❖❚■▼■❩❆➬➹❖ P❘➪❚■❈❆ ❯❙❆◆❉❖
❖ ▲❆●❘❆◆●■❆◆❖ ❆❯▼❊◆❚❆❉❖
❏♦sé ▼❛r✐♦ ▼❛rtí♥❡③
❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❆♣❧✐❝❛❞❛
■▼❊❈❈✲❯◆■❈❆▼P
✷✷ ❞❡ ♥♦✈❡♠❜r♦ ❞❡ ✷✵✵✻✳ ❆t✉❛❧✐③❛❞♦✿ ✶✶ ❞❡ ❥✉❧❤♦ ❞❡ ✷✵✵✼❀
✶✷ ❞❡ ❛❣♦st♦ ❞❡ ✷✵✵✽✱ ✷✶ ❞❡ ♦✉t✉❜r♦ ❞❡ ✷✵✵✽✱ ✸✶ ❞❡ ♠❛rç♦
❞❡ ✷✵✵✾✱ ✶✹ ❞❡ ❛❜r✐❧ ❞❡ ✷✵✵✾✱ ✸ ❞❡ ❛❣♦st♦ ❞❡ ✷✵✵✾✳
Pref´
acio
❊st❛s ♥♦t❛s s❡ ❞❡st✐♥❛♠ ❛ ❢♦r♥❡❝❡r ♦s ❢✉♥❞❛♠❡♥t♦s t❡ór✐❝♦s ❞❡ ❆▲●❊◆❈❆◆✱ ♦ ♠ét♦❞♦ ❞❡ ▲❛❣r❛♥❣✐❛♥♦ ❆✉♠❡♥t❛❞♦ ❞♦ Pr♦❥❡t♦ ❚❆◆●❖✳ ✭❱❡❥❛
✇✇✇✳✐♠❡✳✉s♣✳❜r✴∼❡❣❜✐r❣✐♥✴t❛♥❣♦
♦✉
✇✇✇✳✐♠❡✳✉♥✐❝❛♠♣✳❜r✴∼♠❛rt✐♥❡③✴s♦❢t✇❛r❡✳✮
❆♦ ♠❡s♠♦ t❡♠♣♦✱ ♣r♦❝✉r❛♠ ❝♦❧♦❝❛r ♥❛ ♠❡s♠❛ ♣❡rs♣❡❝t✐✈❛ ❛s ❈♦♥❞✐çõ❡s ❞❡ ❖t✐♠❛❧✐❞❛❞❡ ❞❡ Pr♦❣r❛♠❛çã♦ ◆ã♦✲▲✐♥❡❛r ❡ ♦s ❝r✐tér✐♦s ❞❡ ♣❛r❛❞❛ ✉s❛❞♦s ❡♠ ❛❧❣♦✲ r✐t♠♦s ♣rát✐❝♦s✳
❆ ♣❡r❣✉♥t❛ s✉❜❥❛❝❡♥t❡ ❡♠ ❝❛❞❛ ❝❛♣ít✉❧♦ ❞❡st❛s ♥♦t❛s é✿ ❉❡ ❛❝♦r❞♦ ❝♦♠ ❛ ❚❡♦r✐❛✱ ♦ q✉ê ❛❝♦♥t❡❝❡ ♥❛ Prát✐❝❛❄
❆ ❚❡♦r✐❛ é t♦t❛❧♠❡♥t❡ ❞❡s❡♥✈♦❧✈✐❞❛ ♥♦ ♠✉♥❞♦ ❞♦ ❝♦♥tí♥✉♦✱ ❡ss❡♥❝✐❛❧♠❡♥t❡ IRn✱
❡♥q✉❛♥t♦ ❛ Prát✐❝❛ ♦ ❢❛③ ♥♦ ✉♥✐✈❡rs♦ ♣❛r❛❧❡❧♦ ❞♦s ♥ú♠❡r♦s r❡♣r❡s❡♥tá✈❡✐s ❡♠ ♣♦♥t♦ ✢✉t✉❛♥t❡✳ P♦rt❛♥t♦✱ ❤á ❢❡♥ô♥❡♥♦s ❞❛ Prát✐❝❛ q✉❡ ❛ ❚❡♦r✐❛ ♥ã♦ ♣♦❞❡ ♣r❡✈❡r✳ ◆♦ss♦ ♣♦♥t♦ ❞❡ ✈✐st❛ é ❡①♣❧♦r❛r ❛♦ ♠á①✐♠♦ ❛ ❡①♣❧✐❝❛çã♦ ❞♦s ❝♦♠♣♦rt❛♠❡♥t♦s ♣rát✐❝♦s q✉❡ s✐♠ ♣♦❞❡♠ s❡r ❡①♣❧✐❝❛❞♦s ♣❡❧❛ ❚❡♦r✐❛✳ ❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱ ❝♦♥s✐❞❡r❛♠♦s q✉❡ ❛ ❚❡♦r✐❛ r❡❧❡✈❛♥t❡ é ❛ q✉❡ ❝♦♥tr✐❜✉✐ ❛ ❡①♣❧✐❝❛r ❢❡♥ô♠❡♥♦s ♣rát✐❝♦s✳ ❉✐❛♥t❡ ❞❡ ❝❛❞❛ r❡s✉❧t❛❞♦ t❡ór✐❝♦ ♥♦s ♣❡r❣✉♥t❛r❡♠♦s✿ ❉❡ ❛❝♦r❞♦ ❝♦♠ ✐st♦✱ q✉ê ❞❡✈❡ ❛❝♦♥t❡❝❡r ❝♦♠ ♣r♦❣r❛♠❛s ❝♦♠♣✉t❛❝✐♦♥❛✐s q✉❡ ✐♠♣❧❡♠❡♥t❛♠ ❡st❡ ❛❧❣♦r✐t♠♦❄ ❙❡ ❛ r❡s♣♦st❛ ❢♦r ♣♦✉❝♦ s✐❣♥✐✜❝❛t✐✈❛✱ ♦ r❡s✉❧t❛❞♦ t❡ór✐❝♦ s❡rá ✐rr❡❧❡✈❛♥t❡ ♦✉ ✐♥❝♦♠♣❧❡t♦✳
◆❛ ❛♥á❧✐s❡ ❞♦ ▼ét♦❞♦ ❞❡ ▲❛❣r❛♥❣✐❛♥♦ ❆✉♠❡♥t❛❞♦ s❡❣✉✐♠♦s✱ ✐♠♣❧✐❝✐t❛♠❡♥t❡✱ ♦ s❡❣✉✐♥t❡ Pr♦t♦❝♦❧♦✱ ♦ q✉❛❧ ♣♦❞❡ s❡r ❛♣❧✐❝❛❞♦ à ❛♥á❧✐s❡ t❡ór✐❝❛ ❞❡ q✉❛❧q✉❡r ❛❧❣♦r✐t♠♦ ❞❡ Pr♦❣r❛♠❛çã♦ ◆ã♦ ▲✐♥❡❛r✿
