❛ ❞❡ ♣#❡❢❡#❡♥❝✐❛' '♦❜#❡ ❡❧ ❝♦♥❥✉♥(♦ ❞❡ ❝♦♥'✉♠♦ ❞❡❧ ❝♦♥'✉♠✐❞♦# i② βi ❡'

❡❧ ❝♦♥❥✉♥(♦ ❞❡ #❡'(#✐❝❝✐♦♥❡' ♣#❡'✉♣✉❡'(❛#✐❛'✳ !♦# ♦(#❛ ♣❛#(❡✱ ❝❛❞❛ ❡❧❡♠❡♥(♦ ❞❡

Yj ❡' ❡❧ ❝♦♥❥✉♥(♦ ❞❡ ♣#♦❞✉❝❝✐4♥ ❞❡❧ ♣#♦❞✉❝(♦# j✱πj ❡' ❧❛ ❢✉♥❝✐4♥ ❞❡ ❜❡♥❡✜❝✐♦'

❞❡❧ ♣#♦❞✉❝(♦# j ② ω '♦♥ ❧❛' ❞♦(❛❝✐♦♥❡' ✐♥✐❝✐❛❧❡'✳ ❍❛② ✉♥ ♥H♠❡#♦ ❣#❛♥❞❡ m

❞❡ ❝♦♥'✉♠✐❞♦#❡' ② n ❞❡ ♣#♦❞✉❝(♦#❡'✳ ❊♥ ❡'(❡ (❡①(♦✱ ❡❧ ♦❜❥❡(✐✈♦ ❡' ♠♦❞❡❧❛# ❧❛

❝♦♥❞✉❝(❛ ❞❡ ✉♥ ❛❣❡♥(❡ ❝♦♥'✉♠✐❞♦# i② ❡♥ ❛❞❡❧❛♥(❡ '❡ ♦♠✐(✐#J ❡❧ '✉❜6♥❞✐❝❡✳

❊❧ ♠♦❞❡❧♦ '✉♣♦♥❡ /✉❡ ❧♦' ❛❣❡♥(❡' ❡❝♦♥4♠✐❝♦' ❞❡'❛##♦❧❧❛♥ '✉' ❛❝❝✐♦♥❡' '♦❜#❡

❧❛ ❜❛'❡ ❞❡ ✉♥❛ ❞♦(❛❝✐4♥ ✐♥✐❝✐❛❧ ❞❡ #❡❝✉#'♦' ❞❛❞❛ ❞✐'♣♦♥✐❜❧❡' ♣❛#❛ ❧❛ ❡❝♦♥♦♠6❛✳

❊'(♦' #❡❝✉#'♦' ✐♥❝❧✉②❡♥ (♦❞♦' ❛/✉❡❧❧♦' ❛❝(✐✈♦' #❡❛❧❡'✱ ❝♦♠♦ ❡❞✐✜❝✐♦'✱ ♠❛/✉✐♥❛#✐❛✱

(✐❡##❛✱ #❡❝✉#'♦' ♥❛(✉#❛❧❡' ♦ ❜✐❡♥❡' ♣#♦❞✉❝✐❞♦' ♦ ❛❧♠❛❝❡♥❛❞♦' /✉❡ '♦♥ ❧❛ ❤❡#❡♥❝✐❛

❞❡❧ ♣❛'❛❞♦ ② '✉♣♦♥❡♥ ❧❛ ❜❛'❡ '♦❜#❡ ❧❛ /✉❡ '❡ ❞❡'❛##♦❧❧❛ ❧❛ ♣#♦❞✉❝❝✐4♥✳ ▲♦'

#❡❝✉#'♦' (♦(❛❧❡' '❡ ♣#❡'❡♥(❛♥ ♣♦#ω∈R^l② ❡' ♣♦'✐❜❧❡✱ ❡♥(♦♥❝❡'✱ ✈❡# ❧❛ ❝❛♣❛❝✐❞❛❞

♣#♦❞✉❝(✐✈❛ ❞❡ ✉♥❛ '♦❝✐❡❞❛❞ ❝♦♠♦ ❧❛ ❝♦♠❜✐♥❛❝✐4♥ ❞❡ (#❡' (✐♣♦' ❞❡ #❡❝✉#'♦'✿ ❧♦'