✭❛✮ ❊st✉❞❛r ❡♠ q✉é ❝♦♥❞✐çõ❡s ♦ ❛❧❣♦r✐t♠♦ ❡♥❝♦♥tr❛ ♠✐♥✐♠✐③❛❞♦r❡s ❣❧♦❜❛✐s✳ ❆❧✲ ❣✉♥s ❛❧❣♦r✐t♠♦s✱ ❝♦♠♦ ♦ ❞❡ ▲❛❣r❛♥❣✐❛♥♦ ❆✉♠❡♥t❛❞♦✱ ❝♦♥✈❡r❣❡♠ ❛ ♠✐♥✐♠✐③❛❞♦r❡s ❣❧♦❜❛✐s s❡ s❡ s✉♣õ❡ q✉❡ ♠✐♥✐♠✐③❛❞♦r❡s ❣❧♦❜❛✐s ❞❡ ♣r♦❜❧❡♠❛s ♠❛✐s s✐♠♣❧❡s ♣♦❞❡♠ s❡r ❡♥❝♦♥tr❛❞♦s✳ ❆❝r❡❞✐t❛♠♦s q✉❡ ést❛ é ✉♠❛ ✈✐rt✉❞❡ t❡ór✐❝❛ ❝♦♠ ✐♠♣❧✐❝❛çõ❡s ♣rát✐❝❛s✳
✭❜✮ ❊st✉❞❛r ❛s ♣r♦♣r✐❡❞❛❞❡s ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛ ❞♦ ❛❧❣♦r✐t♠♦ ❞♦ ♣♦♥t♦ ❞❡ ✈✐st❛ ❞❛ ❛❞♠✐ss✐❜✐❧✐❞❛❞❡✳ ❆ ♣❡r❣✉♥t❛ é s❡ ♦ ♠ét♦❞♦ ❝♦♥✈❡r❣❡ ❛ ♣♦♥t♦s q✉❡ s❛t✐s❢❛③❡♠ ❛s r❡str✐çõ❡s ❞♦ ♣r♦❜❧❡♠❛✳ ❉❡✈❡ ❤❛✈❡r ✉♠❛ r❡s♣♦st❛ ♠❡s♠♦ q✉❡ ♦ ♣r♦❜❧❡♠❛ ♥ã♦ ♣♦ss✉❛ ♣♦♥t♦s ✈✐á✈❡✐s✱ ♣♦r ❡①❡♠♣❧♦✱ ♦ ❛❧❣♦r✐t♠♦ ♣♦❞❡ ❝♦♥✈❡r❣❡r ❛ ♣♦♥t♦s ❡st❛❝✐♦✲
♥ár✐♦s ❞❡ ✉♠❛ ♠❡❞✐❞❛ ❞❡ ✐♥✈✐❛❜✐❧✐❞❛❞❡✳ ❙❡ s❡ s✉♣õ❡ ❧✐♠✐t❛çã♦ ❞❛ s❡qüê♥❝✐❛ ❣❡r❛❞❛ ♣❡❧♦ ❛❧❣♦r✐t♠♦✱ ❞❡✈❡ ❤❛✈❡r ❝♦♥❞✐çõ❡s s✉✜❝✐❡♥t❡s r❛③♦á✈❡✐s r❡❧❛t✐✈❛s ❛♦ ♣r♦❜❧❡♠❛ q✉❡ ❣❛r❛♥t❛♠ q✉❡ ❛ ❧✐♠✐t❛çã♦ ❞❛ s❡qüê♥❝✐❛ ❛❝♦♥t❡❝❡✳
✭❝✮ Pr♦✈❛r q✉❡ s❡ ✉♠ ♣♦♥t♦ ❧✐♠✐t❡ ✈✐á✈❡❧ s❛t✐s❢❛③ ✉♠❛ ❝♦♥❞✐çã♦ ❞❡ q✉❛❧✐✜❝❛çã♦✱ ❛s ❝♦♥❞✐çõ❡s ❑❑❚ sã♦ s❛t✐s❢❡✐t❛s ♥❡ss❡ ♣♦♥t♦✳ ❆ ❝♦♥❞✐çã♦ ❞❡ q✉❛❧✐✜❝❛çã♦ ❞❡✈❡ s❡r tã♦ ❢r❛❝❛ q✉❛♥t♦ ♣♦ssí✈❡❧✳ ❉❡✈❡♠ s❡r ✐❞❡♥t✐✜❝❛❞❛s ❝♦♥❞✐çõ❡s s❛t✐s❢❡✐t❛s ♣❡❧❛ s❡qüê♥❝✐❛ ❣❡r❛❞❛ ♣❡❧♦ ❛❧❣♦r✐t♠♦ q✉❡ ♣♦ss❛♠ s❡r ✉s❛❞❛s ❝♦♠♦ ❝r✐tér✐♦ ❞❡ ♣❛r❛❞❛✳
✭❞✮ Pr♦✈❛r q✉❡ s❡ ♦ ❛❧❣♦r✐t♠♦ ❝♦♥✈❡r❣❡ ❛ ✉♠ ♣♦♥t♦ ❡st❛❝✐♦♥ár✐♦ ❝♦♠ ❝♦♥❞✐✲ çõ❡s ❛❞✐❝✐♦♥❛✐s tã♦ ❢r❛❝❛s q✉❛♥t♦ ♣♦ssí✈❡❧✱ ❜♦❛s ♣r♦♣r✐❡❞❛❞❡s ❞❡ ❡st❛❜✐❧✐❞❛❞❡ ♦✉ ✈❡❧♦❝✐❞❛❞❡ sã♦ s❛t✐s❢❡✐t❛s✳
❖ ♣r♦❜❧❡♠❛ ❞❡ Pr♦❣r❛♠❛çã♦ ◆ã♦✲▲✐♥❡❛r ❛ q✉❡ ♥♦s r❡❢❡r✐♠♦s ♥❡st❡ t❡①t♦ é ♦ ❞❡ ♠✐♥✐♠✐③❛r ✉♠❛ ❢✉♥çã♦ ❝♦♥tí♥✉❛ ❝♦♠ r❡str✐çõ❡s ❞❡ ✐❣✉❛❧❞❛❞❡ ❡ ❞❡s✐❣✉❛❧❞❛❞❡✱ ❝♦♥tí♥✉❛s✳ ❊♠ ♥❡♥❤✉♠ ♠♦♠❡♥t♦ ❢❛r❡♠♦s s✉♣♦s✐çõ❡s ❞❡ ❝♦♥✈❡①✐❞❛❞❡✳ ❖ ▼ét♦❞♦ ❞❡ ▲❛❣r❛♥❣✐❛♥♦ ❆✉♠❡♥t❛❞♦ ♣♦ss✉✐ ✉♠❛ t❡♦r✐❛ ❡①tr❡♠❛♠❡♥t❡ r✐❝❛ q✉❛♥❞♦ ❛♣❧✐❝❛❞♦ ❛ ♣r♦❜❧❡♠❛s ❝♦♥✈❡①♦s ✭✈❡r✱ ♣♦r ❡①❡♠♣❧♦✱ ♦ s✉r✈❡② ❞❡ ■✉s❡♠ ❬✶✽❪✮ ♠❛s ❡❧❛ ♥ã♦ s❡rá ❝♦♥s✐❞❡r❛❞❛ ❛q✉✐✳
❖ ▼ét♦❞♦ ❞❡ ▲❛❣r❛♥❣✐❛♥♦ ❆✉♠❡♥t❛❞♦ ♥ã♦ é ✉♠ ♠ét♦❞♦ ❞❡ ♠♦❞❛ ♥❛ ♣❡sq✉✐s❛ ❝♦rr❡♥t❡ ✭❛♥♦ ✷✵✵✻✮ ❞❡ Pr♦❣r❛♠❛çã♦ ◆ã♦✲▲✐♥❡❛r✳ ❊♥tr❡t❛♥t♦✱ ❛s ♠♦❞❛s ♣❛ss❛♠✱ ❡ ❡st❡ ♠ét♦❞♦ ♣♦ss✉✐ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❝❛r❛❝t❡ríst✐❝❛s ❞✐❢í❝❡✐s ❞❡ ❡♠✉❧❛r ♣♦r ♦✉tr♦s ❛❧❣♦r✐t♠♦s ❞❡ P◆▲✿
✶✳ ❆❧t♦ ❣r❛✉ ❞❡ ♠♦❞✉❧❛r✐❞❛❞❡✿ ❛ ❡✜❝✐ê♥❝✐❛ ❞♦ ♠ét♦❞♦ ❞❡♣❡♥❞❡ ❡♠ ❣r❛♥❞❡ ♠❡✲ ❞✐❞❛ ❞❛ ❡✜❝✐ê♥❝✐❛ ❞❡ ❛❧❣♦r✐t♠♦s ♣❛r❛ ♠✐♥✐♠✐③❛r ❡♠ ❝❛✐①❛s✳ ❖ ♣r♦❜❧❡♠❛ ❞❡ ♠✐♥✐♠✐③❛r ❡♠ ❝❛✐①❛s é r❡❧❛t✐✈❛♠❡♥t❡ s✐♠♣❧❡s ❡ ♦s ♣r♦❣r❡ss♦s ♥❡st❛ ár❡❛ sã♦ ♣❡r♠❛♥❡♥t❡s✳ ❚❛✐s ♣r♦❣r❡ss♦s s❡ r❡✢❡t❡♠ r❛♣✐❞❛♠❡♥t❡ q✉❛♥❞♦ ✐♥❝♦r♣♦r❛❞♦s à ♠❡t♦❞♦❧♦❣✐❛ ❞❡ ▲❛❣r❛♥❣✐❛♥♦ ❆✉♠❡♥t❛❞♦✳
✷✳ ❯♠❛ ❝♦♥s❡qüê♥❝✐❛ ❞♦ ít❡♠ ❛♥t❡r✐♦r q✉❡ ♠❡r❡❝❡ ❞❡st❛q✉❡ s❡ r❡❢❡r❡ ❛♦s ♣r♦❜❧❡✲ ♠❛s ❞❡ ❣r❛♥❞❡ ♣♦rt❡✱ ♦♥❞❡ ♦ ♥ú♠❡r♦ ❞❡ ✈❛r✐á✈❡✐s ♦✉ ♦ ♥ú♠❡r♦ ❞❡ r❡str✐çõ❡s é ♠✉✐t♦ ❣r❛♥❞❡✳ ◆❡st❡ ❝❛s♦✱ ❛ ❞✐✜❝✉❧❞❛❞❡ s❡ tr❛♥s❧❛❞❛ t♦t❛❧♠❡♥t❡ ❛♦ s✉❜♣r♦✲ ❜❧❡♠❛ ❞❡ ♠✐♥✐♠✐③❛r ❝♦♠ r❡str✐çõ❡s s✐♠♣❧❡s✳ P❛r❛ ❡st❡ s✉❜♣r♦❜❧❡♠❛ é ✉s✉❛❧ q✉❡ ❜♦♥s ❛❧❣♦r✐t♠♦s ♣❛r❛ ❧✐❞❛r ❝♦♠ ♦ ❣r❛♥❞❡ ♣♦rt❡ ♣♦ss❛♠ s❡r ❞❡✜♥✐❞♦s✳ ✸✳ ❖ ♠ét♦❞♦ ♣♦❞❡ s❡r ✉s❛❞♦ q✉❛♥❞♦ ❛s r❡str✐çõ❡s sã♦ ♥❛t✉r❛❧♠❡♥t❡ ❞✐✈✐❞✐❞❛s
❡♠ ❞♦✐s ❣r✉♣♦s✿ ❢á❝❡✐s ❡ ❞✐❢í❝❡✐s✳ ◆❡st❡ ❝❛s♦✱ ❛s r❡str✐çõ❡s ❢á❝❡✐s ♣❡r♠❛♥❡❝❡♠ ❡♠ ✉♠ ♥í✈❡❧ ✐♥❢❡r✐♦r ❡ ❛ té❝♥✐❝❛ ❞❡ ▲❛❣r❛♥❣✐❛♥♦ ❆✉♠❡♥t❛❞♦ s❡ ❛♣❧✐❝❛ às r❡s✲ tr✐çõ❡s ❞✐❢í❝❡✐s✳ ❖s s✉❜♣r♦❜❧❡♠❛s ♣♦❞❡♠ s❡r r❡s♦❧✈✐❞♦s ♣♦r q✉❛❧q✉❡r ♠ét♦❞♦ s❛❜✐❞❛♠❡♥t❡ ❡✜❝✐❡♥t❡ ❝♦♠ ❡ss❡ t✐♣♦ ❞❡ r❡str✐çõ❡s s✐♠♣❧❡s✳ ❖ ❝❛s♦ ❞❡ ❝❛✐①❛s ♥❛❞❛ ♠❛✐s é q✉❡ ✉♠ ❝❛s♦ ♣❛rt✐❝✉❧❛r ❞❡ r❡str✐çã♦ s✐♠♣❧❡s✳
Pref´acio v ✺✳ ❆ ❝♦♥✈❡r❣ê♥❝✐❛ ❛ ♠✐♥✐♠✐③❛❞♦r❡s ❣❧♦❜❛✐s ❞❡♣❡♥❞❡ ❛♣❡♥❛s ❞❡ q✉❡ ❡ss❛ ♣r♦♣r✐✲
❡❞❛❞❡ ❡st❡❥❛ ♣r❡s❡♥t❡ ♣❛r❛ ♦s s✉❜♣r♦❜❧❡♠❛s s✐♠♣❧❡s✳