#❡❝✉#'♦' ♠❛(❡#✐❛❧❡' ω✱ ❧♦' #❡❝✉#'♦' (❡❝♥♦❧4❣✐❝♦' ✭#❡❝♦❣✐❞♦' ❡♥ ❡❧ ❝♦♥❥✉♥(♦ ❞❡

♣#♦❞✉❝❝✐4♥✮ ② ❡❧ ❝❛♣✐(❛❧ ❤✉♠❛♥♦✱ #❡❝♦❣✐❞♦' ❡♥ ❧♦' ❝♦♥❥✉♥(♦' ❞❡ ❝♦♥'✉♠♦ ✭❱✐❧❧❛#✱

✶✾✾✻✿ ✶✶✾✮✳

▲❛ ❡❝♦♥♦♠6❛ ❡'(J ❝♦♥❢♦#♠❛❞❛ ❛'6 ♣♦# ❝♦♥'✉♠✐❞♦#❡' ② ♣#♦❞✉❝(♦#❡' ❝❛❞❛ ✉♥♦

❝♦♥ ✉♥❛ ❢✉♥❝✐4♥ ♦❜❥❡(✐✈♦ ❛ ♦♣(✐♠✐③❛# ❥✉♥(♦ ❝♦♥ ✉♥❛' ❞♦(❛❝✐♦♥❡' ✐♥✐❝✐❛❧❡' ❞❡

#❡❝✉#'♦'✳ ❯♥❛ ✈❡③ ✉❜✐❝❛❞♦ ❤✐'(4#✐❝❛ ② ❛♥❛❧6(✐❝❛♠❡♥(❡ ❡❧ ♦❜❥❡(✐✈♦ ❜✉'❝❛❞♦ ♣♦#

❡'(❡ (❡①(♦✱ ♣❛'❡♠♦' ❛❤♦#❛ ❛ ♠♦❞❡❧❛# ❡❧ ❝♦♠♣♦#(❛♠✐❡♥(♦ ❞❡❧ ❛❣❡♥(❡ ❝♦♥'✉♠✐❞♦#

❡♥ ✉♥❛ ❡❝♦♥♦♠6❛ (✐♣♦ ❆##♦✇ ✲ ❉❡❜#❡✉ ♣❛#❛ ✈❡# ❝4♠♦ ❧♦' ❝♦♥'✉♠✐❞♦#❡' #❡'✉❡❧✈❡♥

'✉' ♣#♦❜❧❡♠❛' ✐♥❞❡♣❡♥❞✐❡♥(❡♠❡♥(❡✱ '✐♥ ❝♦♥♦❝❡#'❡✱ ❝♦♥ ✉♥ '4❧♦ ♠❡❝❛♥✐'♠♦ ❞❡

✼

✐♥❢♦%♠❛❝✐)♥ ❡ ✐♥✢✉❡♥❝✐❛ ❡♥-%❡ ❛❣❡♥-❡/✿ ❧♦/ ♣%❡❝✐♦/✳

❈❛♣#$✉❧♦ ✶

❈❖◆❙❯▼■❉❖❘❊❙

✶✳✶✳ ■♥$%♦❞✉❝❝✐+♥

▲❛ ✉♥✐❞❛❞ ❢✉♥❞❛♠❡♥)❛❧ ❞❡ ❞❡❝✐,✐-♥ ❡♥ ❧❛ )❡♦/0❛ ❡❝♦♥-♠✐❝❛ ❡, ❡❧ ❝♦♥,✉♠✐❞♦/

② ♣♦/ ❧♦ )❛♥)♦✱ ♥✉❡,)/♦ ❛♥4❧✐,✐, ✈❛ ❛ ❡♠♣❡③❛/ ♣♦/ ❞❡✜♥✐/ ❧❛ ❞❡♠❛♥❞❛ ❞❡ ❧♦,

❝♦♥,✉♠✐❞♦/❡, ❡♥ ✉♥❛ ❡❝♦♥♦♠0❛ ❞❡ ♠❡/❝❛❞♦✳ 9♦/ ❡❝♦♥♦♠0❛ ❞❡ ♠❡/❝❛❞♦✱ ✈❛♠♦,

❛ ❡♥)❡♥❞❡/ ✉♥ ❡,❝❡♥❛/✐♦ ❡♥ ❡❧ ❝✉❛❧ )♦❞♦, ❧♦, ❜✐❡♥❡, ② ,❡/✈✐❝✐♦, ;✉❡ ❧♦, ❛❣❡♥)❡,

❞❡,❡❛♥ ❛❞;✉✐/✐/ ❡,)4♥ ❞✐,♣♦♥✐❜❧❡, ❛ ✉♥ ♣/❡❝✐♦ ❝♦♥♦❝✐❞♦ ②✴♦ ❡,)4♥ ❞✐,♣♦♥✐❜❧❡,

♣❛/❛ ✐♥)❡/❝❛♠❜✐❛/✱ ♣♦/ ♦)/♦, ❜✐❡♥❡,✱ ❛ ✉♥❛ )❛,❛ ❝♦♥♦❝✐❞❛ ✭▼❛,✲❈♦❧❡❧❧✱ ❲❤✐♥,)♦♥

② ●/❡❡♥✱ ✶✾✾✺✿✶✼✮✳

❙❡ ❞❡✜♥❡ ✉♥ ❝♦♥,✉♠✐❞♦/✱ ❝♦♠♦ ✉♥ ✐♥❞✐✈✐❞✉♦ ♦ ❣/✉♣♦ ❞❡ ✐♥❞✐✈✐❞✉♦, ✭✉♥❛

❢❛♠✐❧✐❛✮ ❝♦♥ ✉♥ ♣/♦♣-,✐)♦ ❞❡ ❝♦♥,✉♠♦ ✉♥✐✜❝❛❞♦✿ ❞❡❝❧❛/❛/ ❧❛ ♣/❡❢❡/❡♥❝✐❛ ,♦❜/❡

❝❛❞❛ ♣❧❛♥ ❞❡ ❝♦♥,✉♠♦ ② ❡❧❡❣✐/ ❧♦, ♣❧❛♥❡, ❞❡ ❝♦♥,✉♠♦ ❞❡ ❧♦, ❞✐,)✐♥)♦, ❜✐❡♥❡, ;✉❡

❡,)4♥ ❞✐,♣♦♥✐❜❧❡, ♣❛/❛ ❝♦♠♣/❛/ ❡♥ ❡❧ ♠❡/❝❛❞♦✱ ❞❡ )❛❧ ❢♦/♠❛ ;✉❡ ,✉, ♣/❡❢❡/❡♥❝✐❛, ,❡ ,❛)✐,❢❛❣❛♥ ❛❧ ♠4①✐♠♦✳ ❇❛❥♦ ❝✐❡/)❛, ❝♦♥❞✐❝✐♦♥❡, ;✉❡ ,❡ ♣/❡❝✐,❛/4♥ ♠4, ❛❞❡❧❛♥)❡✱

❡❧ ♣/♦❜❧❡♠❛ ❞❡❧ ❝♦♥,✉♠✐❞♦/ ♣✉❡❞❡ ♣❧❛♥)❡❛/,❡ ❝♦♠♦ ❧❛ ♠❛①✐♠✐③❛❝✐-♥ ❞❡ ✉♥❛

❢✉♥❝✐-♥ ❧❧❛♠❛❞❛ ❞❡ ✉)✐❧✐❞❛❞ ✭▲♦③❛♥♦✱ ▼♦♥,❛❧✈❡ ② ❱✐❧❧❛✱ ✶✾✾✾✿✶✹✮^✶✳

❯♥ ♣❧❛♥ ❞❡ ❝♦♥,✉♠♦ ❡, ✉♥❛ ❡,♣❡❝✐✜❝❛❝✐-♥ ❞❡ ❧❛, ❝❛♥)✐❞❛❞❡, ❞❡ ❜✐❡♥❡, ❞❡

❝♦♥,✉♠♦ ❛ ❞❡♠❛♥❞❛/ ❛❧ ❝♦♠✐❡♥③♦ ❞❡❧ ♣❡/0♦❞♦ ② ❛❧ ❝♦♥❥✉♥)♦ ❞❡ )♦❞♦, ❧♦, ♣❧❛♥❡,

❞❡ ❝♦♥,✉♠♦✱ ,❡ ❧❡ ❞❡♥♦♠✐♥❛ ❝♦♥❥✉♥)♦ ❞❡ ❝♦♥,✉♠♦✳ ❊❧ ❝♦♥❥✉♥)♦ ❞❡ ❝♦♥,✉♠♦ ❡,)4

✐♥❝❧✉✐❞♦ ❛,0 ❡♥ ❡❧ ❡,♣❛❝✐♦ ❞❡ ♠❡/❝❛♥❝0❛, ✭▼♦♥,❛❧✈❡✱ ✷✵✵✾✿✸✸✼✮✳ ❊♥ ❡,)❡ ,❡♥)✐❞♦✱

❡❧ ♠❡❝❛♥✐,♠♦ ❡①♣✉❡,)♦ ❡, ❝♦♠♣❧❡)❛♠❡♥)❡ ❞❡,❝❡♥)/❛❧✐③❛❞♦ ❡ ✐♠♣❡/,♦♥❛❧✿ ♥✐♥❣V♥

❛❣❡♥)❡ ♥❡❝❡,✐)❛ ,❛❜❡/ ❛❧❣♦ ❞❡ ❧♦, ❞❡♠4,✱ ♥✐ ❞❡ ,✉, ♣/❡❢❡/❡♥❝✐❛, ♥✐ ❞❡❧ ❝♦♥❥✉♥)♦

✶◆♦ ✈❛♠♦% ❛ &❡♥❡) ❡♥ ❝✉❡♥&❛ ❡❧ ❝❛%♦ ❡♥ -✉❡ ❧♦% ✐♥❞✐✈✐❞✉♦% ❡%♣❡❝✐✜❝❛♥ ❧❛% ❝❛♥&✐❞❛❞❡% ❞❡

%❡)✈✐❝✐♦% ❛ ♦❢)❡❝❡)✳ 4♦) ❧♦ &❛♥&♦✱ ❡♥ ❧❛ ❝❡%&❛ ❞❡ ❝♦♥%✉♠♦ %♦❧♦ ❡①✐%&✐)7♥ ❡♥&)❛❞❛% ♥♦ ♥❡❣❛&✐✈❛%✳

✽

❈❆"❮❚❯▲❖ ✶✳ ❈❖◆❙❯▼■❉❖❘❊❙ ✾

❞❡ ❝♦♥&✉♠♦ ❞❡ ❧♦& ♦*+♦&✱ ♥✐ ❞❡ &✉& ❞♦*❛❝✐♦♥❡&✱ ❡*❝✳

❙❡ ❛&✉♠✐+1 2✉❡ ❡❧ ♥3♠❡+♦ ❞❡ ❜✐❡♥❡& ❡& ✜♥✐*♦ ❡ ✐❣✉❛❧ ❛ L✱ ✐♥❞❡①❛❞♦& ❝❛❞❛ ✉♥♦

❞❡ ❧♦& ❜✐❡♥❡& ❝♦♥ ✉♥ l = 1,2, ..., L✳ ❙❡ &✉♣♦♥❡ *❛♠❜✐9♥ 2✉❡ ❧♦& ❛❣❡♥*❡& *✐❡♥❡♥

✐♥❢♦+♠❛❝✐;♥ &♦❜+❡ *♦❞❛& ❧❛& ❝❛+❛❝*❡+<&*✐❝❛& ❞❡ ❧♦& ❜✐❡♥❡& ② ❞❡ ❧♦& +❡&✉❧*❛❞♦& ❞❡ &✉&