✻✳ ❈❛r❛❝t❡ríst✐❝❛s q✉❡ ❞✐✜❝✉❧t❛♠ ❛ ❛♣❧✐❝❛çã♦ ❞❡ ♦✉tr♦s ♠ét♦❞♦s ✭❝♦♠♦ ♦ ❡①❝❡ss♦ ❞❡ r❡str✐çõ❡s ❝♦♠ ❛ ❝♦♥s❡q✉❡♥t❡ ❞✐♠✐♥✉✐çã♦ ❞❡ ❣r❛✉s ❞❡ ❧✐❜❡r❞❛❞❡✮ ♥ã♦ ❛❢❡t❛♠ ♦ ♠ét♦❞♦ ❞❡ ▲❛❣r❛♥❣✐❛♥♦ ❆✉♠❡♥t❛❞♦✳
✼✳ ❖ ♠ét♦❞♦ ♣♦❞❡ s❡r ✉s❛❞♦ ❝♦♠ ♣❧❡♥❛ t❡♦r✐❛ ❞❡ ❝♦♥✈❡r❣ê♥❝✐❛ ♣❛r❛ ❛❧❣✉♥s ♣r♦✲ ❜❧❡♠❛s ♥ã♦ ❞✐❢❡r❡♥❝✐á✈❡✐s ❞❡ ✐♠♣♦rtâ♥❝✐❛ ♣rát✐❝❛✱ ❝♦♠♦ ♦ ❞❡ ❖t✐♠✐③❛çã♦ ❞♦ ▼❡♥♦r ❱❛❧♦r ❖r❞❡♥❛❞♦✳
❊✈✐❞❡♥t❡♠❡♥t❡✱ ❛ r❡♣✉t❛çã♦ ❞❡ ✉♠ ♠ét♦❞♦ ❞❡ ❖♣t✐♠✐③❛çã♦ ❡stá ❧✐❣❛❞❛ à q✉❛❧✐✲ ❞❛❞❡ ❞♦s ♣r♦❣r❛♠❛s q✉❡ ♦ ✐♠♣❧❡♠❡♥t❛♠✳ P♦r ❡ss❡ ♠♦t✐✈♦✱ ♥♦ss♦ s♦❢t✇❛r❡ ❆❧❣❡♥❝❛♥✱ ❞♦ ♣r♦❥❡t♦ ❚❆◆●❖✱ é ♦❜❥❡t♦ ❞❡ ❛t✉❛❧✐③❛çã♦ ♣❡r♠❛♥❡♥t❡✱ ♠✉✐t❛s ✈❡③❡s ♦r✐❡♥t❛❞❛ ♣❡❧❛s ✐♥q✉✐❡t✉❞❡s ❡ ❞✐✜❝✉❧❞❛❞❡s ❞♦s ✉s✉ár✐♦s✳ ▼✉✐t♦s ♣r♦❜❧❡♠❛s ♣rát✐❝♦s ❞❡ Pr♦✲ ❣r❛♠❛çã♦ ◆ã♦✲▲✐♥❡❛r ♥ã♦ sã♦ ❢á❝❡✐s ❡ s✉❛ ❛❜♦r❞❛❣❡♠ ✐♥❣ê♥✉❛ ❝♦st✉♠❛ r❡❞✉♥❞❛r ❡♠ ❢r❛❝❛ss♦✳ ❊♥tr❡t❛♥t♦✱ ♠✉✐t❛s ✈❡③❡s ♦ s✉❝❡ss♦ ✈❡♠ ❞❡ ♣❡q✉❡♥❛s ♠❛♥✐♣✉❧❛çõ❡s q✉❡✱ ♣❛r❛ ♦ ✉s✉ár✐♦ ❡①♣❡r✐♠❡♥t❛❞♦✱ ❛❝❛❜❛♠ s❡♥❞♦ ❢❛♠✐❧✐❛r❡s✳
Sum´
ario
✶ ■♥tr♦❞✉çã♦ ✶
✷ ▼ét♦❞♦ ❞❡ P❡♥❛❧✐❞❛❞❡ ❊①t❡r♥❛ ✸
✷✳✶ P❡♥❛❧✐❞❛❞❡ ❊①t❡r♥❛ P✉r❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✷✳✷ P❡♥❛❧✐❞❛❞❡ ❊①t❡r♥❛ ❝♦♠ ❈♦♥tr♦❧❡ ❞❡ ❆❞♠✐ss✐❜✐❧✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✼
✸ ▼ét♦❞♦ ❞❡ P❡♥❛❧✐❞❛❞❡ ❉❡s❧♦❝❛❞❛ ✶✶
✸✳✶ P❡♥❛❧✐❞❛❞❡ ❉❡s❧♦❝❛❞❛ ❝♦♠ ❈♦♥tr♦❧❡ ❞❡ ❆❞♠✐ss✐❜✐❧✐❞❛❞❡ ✳ ✳ ✳ ✳ ✶✻ ✸✳✷ ❈á❧❝✉❧♦ ❞♦s ❉❡s❧♦❝❛♠❡♥t♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾
✹ ❈♦♥❞✐çõ❡s ❞❡ ❖t✐♠❛❧✐❞❛❞❡ ❡ ❈r✐tér✐♦s ❞❡ P❛r❛❞❛ ✷✸
✹✳✶ ❈♦♥❞✐çõ❡s ❞❡ ◗✉❛❧✐✜❝❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✹✳✷ ❈P▲❉ é ✉♠❛ ❈♦♥❞✐çã♦ ❞❡ ◗✉❛❧✐✜❝❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✹✳✸ ❈r✐tér✐♦s ❞❡ P❛r❛❞❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷
✺ ❙✉❜♣r♦❜❧❡♠❛s ■rr❡str✐t♦s ✸✼
✺✳✶ ❆❧❣♦r✐t♠♦ ●❡r❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✺✳✷ ❇♦❛ ❉❡✜♥✐çã♦ ❡ ❈♦♥✈❡r❣ê♥❝✐❛ ●❧♦❜❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵ ✺✳✸ ❈♦♥✈❡r❣ê♥❝✐❛ ▲♦❝❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ ✺✳✹ ❈á❧❝✉❧♦ ❞❛s ❉✐r❡çõ❡s ❞❡ ❇✉s❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵ ✺✳✹✳✶ ◆❡✇t♦♥✲❚r✉♥❝❛❞♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶ ✺✳✹✳✷ ◗✉❛s❡✲◆❡✇t♦♥ ❡ Pr❡❝♦♥❞✐❝✐♦♥❛❞♦r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷
✻ ❙✉❜♣r♦❜❧❡♠❛s ❝♦♠ ❘❡str✐çõ❡s ❙✐♠♣❧❡s ✺✼
✻✳✶ Pr✐♥❝í♣✐♦ ❋♦rt❡ ❞❛s ❘❡str✐çõ❡s ❆t✐✈❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✽ ✻✳✷ Pr✐♥❝í♣✐♦ Prát✐❝♦ ❞❡ ❘❡str✐çõ❡s ❆t✐✈❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✾ ✻✳✷✳✶ ❊sq✉❡♠❛ ❜ás✐❝♦ ❞❡ ●❡♥❝❛♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✵
✼ ❈♦♥✈❡r❣ê♥❝✐❛ ●❧♦❜❛❧ ✻✾
✼✳✶ ❈♦♥✈❡r❣ê♥❝✐❛ ❛ ♣♦♥t♦s ❛❞♠✐ssí✈❡✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✶ ✼✳✷ ❈♦♥✈❡r❣ê♥❝✐❛ ❛ ♣♦♥t♦s ❑❑❚ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✺
✽ ▲✐♠✐t❛çã♦ ❞♦ P❛râ♠❡tr♦ ❞❡ P❡♥❛❧✐❞❛❞❡ ✽✶ ✽✳✶ ❘❡str✐çõ❡s ❞❡ ■❣✉❛❧❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✶ ✽✳✶✳✶ ❍✐♣ót❡s❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✸ ✽✳✶✳✷ ❚❡♦r❡♠❛ ♣r✐♥❝✐♣❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✹ ✽✳✷ ❘❡str✐çõ❡s ❞❡ ❉❡s✐❣✉❛❧❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✽ ✽✳✷✳✶ ❍✐♣ót❡s❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✽ ✽✳✷✳✷ ❚❡♦r❡♠❛ ❞❡ ❧✐♠✐t❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✾
✾ ■♠♣❧❡♠❡♥t❛çã♦ ❡ ❯s♦ ❞❡ ❆▲●❊◆❈❆◆ ✾✸