❛❝❝✐♦♥❡&✱ ❡& ❞❡❝✐+❀ ❤❛② ✐♥❢♦+♠❛❝✐;♥ ❝♦♠♣❧❡*❛ ❛❞❡♠1& ❞❡ 2✉❡ ♥♦ ❤❛② ♣+♦❜❧❡♠❛&

❡♥ ❡❧ ❝✉♠♣❧✐♠✐❡♥*♦ ❞❡ ❧♦& ❝♦♥*+❛*♦& ② *♦❞❛& ❧❛& ❝♦♥*✐♥❣❡♥❝✐❛& &❡ *✐❡♥❡♥ ❡♥ ❝✉❡♥*❛

✭❙*✐❣❧✐*③✱ ✷✵✵✵✮✳

E❛+❛ ❡❧ ❛♥1❧✐&✐& 2✉❡ &❡ ✈❛ ❛ ♣+❡&❡♥*❛+✱ ✉♥ ❜✐❡♥ ❡& ❛❧❣♦ 2✉❡ &❡ ♣✉❡❞❡ ✐♥*❡+✲

❝❛♠❜✐❛+ ② ❞❛ &❛*✐&❢❛❝❝✐;♥✳ E✉❡❞❡♥ &❡+ *❛♥❣✐❜❧❡& ♦ ✐♥*❛♥❣✐❜❧❡&✳ ❚❛❧ ❝♦♠♦ &❡ ❛♥♦*;

❛♥*❡&✱ &❡ ✈❛ ❛ ❞❡✜♥✐+ ✉♥❛ ❝❡&*❛✱ ♣❧❛♥ ♦ ❝❛♥❛&*❛ ❞❡ ❝♦♥&✉♠♦ ❝♦♠♦ ✉♥❛ ❧✐&*❛ ❞❡

❧♦& ❞✐❢❡+❡♥*❡& ❜✐❡♥❡& ② &❡ ❞❡♥♦*❛+1 ♣♦+x❞♦♥❞❡

x=















 x1

x2

. : xL

















❙❡ *✐❡♥❡ ❛&< 2✉❡ ✉♥❛ ❝❛♥❛&*❛ ♦ ❝❡&*❛ ❞❡ ❜✐❡♥❡& ❡& ✉♥ ♣✉♥*♦ ❡♥ R^l₊ ❡❧ ❝✉❛❧ ❡&

♣+❡❝✐&❛♠❡♥*❡ ❡❧ ❡&♣❛❝✐♦ ❞❡ ❧♦& ❜✐❡♥❡&^✷✳ E♦+ ❧♦ *❛♥*♦✱ ❧❛lth❡♥*+❛❞❛ ❞❡❧ ✈❡❝*♦+ ❞❡

❜✐❡♥❡& ❡&♣❡❝✐✜❝❛+1 ❧❛ ❝❛♥*✐❞❛❞ ❞❡❧ ❜✐❡♥l ❝♦♥&✉♠✐❞♦✳

▲♦& ❜✐❡♥❡&✱ ❛❞❡♠1&✱ &♦♥ ❞✐✈✐&✐❜❧❡& ② ❡&*1♥ ❞❡✜♥✐❞♦& ♣♦+ ❢❡❝❤❛✱ *✐❡♠♣♦ ② ❡&✲

♣❛❝✐♦ ② ♣♦+ ❧♦ *❛♥*♦✱ ❡❧ ♠✐&♠♦ ❜✐❡♥ ❞❡✜♥✐❞♦ ❡♥ ❞♦& ❢❡❝❤❛& ❞✐&*✐♥*❛& ❡& ✉♥ ❜✐❡♥

❞✐&*✐♥*♦ ✭❆++♦✇ ② ❉❡❜+❡✉✱ ✶✾✺✹✿✷✻✻✮✳ ❆&< ♣✉❡&✱ ♣♦❞❡♠♦& ❞❡✜♥✐+ ✉♥❛ ♠❡+❝❛♥❝<❛

❝♦♠♦ ✉♥ ❜✐❡♥ ♦ &❡+✈✐❝✐♦ ❝♦♠♣❧❡*❛♠❡♥*❡ ❡&♣❡❝✐✜❝❛❞♦ ❢<&✐❝❛✱ *❡♠♣♦+❛❧ ② ❡&♣❛❝✐❛❧✲

♠❡♥*❡✳

✶✳✷✳ ❊❧ ❝♦♥❥✉♥*♦ ❞❡ ❝♦♥-✉♠♦ ♦ ❞❡ ❡❧❡❝❝✐0♥

❊❧ ❝♦♥❥✉♥*♦ ❞❡ ❝♦♥&✉♠♦ ❡& ✉♥ &✉❜❝♦♥❥✉♥*♦ ❞❡❧ ❡&♣❛❝✐♦ ❞❡ ❜✐❡♥❡& R^l₊ ② ❧♦

✈❛♠♦& ❛ ❞❡♥♦*❛+ ❝♦♠♦ X ⊆R^l₊ ❝✉②♦& ❡❧❡♠❡♥*♦& &♦♥ ❧❛& ❝❛♥❛&*❛& ❞❡ ❝♦♥&✉♠♦

2✉❡ ❡❧ ✐♥❞✐✈✐❞✉♦ ♣✉❡❞❡ ♣♦&✐❜❧❡♠❡♥*❡ ❝♦♥&✉♠✐+ ❞❛❞❛ ❧❛& +❡&*+✐❝❝✐♦♥❡& ❢<&✐❝❛&^✸ ②

❡❝♦♥;♠✐❝❛& ✐♠♣✉❡&*❛&✳

✷◆!"❡$❡ ❞❡ ♥✉❡✈♦ *✉❡ ♥♦ $❡ ✈❛♥ ❛ "❡♥❡, ❡♥ ❝✉❡♥"❛ ❡♥",❛❞❛$ ♥❡❣❛"✐✈❛$ ❡♥ ❡❧ ❡$♣❛❝✐♦ ❞❡ ❜✐❡♥❡$

②❛ *✉❡ ♥♦ $❡ ✈❛♥ ❛ "♦♠❛, ❞5❜✐"♦$ ♦ $❛❧✐❞❛$ ❞❡ ❜✐❡♥❡$ ♦ $❡,✈✐❝✐♦$ ♣♦, ♣❛,"❡ ❞❡❧ ❝♦♥$✉♠✐❞♦, ❝♦♠♦✱

♣♦, ❡❥❡♠♣❧♦✱ $✉ ♦❢❡,"❛ ❞❡ ",❛❜❛❥♦✳

✸:♦, ,❡$",✐❝❝✐♦♥❡$ ❢;$✐❝❛$ ❡♥"✐5♥❞❛$❡✱ ♣♦, ❡❥❡♠♣❧♦✱ *✉❡ ♥♦ ❡$ ♣♦$✐❜❧❡ ❝♦♥$✉♠✐, ♠<$ ❞❡ ✷✹