✾✳✶ ❆❧❣♦r✐t♠♦ ♣❛r❛ ♦ ❙✉❜♣r♦❜❧❡♠❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✸ ✾✳✷ P❛râ♠❡tr♦ ❞❡ P❡♥❛❧✐❞❛❞❡ ■♥✐❝✐❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✹ ✾✳✸ ❈r✐tér✐♦s ❞❡ P❛r❛❞❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✺ ✾✳✸✳✶ P❛r❛❞❛ ♥♦s s✉❜♣r♦❜❧❡♠❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✺ ✾✳✸✳✷ ❉❡❝✐sõ❡s ❡♠❡r❣❡♥❝✐❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✼ ✾✳✸✳✸ P❛r❛❞❛ ✜♥❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✾ ✾✳✹ ❯s❛♥❞♦ ❆❧❣❡♥❝❛♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✵
✾✳✹✳✶ ❙✉❜r♦t✐♥❛ ■♥✐♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✷ ✾✳✹✳✷ ❙✉❜r♦t✐♥❛s ❡✈❛❧❢ ❡ ❡✈❛❧❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✹ ✾✳✹✳✸ ❙✉❜r♦t✐♥❛s ❡✈❛❧❝ ❡ ❡✈❛❧❥❛❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✺ ✾✳✹✳✹ Pr♦❣r❛♠❛ Pr✐♥❝✐♣❛❧✱ s✉❜r♦t✐♥❛ ♣❛r❛♠ ❡ s✉❜r♦t✐♥❛ ❡♥❞♣✶✵✽ ✾✳✹✳✺ ❘❡s✉❧t❛❞♦ ❞❡ ❆❧❣❡♥❝❛♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✷
Cap´ıtulo 1
Introdu¸
c˜
ao
❖ ♣r♦❜❧❡♠❛ ❞❡ Pr♦❣r❛♠❛çã♦ ◆ã♦✲▲✐♥❡❛r ✭P◆▲✮ ❝♦♥s✐st❡ ❡♠
▼✐♥✐♠✐③❛rf(x)
s✉❥❡✐t❛ ❛ hi(x) = 0, i = 1, . . . , m✱ gi(x) ≤ 0, i = 1, . . . , p✱ x ∈ Ω✱ ♦♥❞❡ Ω é ✉♠
s✉❜❝♦♥❥✉♥t♦ ❞❡IRn✱ ❣❡r❛❧♠❡♥t❡ s✐♠♣❧❡s✳ ▼✉✐t❛s ✈❡③❡sΩé ♦ ♣ró♣r✐♦IRn ❡✱ ❢r❡qü❡♥✲
t❡♠❡♥t❡✱Ωé ✉♠ ♣❛r❛❧❡❧❡♣í♣❡❞♦n✲❞✐♠❡♥s✐♦♥❛❧ ✭♦✉ ❝❛✐①❛✮✳ ❆s ❢✉♥çõ❡sf✱ hi✱gi sã♦
❝♦♥tí♥✉❛s ❡✱ ❡♠ ❣❡r❛❧✱ ❞❡r✐✈á✈❡✐s✳
❊st❡ ♣r♦❜❧❡♠❛ t❡♠ ♠✉✐t❛s ❛♣❧✐❝❛çõ❡s ♣rát✐❝❛s ❡ ❡①✐st❡♠ ♠✉✐t♦s ♠ét♦❞♦s ♣❛r❛ r❡s♦❧✈ê✲❧♦✳ ❖ ▼ét♦❞♦ ❞❡ ▲❛❣r❛♥❣✐❛♥♦ ❆✉♠❡♥t❛❞♦ é ✉♠❛ ✈❡rsã♦ ♠♦❞❡r♥❛ ❞❡ ✉♠❛ ❞❛s ✐❞é✐❛s ♠❛✐s ❛♥t✐❣❛s s♦❜r❡ ❝♦♠♦ ❧✐❞❛r ❝♦♠ ❡❧❡✳ ❚❛❧ ✐❞é✐❛ ❝♦♥s✐st❡ ❡♠ ❡❧✐♠✐♥❛r ❛s r❡str✐çõ❡s hi(x) = 0 ❡ gi(x) ≤ 0✱ ✐♥❝❧✉í♥❞♦ ❡st❛s r❡str✐çõ❡s ♥❛ ❢✉♥çã♦ ♦❜❥❡t✐✈♦
❞❡ ♠❛♥❡✐r❛ q✉❡ ♦ ♣r♦❜❧❡♠❛ ❛ss✐♠ tr❛♥s❢♦r♠❛❞♦ t❡♥❤❛ s♦❧✉çõ❡s ✐❣✉❛✐s ♦✉ ♣❛r❡❝✐❞❛s às ❞♦ ♣r♦❜❧❡♠❛ ♦r✐❣✐♥❛❧✳ ❉❛❞♦ ✉♠ ♣❛râ♠❡tr♦ ✏❞❡ ♣❡♥❛❧✐❞❛❞❡✑ ρ >0✱ ❛ ❢✉♥çã♦ q✉❡
❞❡♥♦♠✐♥❛♠♦s ▲❛❣r❛♥❣✐❛♥♦ ❆✉♠❡♥t❛❞♦ s❡ ❞❡✜♥❡ ❛ss✐♠✿
Lρ(x, λ, µ) =f(x) + ρ 2
Xm
i=1
hi(x) + λi
ρ
2
+ p
X
i=1
max
0, gi(x) + µi
ρ
2
.