❤♦,❛$✳

❈❆"❮❚❯▲❖ ✶✳ ❈❖◆❙❯▼■❉❖❘❊❙ ✶✵

❆#$✱ ❡❧ ❝♦♥❥✉♥-♦ ❞❡ ❝♦♥#✉♠♦ ❡#-❛12 ❞❡✜♥✐❞♦ ❝♦♠♦

X⊆R^l₊=

x∈R^l₊:xl≥0∀l= 1,2, ..., L

❊♥ ❡❧ ❡#♣❛❝✐♦R²₊ ♣❛1❛ ❞♦# ❜✐❡♥❡#✱ ❡❧ ❝♦♥❥✉♥-♦ ❞❡ ❝♦♥#✉♠♦ #❡1$❛✿

●!"✜❝♦ ✶

❯♥❛ ❝❛1❛❝-❡1$#-✐❝❛ :✉❡ -✐❡♥❡ ❡❧ ❝♦♥❥✉♥-♦ ❞❡ ❝♦♥#✉♠♦ ❡# :✉❡ ❡# ✉♥ ❝♦♥❥✉♥-♦

❝♦♥✈❡①♦✳ ❊# ❞❡❝✐1✱ ♣❛1❛ ❞♦# ❝❡#-❛# ❞❡ ❜✐❡♥❡# ❝♦♠♦ x, x^′ ∈X ⊆R^l₊ #❡ ❝✉♠♣❧❡

:✉❡ ✉♥❛ ❝♦♠❜✐♥❛❝✐>♥ ❞❡ ❧❛ ❢♦1♠❛x^′′=λx+ (1−λ)x^′ -❛♠❜✐@♥ ❡# ✉♥❛ ❝❡#-❛ ❞❡

X⊆R^l₊ ♣❛1❛ -♦❞♦λ∈[0,1]✳

❊❧ ❝♦♥❥✉♥-♦ ❞❡ ❝♦♥#✉♠♦X -❛♠❜✐@♥ ❝✉♠♣❧❡ ❝♦♥ ❧❛ ♣1♦♣✐❡❞❛❞ ❞❡ #❡1 ❝❡11❛❞♦✳

❊# ❞❡❝✐1✱ #✐ ❡①✐#-❡ ✉♥❛ #✉❝❡#✐>♥ ❞❡ ♣❧❛♥❡# ♣♦#✐❜❧❡# ❞❡ ❝♦♥#✉♠♦{xⁿ_i}:✉❡ ❝♦♥✈❡1❣❡

❛ ❛❧❣B♥ ✈❡❝-♦1x❡♥ ❡❧ ❡#♣❛❝✐♦ ❞❡ ❜✐❡♥❡#✱ ❡♥-♦♥❝❡# ❡#❡ ✈❡❝-♦1 -❛♠❜✐@♥ ❡# ✉♥ ♣❧❛♥

❞❡ ❝♦♥#✉♠♦ ♣♦#✐❜❧❡✳ C♦1 B❧-✐♠♦✱ ❞✐1❡♠♦# :✉❡ X -✐❡♥❡ ✉♥❛ ❝♦-❛ ✐♥❢❡1✐♦1 ♣❛1❛≤✱

❡# ❞❡❝✐1 ❤❛② ✉♥ ❝♦♥#✉♠♦ ♠$♥✐♠♦ ❞❡ #✉❜#✐#-❡♥❝✐❛ ♣❛1❛ ❡❧ ❝♦♥#✉♠✐❞♦1^✹✳

✹❊! ❞❡❝✐&✱ ✉♥ ♣❧❛♥ ❞❡ ❝♦♥!✉♠♦ ♣♦!✐❜❧❡ ❡! ✉♥♦ 0✉❡ ♣♦!❡❛ ♠❡♥♦& ❝❛♥1✐❞❛❞ ❞❡ ❜✐❡♥❡! ❛ ❧❛!

0✉❡ ♣♦!❡❡ ❡❧ ❝♦♥!✉♠✐❞♦& ❝♦♠♦ ❞♦1❛❝✐2♥ ✐♥✐❝✐❛❧✳ ❉❡ ❡!1❛ ❢♦&♠❛ ❡❧ ♠6♥✐♠♦ ❝♦♥❥✉♥1♦ ❞❡ ❝♦♥!✉♠♦

♣♦!✐❜❧❡ ❞❡❧ ❝♦♥!✉♠✐❞♦& ♥♦ ✈❛ ❛ !❡& ❡①❛❝1❛♠❡♥1❡ ❧❛ &✐0✉❡③❛ ❞❡❧ ❝♦♥!✉♠✐❞♦& ✭▲♦③❛♥♦✱ ▼♦♥!❛❧✈❡