❖s ♣❛râ♠❡tr♦s λi ∈ IR ❡ µi ≥ 0 s❡ ❝❤❛♠❛♠ ✏♠✉❧t✐♣❧✐❝❛❞♦r❡s✑✳ ❆q✉✐ ❞❡✜♥✐♠♦s
L ❝♦♠ ✉♠ ú♥✐❝♦ ♣❛râ♠❡tr♦ ❞❡ ♣❡♥❛❧✐❞❛❞❡ ρ✱ ❡♠❜♦r❛✱ às ✈❡③❡s✱ ♣r❡✜r❛✲s❡ ❞❡✜♥✐r ✉♠ ♣❡♥❛❧✐③❛❞♦r ❞✐❢❡r❡♥t❡ ρi ♣❛r❛ ❝❛❞❛ r❡str✐çã♦✳ ◗✉❛♥❞♦ s❡ ✐♥✐❝✐❛ ❛ ❡①❡❝✉çã♦ ❞♦
▼ét♦❞♦ ❞❡ ▲❛❣r❛♥❣✐❛♥♦ ❆✉♠❡♥t❛❞♦✱ ♦s ♠✉❧t✐♣❧✐❝❛❞♦r❡s ✐♥✐❝✐❛✐s ❡ ♦ ♣❡♥❛❧✐③❛❞♦r sã♦ ❞❛❞♦s✱ ❞❡ ♠❛♥❡✐r❛ q✉❡ ♦s ♣❛ss♦s ❢✉♥❞❛♠❡♥t❛✐s ❞♦ ♠ét♦❞♦ sã♦✿
✶✳ ▼✐♥✐♠✐③❛r Lρ(x, λ, µ)s✉❥❡✐t❛ ❛ x∈Ω✳
✷✳ ❉❡❝✐❞✐r s❡ ♦ ♣♦♥t♦ ♦❜t✐❞♦ ♥♦ ♣r✐♠❡✐r♦ ♣❛ss♦ ♣♦❞❡ s❡r ❛❝❡✐t♦ ❝♦♠♦ s♦❧✉çã♦ ❞♦ ♣r♦❜❧❡♠❛ ♦r✐❣✐♥❛❧✳ ❊♠ ❝❛s♦ ❛✜r♠❛t✐✈♦✱ ♣❛r❛r ❛ ❡①❡❝✉çã♦ ❞♦ ❛❧❣♦r✐t♠♦✳
✸✳ ❉❡ ❛❝♦r❞♦ ❝♦♠ ♦s r❡s✉❧t❛❞♦s ❞♦s ♣❛ss♦s ❛♥t❡r✐♦r❡s✱ ❛t✉❛❧✐③❛r ♦s ♠✉❧t✐♣❧✐❝❛✲ ❞♦r❡s ❡ ♦ ♣❛râ♠❡tr♦ ❞❡ ♣❡♥❛❧✐❞❛❞❡✳
✹✳ ❱♦❧t❛r ❛♦ ♣r✐♠❡✐r♦ ♣❛ss♦✳
➱ ❢á❝✐❧ ♣❡r❝❡❜❡r q✉❡✱ ♣❡❧♦ ♠❡♥♦s ♥♦ ❝❛s♦ ❡♠ q✉❡ ♦s ♠✉❧t✐♣❧✐❝❛❞♦r❡s sã♦ ♥✉❧♦s✱ ❛ ❡str❛té❣✐❛ é ♠✉✐t♦ s❡♥s❛t❛✳
❖s ♠ét♦❞♦s ❞❡ ▲❛❣r❛♥❣✐❛♥♦ ❆✉♠❡♥t❛❞♦ s❡❣✉❡♠✱ q✉❛s❡ s❡♠♣r❡✱ ♦ ❡sq✉❡♠❛ ❛❝✐♠❛✳ ❉❡✈❡♠ s❡r ❞✐❢❡r❡♥❝✐❛❞♦s ❞❡ ♦✉tr♦s ♠ét♦❞♦s ♣❛r❛ P◆▲ q✉❡ ✉s❛♠ ♦ ▲❛❣r❛♥✲ ❣✐❛♥♦ ❆✉♠❡♥t❛❞♦ ❝♦♠♦ ❢✉♥çã♦ ❞❡ ♠ér✐t♦ ❛✉①✐❧✐❛r✳
❆s ❞❡✜♥✐çõ❡s ❜ás✐❝❛s ✉s❛❞❛s ❛♦ ❧♦♥❣♦ ❞❡st❡ t❡①t♦ s❡rã♦ ❛s ❞❡ ♠✐♥✐♠✐③❛❞♦r ❣❧♦❜❛❧ ❡ ❧♦❝❛❧✳
❉✐r❡♠♦s q✉❡ x∗ é ♠✐♥✐♠✐③❛❞♦r ❣❧♦❜❛❧ ❞❡ f(x) s✉❥❡✐t❛ ❛ x∈ D q✉❛♥❞♦ x∗ ∈ D ❡ f(x)≥f(x∗) ♣❛r❛ t♦❞♦x∈D✳
❖ ♣♦♥t♦x∗ ∈D s❡rá ♠✐♥✐♠✐③❛❞♦r ❧♦❝❛❧ ❞❡ss❡ ♣r♦❜❧❡♠❛ q✉❛♥❞♦ ❡①✐st❡ε >0t❛❧
q✉❡ f(x)≥f(x∗) s❡♠♣r❡ q✉❡ x∈D ❡ ❛ ❞✐stâ♥❝✐❛ ❡♥tr❡ x ❡x∗ é ♠❡♥♦r q✉❡ ε✳ ◆♦t❛çõ❡s✿
k · k s❡rá s❡♠♣r❡ ❛ ♥♦r♠❛ ❊✉❝❧✐❞✐❛♥❛✳ ▼✉✐t❛s ✈❡③❡s ♣♦❞❡rá s❡r s✉❜st✐t✉í❞❛ ♣♦r
✉♠❛ ♥♦r♠❛ ❛r❜✐trár✐❛✳