② ❱✐❧❧❛✱✶✾✾✾✿✹✽✮✳

❈❆"❮❚❯▲❖ ✶✳ ❈❖◆❙❯▼■❉❖❘❊❙ ✶✶

✶✳✸✳ ❈❛%❛❝'❡%)*'✐❝❛* ❞❡❧ ♠❡%❝❛❞♦

❈♦♠♦ $❡ ♠❡♥❝✐♦♥) ❛♥+❡,✐♦,♠❡♥+❡✱ ❡❧ ❝♦♥$✉♠✐❞♦, ❡♥❢,❡♥+❛ ❛❞❡♠2$ ❞❡ ,❡$+,✐❝✲

❝✐♦♥❡$ ❢4$✐❝❛$✱ ,❡$+,✐❝❝✐♦♥❡$ ❡❝♦♥)♠✐❝❛$✳ ❊$ ❞❡❝✐,✱ ❡❧ ❝♦♥❥✉♥+♦ ❞❡ ❝♦♥$✉♠♦ ❡$+2

❧✐♠✐+❛❞♦ ♣♦, ❧❛ ❝❛♥+✐❞❛❞ ❞❡ ❝❡$+❛$ ❞❡ ❜✐❡♥❡$ :✉❡ ;❧ ♣✉❡❞❛ ❝♦$+❡❛,$❡✳

<❛,❛ ❢♦,♠❛❧✐③❛, ❡$+❛$ ,❡$+,✐❝❝✐♦♥❡$✱ ❡$ ♥❡❝❡$❛,✐♦ ✐♥+,♦❞✉❝✐, ❞♦$ $✉♣✉❡$+♦$✳

<,✐♠❡,♦✱ ✈❛♠♦$ ❛ $✉♣♦♥❡, :✉❡ ❧❛ ❝❛♥+✐❞❛❞ L ❜✐❡♥❡$ $❡ ❝♦♠❡,❝✐❛♥ +♦❞♦$ ❡♥ ❡❧

♠❡,❝❛❞♦ ❛ ♣,❡❝✐♦$ ❡♥ ♣❡$♦$ ❝♦,,✐❡♥+❡$ :✉❡ $♦♥ ❝♦♥♦❝✐❞♦$ ② ♣@❜❧✐❝❛♠❡♥+❡ ❝♦✲

+✐③❛❞♦$✳ ❊$+❡ ♣,✐♥❝✐♣✐♦ ❡$ ❝♦♥♦❝✐❞♦ ❝♦♠♦ ❡❧ ♣,✐♥❝✐♣✐♦ ❞❡ ♠❡,❝❛❞♦$ ❝♦♠♣❧❡+♦$

✭▼❛$✲❈♦❧❡❧❧✱ ❲❤✐♥$+♦♥ ② ●,❡❡♥✱ ✶✾✾✺✿✷✵✮✳

❋♦,♠❛❧♠❡♥+❡ ❧♦$ ♣,❡❝✐♦$ ❡$+2♥ ,❡♣,❡$❡♥+❛❞♦$ ♣♦, ✉♥ ✈❡❝+♦, ❞❡ ♣,❡❝✐♦$

p=















 p1

p2

. : pL

















∈R^l₊

❈♦♥ ❞✐❝❤♦ ✈❡❝+♦, ❡$ ♣♦$✐❜❧❡ ❝♦♥♦❝❡, ❡♥+♦♥❝❡$ ❡❧ ❝♦$+♦ ❡♥ ♣❡$♦$ ❞❡ ❝❛❞❛ ✉♥♦

❞❡ ❧♦$ L❜✐❡♥❡$✳ ❙❡ ✈❛ ❛$✉♠✐, ❞❡ ❛❤♦,❛ ❡♥ ❛❞❡❧❛♥+❛ :✉❡ p≫0✱ ❡$+♦ ❡$pl >0

♣❛,❛ +♦❞♦l✳ ▼2$ ❡$♣❡❝4✜❝❛♠❡♥+❡

p∈R^l₊₊

❊❧ $❡❣✉♥❞♦ $✉♣✉❡$+♦ ❡$ :✉❡ ❧♦$ ♣,❡❝✐♦$ ♥♦ ❡$+2♥ ✐♥✢✉❡♥❝✐❛❞♦$ ♣♦, ❡❧ ❝♦♥$✉✲

♠✐❞♦, ② ♣♦, ❧♦ +❛♥+♦✱ ❧♦$ ❝♦♥$✉♠✐❞♦,❡$ $♦♥ ♣,❡❝✐♦ ❛❝❡♣+❛♥+❡$✳ ❊$+❡ ♣,✐♥❝✐♣✐♦ ❡$

❝♦♥♦❝✐❞♦ ❝♦♠♦ ❡❧ ♣,✐♥❝✐♣✐♦ ❞❡ ❝♦♥❞✉❝+❛ ♣❛,❛♠;+,✐❝❛ ,❡$♣❡❝+♦ ❛ ❧♦$ ♣,❡❝✐♦$✳ ❊$+❡

$✉♣✉❡$+♦ ✈❛ ❞❡ ❧❛ ♠❛♥♦ ❞❡ $✉♣♦♥❡, :✉❡ ❧❛ ❞❡♠❛♥❞❛ ❞❡❧ ❝♦♥$✉♠✐❞♦, ❡$ ❛♣❡♥❛$

✉♥❛ ♣❡:✉❡Q❛ ❢,❛❝❝✐)♥ ❞❡ ❧❛ ❞❡♠❛♥❞❛ ❞❡❧ ♠❡,❝❛❞♦✱ ②❛ :✉❡ $❡ $✉♣♦♥❡ :✉❡ ❤❛② ✉♥

❣,❛♥ ♥@♠❡,♦m❞❡ ❝♦♥$✉♠✐❞♦,❡$ ✭▼❛$✲❈♦❧❡❧❧✱ ❲❤✐♥$+♦♥ ② ●,❡❡♥✱ ✶✾✾✺✿✷✵✮✳

<♦, @❧+✐♠♦✱ ② ♣♦, ❝♦♠♦❞✐❞❛❞✱ ✈❛♠♦$ ❛ $✉♣♦♥❡, :✉❡ ❡❧ ❝♦♥❥✉♥+♦ ❞❡ ❡❧❡❝❝✐)♥

❞❡❧ ❝♦♥$✉♠✐❞♦, ❞❡♣❡♥❞❡ ❞❡ ❞♦$ ❝♦$❛$✿ ❧♦$ ♣,❡❝✐♦$ ❞❡ ♠❡,❝❛❞♦ p② ❧❛ ,❡♥+❛ ❞❡❧

❝♦♥$✉♠✐❞♦,^✺❡♥ ♣❡$♦$ :✉❡ ❞❡♥♦+❛,❡♠♦$m∈R✳ ❚❡♥✐❡♥❞♦ ❡♥ ❝✉❡♥+❛ ❧♦$ ❛♥+❡,✐♦,❡$

$✉♣✉❡$+♦$ ♣❛$❡♠♦$ ❛❤♦,❛ ❛ ❞❡✜♥✐, ❡❧ ❝♦♥❥✉♥+♦ ❞❡ ,❡$+,✐❝❝✐)♥ ♣,❡$✉♣✉❡$+❛,✐❛ ♦ ❞❡

,❡$+,✐❝❝✐♦♥❡$ ❡❝♦♥)♠✐❝❛$✳

✺◆✉❡#$%♦ ♠♦❞❡❧♦ ✈❛ ❛ #✉♣♦♥❡% %❡♥$❛ ❝♦♠♦ ❡/✉✐✈❛❧❡♥$❡ ❛ %✐/✉❡③❛ ❛✉♥/✉❡ ❡❧❧♦ ♥♦ #❡❛ ❡#$%✐❝✲

$❛♠❡♥$❡ ❝✐❡%$♦✳

❈❆"❮❚❯▲❖ ✶✳ ❈❖◆❙❯▼■❉❖❘❊❙ ✶✷

✶✳✹✳ ❈♦♥❥✉♥(♦ ❞❡ +❡,(+✐❝❝✐/♥ ♣+❡,✉♣✉❡,(❛+✐❛

■♥✐❝✐❛❧♠❡♥*❡ +❡ ❞❡❜❡ ❛❝❧❛.❛. /✉❡ ❝❛❞❛ ❡❧❡♠❡♥*♦ ❞❡❧ ❝♦♥❥✉♥*♦ ❞❡ .❡+*.✐❝❝✐♦♥❡+

♣.❡+✉♣✉❡+*❛.✐❛+β✱ ❡+ ❛♥*❡ *♦❞♦ ✉♥ +✉❜❝♦♥❥✉♥*♦ ❞❡❧ ❝♦♥❥✉♥*♦ ❞❡ ❝♦♥+✉♠♦ ❞❡ X∈ R^l₊✳ ❙♦❜.❡ ❡+*❡ ❝♦♥❥✉♥*♦ ❞❡ ❝♦♥+✉♠♦✱ +❡ ♣✉❡❞❡♥ ✐♠♣♦♥❡. ❞✐❢❡.❡♥*❡+ .❡+*.✐❝❝✐♦♥❡+

/✉❡ +♦♥ ♣.❡❝✐+❛♠❡♥*❡ ✐❣✉❛❧❡+ ❛β✳

β ❡+ ❛+9 ✉♥ +✉❜❝♦♥❥✉♥*♦ ❞❡❧ ❝♦♥❥✉♥*♦ ❞❡ ♣♦❞❡. ❞❡ X ∈R^l₊✳ ❊+ ❞❡❝✐.✿

β⊆P(X)\ {∅}

❊❧ +✉❜❝♦♥❥✉♥*♦ ❞❡ ♣♦❞❡.✱ ❡+ ✐❣✉❛❧ ❛ P(X) ={β/β⊆X} ❞♦♥❞❡∅∈/ P(X).

❊+*❛ <❧*✐♠❛ ❝♦♥❞✐❝✐=♥ +❡ ✐♠♣♦♥❡ ♣❛.❛ /✉❡ ❛ +✉ ✈❡③ β 6=∅② ❛+9 ❡❧ ❝♦♥+✉♠✐❞♦.

*❡♥❣❛ ♣♦+✐❜✐❧✐❞❛❞ ❞❡ ❡❧❡❣✐. ❛❧❣♦✳ ❙❡ *✐❡♥❡ ❛+9 /✉❡ ✉♥ +✉❜❝♦♥❥✉♥*♦ ❞❡ X ♣✉❡❞❡

❡+*❛. ❡♥β ♣❡.♦ ✉♥ ❡❧❡♠❡♥*♦ ❞❡ X ♣✉❡❞❡ ♥♦ ❡+*❛. ❡♥β✳ ❊+ ❞❡❝✐.✿

❙✐ B ∈ β✱ ❡♥*♦♥❝❡+ B ∈ X ∈ R^l₊ ② B 6= ∅✳ ❙✐❡♥❞♦ B ❡♥ ❡+*❡ ❝❛+♦ ✉♥

♣.❡+✉♣✉❡+*♦ ②β ❡❧ ❝♦♥❥✉♥*♦ ❞❡ *♦❞♦+ ❧♦+ ♣.❡+✉♣✉❡+*♦+ ✐❣✉❛❧ ❛

β =

B⊆R^l₊/∃(p, m)∈R^l₊₊×R₊:B(p, m) =B

❚❡♥✐❡♥❞♦ ❡♥ ❝✉❡♥*❛ ❧♦ ❛♥*❡.✐♦.✱ ② +✉♣♦♥✐❡♥❞♦ ❝♦♠♦ ❞❛❞♦+ ✉♥ ✈❡❝*♦. ❞❡ ♣.❡❝✐♦+

p ∈ R^l₊₊ ② ✉♥❛ .❡♥*❛ m ∈ R✱ ❡+ ♣♦+✐❜❧❡ ❝♦♥+*.✉✐. ❡❧ +✐❣✉✐❡♥*❡ ❝♦♥❥✉♥*♦ ❞❡

.❡+*.✐❝❝✐=♥ ♣.❡+✉♣✉❡+*❛.✐❛ /✉❡ ❧❧❛♠❛.❡♠♦+ B✱ ♣❛.❛ ✉♥B∈β✳

B(p, m) =

x∈R^l₊/p.x≤m

❙✐❡♥❞♦x✉♥ ♣❧❛♥ ❞❡ ❝♦♥+✉♠♦ ❞❡❧ ❝♦♥❥✉♥*♦ ❞❡ ❝♦♥+✉♠♦ X✳ ❊+ ❞❡❝✐.✿

x∈X⊆R^l₊

❙❡ ❡+*❛❜❧❡❝❡ ✉♥ +✉♣✉❡+*♦ ❛❞✐❝✐♦♥❛❧ ② ❡+ /✉❡ ♥♦ +❡ ✈❛ ❛ *.❛❜❛❥❛. ❝♦♥ ❛❣❡♥*❡+

/✉❡❜.❛❞♦+✳ ❊+ ❞❡❝✐.✱ ❞❡ ❛❝B ❡♥ ❛❞❡❧❛♥*❡ +❡ ✈❛ ❛ +✉♣♦♥❡. m >0✳

❊❧ ❝♦♥❥✉♥*♦ ❞❡ .❡+*.✐❝❝✐=♥ ♣.❡+✉♣✉❡+*❛.✐❛ ❡+✱ ❡♥*♦♥❝❡+✱ ❡❧ ❝♦♥❥✉♥*♦ +♦❜.❡ ❡❧

❝✉❛❧ ❡❧ ❝♦♥+✉♠✐❞♦. ✈❛ ❛ *♦♠❛. +✉+ ❞❡❝✐+✐♦♥❡+ ❞❡ ❝♦♥+✉♠♦ ② ❡+ ❝♦♥♦❝✐❞♦ ❝♦♠♦

❝♦♥❥✉♥*♦ ♣.❡+✉♣✉❡+*❛.✐♦ ✇❛❧.❛+✐❛♥♦ ♦ ❝♦♠♣❡*✐*✐✈♦^✻✳ ❊+ ❞❡❝✐.✱ +✐ B ∈ β /✉✐❡.❡

❞❡❝✐. /✉❡ ❡❧ ❛❣❡♥*❡ ❡♥❢.❡♥*❛ ❡❧ ♣.♦❜❧❡♠❛ ❞❡ ❡+❝♦❣❡. ✉♥ x*❛❧ /✉❡x∈B⊆X✳

✻❊!"❡ "✐♣♦ ❞❡ (❡!"(✐❝❝✐*♥ ♣(❡!✉♣✉❡!"❛(✐❛ ❞❡❥❛ ❛❢✉❡(❛ ♥♦ ❧✐♥❡❛❧✐❞❛❞❡! ❡♥ ❧❛ (❡!"(✐❝❝✐*♥✱ ✐♥❞✐✈✐✲

!✐❜✐❧✐❞❛❞❡! ❡ ✐♥❝❡("✐❞✉♠❜(❡ ❡♥ ❧❛ "❛!❛ ❞❡ ❝❛♠❜✐♦ ❡♥"(❡ ❜✐❡♥❡! ✭❉❡❛"♦♥ ② ▼✉❡❧❧❜❛✉❡(✱ ✶✾✾✸✿✶✹✮✳

❈❆"❮❚❯▲❖ ✶✳ ❈❖◆❙❯▼■❉❖❘❊❙ ✶✸

●!"✜❝♦ ✷

✶✳✺✳ ❊❧ ❝♦♥-✉♠✐❞♦! ② ❧♦- ❝!✐3❡!✐♦- ❞❡ ❡❧❡❝❝✐5♥

❊#$❛❜❧❡❝✐❞❛# ❧❛# ♣-♦♣✐❡❞❛❞❡# ❞❡❧ ❝♦♥❥✉♥$♦ ❞❡ ❝♦♥#✉♠♦ ② ❞❡ -❡#$-✐❝❝✐4♥ ♣-❡✲

#✉♣✉❡#$❛-✐❛✱ #❡ #✉♣♦♥❡ 7✉❡ ❝❛❞❛ ❛❣❡♥$❡ ❞❡ ❧❛ ❡❝♦♥♦♠9❛ $✐❡♥❡ ❞❡✜♥✐❞♦ ✉♥ ❝-✐$❡-✐♦

❞❡ ♦-❞❡♥❛❝✐4♥ #♦❜-❡ #✉ ❝♦♥❥✉♥$♦ ❞❡ ❡❧❡❝❝✐4♥ ❡❧ ❝✉❛❧ ❡#$; #✉❥❡$♦ ❛ ❝✐❡-$❛# -❡#$-✐❝✲

❝✐♦♥❡#✳ ❊❧ ❛❣❡♥$❡ ❝♦♥#✉♠✐❞♦- ❡#$❛-; ❞❡✜♥✐❞♦ ❡♥$♦♥❝❡# ❝♦♠♦ ✉♥❛ $-✐❛❞❛✿

[X,%, B]

❉♦♥❞❡X ❡# ❡❧ ❝♦♥❥✉♥$♦ ❞❡ ❝♦♥#✉♠♦✱%❡# ❧❛ -❡❧❛❝✐4♥ ❜✐♥❛-✐❛ ❞❡ ♣-❡❢❡-❡♥❝✐❛#

#♦❜-❡ ❡❧ ❝♦♥❥✉♥$♦ ❞❡ ❝♦♥#✉♠♦ ②B ❡# ❡❧ ❝♦♥❥✉♥$♦ ♣-❡#✉♣✉❡#$❛-✐♦✳

@♦- ❧♦ $❛♥$♦✱ #❡ $✐❡♥❡ 7✉❡ $-❡# ❡❧❡♠❡♥$♦# ❝❛-❛❝$❡-✐③❛♥ ❡❧ ♣-♦❜❧❡♠❛ ❞❡ ❞❡❝✐#✐4♥

❞❡❧ ❝♦♥#✉♠✐❞♦- ✭❱✐❧❧❛-✱ ✶✾✾✻✿✷✶✮✿

❊❧ ❝♦♥❥✉♥$♦ ❞❡ ❡❧❡❝❝✐4♥X ∈R^l₊

❊❧ ❝-✐$❡-✐♦ ❞❡ ✈❛❧♦-❛❝✐4♥ %

▲❛# -❡#$-✐❝❝✐♦♥❡# #♦❜-❡ ❡❧ ❝♦♥❥✉♥$♦ ❞❡ ❡❧❡❝❝✐4♥ ❡①♣-❡#❛❞❛# ❡♥ B✳

❊❧ ♦❜❥❡$✐✈♦ ❞❡❧ ❝♦♥#✉♠✐❞♦- ❡#✱ ❡♥$♦♥❝❡#✱ ❡❧❡❣✐- ✉♥ ♣❧❛♥ ❞❡ ❝♦♥#✉♠♦ x ❡♥ ✉♥

#✉❜❝♦♥❥✉♥$♦ ♥♦ ✈❛❝✐4 ❞❡ R^l✱ 7✉❡ ❡# #✉ ❝♦♥❥✉♥$♦ ❞❡ ❝♦♥#✉♠♦ X✱ ❞❡ $❛❧ ♠♦❞♦

7✉❡ ♠❛①✐♠✐❝❡ #✉ ✉$✐❧✐❞❛❞✳ @❛-❛ ♣♦❞❡- -❡❛❧✐③❛- #✉ ❡❧❡❝❝✐4♥ ❞❡❧ ♣❧❛♥ ❞❡ ❝♦♥#✉♠♦

4♣$✐♠♦✱ ❡# ♥❡❝❡#❛-✐♦ $❡♥❡- ✉♥ ❝-✐$❡-✐♦ ❞❡ ❝♦♠♣❛-❛❝✐4♥ ② ♦-❞❡♥❛❝✐4♥ ❞❡ ❧♦# ♣❧❛✲

♥❡# ❞❡ ❝♦♥#✉♠♦ ❞❡❧ ❝♦♥❥✉♥$♦ ❞❡ ❝♦♥#✉♠♦✳ ❊#$❡ ❡# ♣-❡❝✐#❛♠❡♥$❡ ❡❧ ❝-✐$❡-✐♦ ❞❡

♦-❞❡♥❛❝✐4♥%❛❧ 7✉❡ ♥♦# -❡❢❡-9❛♠♦# ❛♥$❡# ✭▲♦③❛♥♦✱ ▼♦♥#❛❧✈❡ ② ❱✐❧❧❛✱ ✶✾✾✾✿✹✼✮✳

❈❆"❮❚❯▲❖ ✶✳ ❈❖◆❙❯▼■❉❖❘❊❙ ✶✹

❆#$✱ ❡❧ ♦❜❥❡+✐✈♦ #✐❣✉✐❡♥+❡ ❡# ❡#+❛❜❧❡❝❡3 ❧❛# ♣3♦♣✐❡❞❛❞❡# #♦❜3❡ ❧❛ 3❡❧❛❝✐6♥ ❜✐✲

♥❛3✐❛ ❞❡ ♣3❡❢❡3❡♥❝✐❛# ❞❡✜♥✐❞❛# #♦❜3❡ ❡❧ ❝♦♥❥✉♥+♦ ❞❡ ❡❧❡❝❝✐6♥ ❞❡ +❛❧ ❢♦3♠❛ ;✉❡

❡❧ ❝♦♥#✉♠✐❞♦3 +❡♥❣❛ ✉♥ ❝3✐+❡3✐♦ ❝❧❛3♦ ❞❡ ❡❧❡❝❝✐6♥ #♦❜3❡ #✉ ❝♦♥❥✉♥+♦ ❞❡ ❝♦♥#✉♠♦

❛#❡;✉✐❜❧❡ ♦ ❝♦#+❡❛❜❧❡✳

✶✳✻✳ ❘❡❧❛❝✐)♥ ❞❡ ♣-❡❢❡-❡♥❝✐❛/

=❛3❛ ❡♠♣❡③❛3 #❡ ❞❡❜❡ ❛❝❧❛3❛3 ❡❧ ❝♦♥❝❡♣+♦ ❞❡ ❧♦ ;✉❡ ❡# ✉♥❛ 3❡❧❛❝✐6♥ ❜✐♥❛3✐❛✳

❊#+❛ ❡# ✉♥ ❝3✐+❡3✐♦ ❞❡ ❝♦♠♣❛3❛❝✐6♥ ❡♥+3❡ ❞♦# ❡❧❡♠❡♥+♦# ②✱ ♣❛3❛ ❡❧ ❝❛#♦ ;✉❡

♥♦# ♦❝✉♣❛✱ ❧❡ ✈❛♠♦# ❛ ✐♠♣♦♥❡3 ✉♥ #✐❣♥✐✜❝❛❞♦ ♣❛3❛ ❞❛3❧❡ #❡♥+✐❞♦ ❛ ❧❛ +❡♦3$❛✳

▲❛ 3❡❧❛❝✐6♥ ❜✐♥❛3✐❛ ;✉❡ ✈❛♠♦# ✐♠♣♦♥❡3 ❛❝B ❡# ✉♥❛ 3❡❧❛❝✐6♥ ❞❡ ♣3❡❢❡3❡♥❝✐❛#✱

❡①♣3❡#❛❞❛ ❝♦♠♦%✱ ;✉❡ ✈❛ ❛ ❡#♣❡❝✐✜❝❛3 ❡♥ ❧❛ +❡♦3$❛ ❞❡❧ ❝♦♥#✉♠✐❞♦3 ➫➫❛❧ ♠❡♥♦#

+❛♥ ❜✉❡♥♦ ❝♦♠♦➫➫✳ =❛3❛ ❡❧ ❝❛#♦✱ ❧❛ 3❡❧❛❝✐6♥%❡#+❛❜❧❡❝❡ ♣♦3 ❡❥❡♠♣❧♦ ;✉❡ ❡♥+3❡

❞♦# ♣❧❛♥❡# ❞❡ ❝♦♥#✉♠♦x¹, x²∈X #❡ ♣✉❡❞❡ 3❡❛❧✐③❛3 ❧❛ #✐❣✉✐❡♥+❡ ❝♦♠♣❛3❛❝✐6♥✿

x¹%x², ∀x¹, x²∈X∈R^l

+

❊# ❞❡❝✐3✱x¹ ❡# ❛❧ ♠❡♥♦# +❛♥ ❜✉❡♥♦ ❝♦♠♦x²✳ ❊# ❞❡❝✐3✱ ❧❛ 3❡❧❛❝✐6♥ ❞❡ ♣3❡❢❡3❡♥❝✐❛

♣3❡+❡♥❞❡ #❡3 ✉♥ ❝♦♥❝❡♣+♦ ♣3B❝+✐❝♦ ♣❛3❛ ❞❡♥♦+❛3 ❧❛ +♦♠❛ ❞❡ ❞❡❝✐#✐♦♥❡# ❞❡❧ ❝♦♥✲

#✉♠✐❞♦3 ✭❱❛3✐❛♥✱ ✶✾✾✾✿✸✻✮✳ ❙♦❜3❡ ❧❛ 3❡❧❛❝✐6♥ ❞❡ ♣3❡❢❡3❡♥❝✐❛# #❡ ✈❛♥ ❛ ✐♠♣♦♥❡3

❧♦# #✐❣✉✐❡♥+❡# ❛①✐♦♠❛#✳

✶✳✻✳✶✳ ❆①✐♦♠❛) ❞❡ ♦,❞❡♥

✐✮ ❘❡✢❡①✐✈❛✿

∀x∈X∈R^l₊ #❡ ❝✉♠♣❧❡ ;✉❡ x%x✳

❊# ❞❡❝✐3✱ +♦❞❛ ❝❛♥❛#+❛ ❡# ❛❧ ♠❡♥♦# +❛♥ ❜✉❡♥❛ ❝♦♠♦ ❡❧❧❛ ♠✐#♠❛ ② ♥♦ ❡# ♥❡❝❡#❛3✐♦

❡#♣❡❝✐✜❝❛3 ❡❧ ❝✉B♥+♦ ❞❡ ❧❛ ♣3❡❢❡3❡♥❝✐❛ #♦❜3❡ ❞✐❝❤❛ ❝❛♥❛#+❛ ②❛ ;✉❡ ❧❛ 3❡❧❛❝✐6♥ ❞❡

♣3❡❢❡3❡♥❝✐❛# ❡# ♦3❞✐♥❛❧✳

✐✐✮ ❚*❛♥,✐-✐✈❛✿

∀x¹x², x³∈X∈R^l₊ #❡ ❝✉♠♣❧❡ ;✉❡ #✐x¹%x² ②x²%x³ ❡♥+♦♥❝❡#✿x¹%x³✳

❊# ❞❡❝✐3✱ ❡# ♣♦#✐❜❧❡ ❝♦♠♣❛3❛3 ❞♦# ❝❛♥❛#+❛# ❝♦♥ ✉♥❛ +❡3❝❡3❛ ② ❝♦♥#❡3✈❛3 ❡❧ ♦3✲

❞❡♥❛♠✐❡♥+♦ ❞❡ ❧❛# ❝❛♥❛#+❛#✳ ❊# ✐♠♣♦3+❛♥+❡ ❛❝❧❛3❛3 ;✉❡ ❝✉❛♥❞♦ ❡❧ ❛❣❡♥+❡ ♥♦ ❡#

❈❆"❮❚❯▲❖ ✶✳ ❈❖◆❙❯▼■❉❖❘❊❙ ✶✺

"#❛♥&✐"✐✈♦ &❡ ♣✉❡❞❡ ❥✉❣❛# ❝♦♥ &✉ ✐♥❝♦♥&✐&"❡♥❝✐❛ ❡♥ ❧❛& ❡❧❡❝❝✐♦♥❡& ② ❛##✉✐♥❛#❧♦^✼✳

❈✉❛♥❞♦ &♦❜#❡%&❡ ❞❡✜♥❡♥ ❧❛& ♣#♦♣✐❡❞❛❞❡& ✐✮ ② ✐✐✮ &❡ ❡&"❛❜❧❡❝❡ ✉♥ ♣#❡♦#❞❡♥

❞❡ ♣#❡❢❡#❡♥❝✐❛& &♦❜#❡ ❡❧ ❝♦♥❥✉♥"♦ ❞❡ ❝♦♥&✉♠♦✳ ❆❞❡♠;&%&❡ ❞✐❝❡ <✉❡ ❡& #❛❝✐♦♥❛❧

&✐ ❡& ❝♦♠♣❧❡"❛ ② "#❛♥&✐"✐✈❛✳ ▲❛ ❝♦♠♣❧❡"✐"✉❞ ❧❛ ❞❡✜♥✐♠♦& ❛ ❝♦♥"✐♥✉❛❝✐>♥✳

✐✐✐✮ ❈♦♠♣❧❡(✐(✉❞✿

▲❛ #❡❧❛❝✐>♥ ❜✐♥❛#✐❛%❡♥R❞❡✜♥✐❞❛ &♦❜#❡ X ∈R^l₊ ❡& ❝♦♠♣❧❡"❛ ✭♦ "♦"❛❧✮ &✐✿

∀x¹, x²∈X ∈R^l

+ &❡ ❝✉♠♣❧❡ <✉❡ x¹%x² >x²%x¹

❊& ❞❡❝✐#✱ ❡❧ ❛❣❡♥"❡ ❡&"❛ ❡♥ ❝❛♣❛❝✐❞❛❞ ❞❡ ❞❡❝✐❞✐# ❡♥"#❡ ❞♦& ❛❧"❡#♥❛"✐✈❛&✳

❊& #❛③♦♥❛❜❧❡ ❛&D ❡❧ ❛①✐♦♠❛ ❞❡ ❧❛ ❝♦♠♣❧❡"✐"✉❞ ❞❡❜✐❞♦ ❛ <✉❡ &✐ ❧♦& ❛❣❡♥"❡&

❝♦♥♦❝❡♥ ❛❞❡❝✉❛❞❛♠❡♥"❡ ❧♦& ❜✐❡♥❡& ✭❧♦ ❝✉❛❧ ❡& &✉♣✉❡&"♦✮ ❞❡❜❡#D❛♥ &❛❜❡# <✉F ❧❡&

❣✉&"❛♥ ♠;& ② <✉F ❧❡& ❣✉&"❛ ♠❡♥♦&✱ &✐❡♥❞♦ ♣#❡❝✐&❛♠❡♥"❡ ❡&"♦ ❧♦ <✉❡ ❡&♣❡❝D✜❝❛ ❡❧

♣#✐♥❝✐♣✐♦ ❞❡ ❝♦♠♣❧❡"✐"✉❞ ✭❈❛#✈❛❥❛❧ ② ❘✐❛&❝♦&✱ ✷✵✵✻✿✹✮✳

✐✈✮ ■♥❞✐❢❡0❡♥❝✐❛✿

❙✐ ♣❛#❛ ❛❧❣M♥ x¹, x² ∈X ∈R^l₊ &❡ ❝✉♠♣❧❡ <✉❡x¹%x² ②x² %x¹ ❡♥"♦♥❝❡&

&❡ ❞✐❝❡ <✉❡✿

x¹∼x²

❉♦♥❞❡∼❡&♣❡❝D✜❝❛ ✉♥❛ #❡❧❛❝✐>♥ ❞❡ ✐♥❞✐❢❡#❡♥❝✐❛ ❡♥"#❡ ❞♦& ❝❡&"❛&✳

✈✮ 30❡❢❡0❡♥❝✐❛ ❡4(0✐❝(❛✿

O❛#❛ ❛❧❣M♥ x¹, x² ∈ X ∈ R^l₊ &❡ ❡&"❛❜❧❡❝❡#; ❧❛ #❡❧❛❝✐>♥ ≻✱ <✉❡ ❡&♣❡❝✐✜❝❛#;

➫➫♠❡❥♦# ❛➫➫✱ ♣❛#❛ ❡❧ ❝❛&♦ &✐❣✉✐❡♥"❡✿

x¹≻x²⇐⇒ ♥♦ ❡& ❝✐❡#"♦ <✉❡x²%x¹

❊&"❛ #❡❧❛❝✐>♥ ❞❡ ♣#❡❢❡#❡♥❝✐❛ ❡&"#✐❝"❛ ❡& ✉♥ ❝♦♥❝❡♣"♦ ♣#;❝"✐❝♦✳ ❙✐ ❡❧ ❝♦♥&✉♠✐❞♦#

♣#❡✜❡#❡ ✉♥❛ ❝❡&"❛ ❛ ♦"#❛✱ &✐❣♥✐✜❝❛ <✉❡ ❡❧❡❣✐#; ❧❛ ❝❡&"❛ <✉❡ ♣#❡✜❡#❡ ♣♦#<✉❡ ❡&";

✼❙✉♣♦♥❣❛♠♦( )✉❡ ❡❧ ❛❣❡♥,❡ ♥♦ ❡( ,-❛♥(✐,✐✈♦ ② ♣❛(❛ ❧❛ (✐❣✉✐❡♥,❡ -❡❧❛❝✐2♥✿x¹%x²②x²%x³

♣❡-♦x³ % x¹✳ ❊♥ ❡(,❛ (✐,✉❛❝✐2♥ ♦,-♦ ❛❣❡♥,❡ ♣♦❞-7❛ ❝♦♠♣-❛-x¹ ② ❝❛♠❜✐9-(❡❧♦ ❛❧ ❛❣❡♥,❡ ❡♥

❝✉❡(,✐2♥ ♣♦-x²✳ ❆ (✉ ✈❡③✱x²❧♦ ♣♦❞-7❛ ❝❛♠❜✐❛- ♣♦-x³♣❛-❛ ❞❡(♣✉=( ❝❛♠❜✐❛-❧❡x³♣♦-x¹♠9(

✉♥ -❡♠❛♥❡♥,❡ ❞❛❞❛ ❧❛ ♣-❡❢❡-❡♥❝✐❛ ❞❡❧ ❛❣❡♥,❡ ❤❛❝✐❛x¹✳ ❊♥ ❡❧ ❧7♠✐,❡✱ (✐ (❡ -❡♣✐,❡ ❧❛ ♦♣❡-❛❝✐2♥

❛♥,❡-✐♦- ❡( ♣♦(✐❜❧❡ )✉✐,❛-❧❡ ,♦❞♦ ❛❧ ❛❣❡♥,❡ ② ❛(7 ❞❡❥❛-❧♦ (✐♥ ♥❛❞❛✳ A♦- B❧,✐♠♦✱ ❡( ♥❡❝❡(❛-✐♦

-❡❝♦-❞❛- )✉❡ ❧❛ ,-❛♥(✐,✐✈✐❞❛❞ ❡①✐❣❡ ❝♦♥♦❝✐♠✐❡♥,♦ ❝♦♠♣❧❡,♦ ❞❡ ❧❛( ❛❧,❡-♥❛,✐✈❛(✳