❆s ❝♦♠♣♦♥❡♥t❡s ❞❡ ✉♠ ✈❡t♦r s❡rã♦ ❞❡♥♦t❛❞❛s ❝♦♠ s✉❜í♥❞✐❝❡s✳ ❙❡v ∈IRn✱ ❞❡♥♦t❛r❡♠♦sv
+♦ ✈❡t♦r ❝✉❥❛s ❝♦♠♣♦♥❡♥t❡s sã♦max{0, v1}, . . . ,max{0, vn}✳
IR é ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s r❡❛✐s✱ IN é ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥❛t✉r❛✐s✱ ❝♦♠❡ç❛♥❞♦ ❝♦♠ 0✳
K⊂
∞IN ✐♥❞✐❝❛ q✉❡ K é ✉♠ s✉❜❝♦♥❥✉♥t♦ ✐♥✜♥✐t♦ ❞❡ IN✱ ♣r❡s❡r✈❛♥❞♦ ❛ ♦r❞❡♠✳
❘✐❣♦r♦s❛♠❡♥t❡ ❢❛❧❛♥❞♦✱ K é ✉♠❛ s❡qüê♥❝✐❛ ❡str✐t❛♠❡♥t❡ ❝r❡s❝❡♥t❡ ❞❡ ♥ú♠❡r♦s ♥❛t✉r❛✐s✳ ❆♥❛❧♦❣❛♠❡♥t❡✱ K1⊂
Cap´ıtulo 2
M´
etodo de Penalidade
Externa
◆❡st❡ ❝❛♣ít✉❧♦ ❝♦♥s✐❞❡r❛♠♦s ❛ ✈❡rsã♦ ♠❛✐s ❜ás✐❝❛ ❞♦ ❝❤❛♠❛❞♦ ▼ét♦❞♦ ❞❡ P❡♥❛❧✐✲ ❞❛❞❡ ❊①t❡r♥❛✳ ❆♣❡s❛r ❞❡st❛ ❡str❛té❣✐❛ r❛r❛♠❡♥t❡ ♣♦❞❡r s❡r ✉s❛❞❛ ♥❛ ♣rát✐❝❛✱ s✉❛s ❝❛r❛❝t❡ríst✐❝❛s t❡ór✐❝❛s ✐❧✉♠✐♥❛♠ ❛❧❣♦r✐t♠♦s ♠❛✐s s♦✜st✐❝❛❞♦s✱ ✐♥❝❧✉s✐✈❡ ♦ ▼ét♦❞♦ ❞❡ ▲❛❣r❛♥❣✐❛♥♦ ❆✉♠❡♥t❛❞♦✱ ♥♦ss♦ ♦❜❥❡t♦ ❞❡ ❡st✉❞♦✳ ❱❡r❡♠♦s t❛♠❜é♠ q✉❡ ❛s ♣r♦✲ ♣r✐❡❞❛❞❡s ❞♦ ▼ét♦❞♦ ❞❡ P❡♥❛❧✐❞❛❞❡ ❊①t❡r♥❛ ❢♦r♥❡❝❡♠ ♣r♦✈❛s s✐♠♣❧❡s ❞❡ r❡s✉❧t❛❞♦s ❜❛st❛♥t❡ ❢♦rt❡s ❞❡ ♦t✐♠❛❧✐❞❛❞❡✳
❙❡❥❛ Ω ⊂IRn ✉♠ ❝♦♥❥✉♥t♦ ❢❡❝❤❛❞♦✳ ❙❡❥❛♠ f : Ω
→ IR✱ h : Ω →IRm✱ g : Ω →
IRp✳
❊s❝r❡✈❡r❡♠♦s h = (h1, . . . , hm)T✱ g = (g1, . . . , gp)T ❡ s✉♣♦r❡♠♦s q✉❡ ❛s ❢✉♥çõ❡s
f, h, g sã♦ ❝♦♥tí♥✉❛s ❡♠ Ω✳
❉❡✜♥✐♠♦s ♦ ♣r♦❜❧❡♠❛ P◆▲ ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛✿ ▼✐♥✐♠✐③❛rf(x)
s✉❥❡✐t❛ ❛h(x) = 0, g(x)≤0, x∈Ω. ❖ ♣r♦❜❧❡♠❛ P◆▲ é ♦ ♣r♦❜❧❡♠❛ ❝❡♥tr❛❧ ❡♥❝❛r❛❞♦ ♥❡st❡ ❧✐✈r♦✳
2.1
Penalidade Externa Pura
❖ ▼ét♦❞♦ ❞❡ P❡♥❛❧✐❞❛❞❡ ❊①t❡r♥❛ ❡stá ❞❡✜♥✐❞♦ ♣❡❧♦ ❆❧❣♦r✐t♠♦ ✷✳✶✳
❆❧❣♦r✐t♠♦ ✷✳✶
❙❡❥❛ {ρk}✉♠❛ s❡qüê♥❝✐❛ ❞❡ ♥ú♠❡r♦s ♣♦s✐t✐✈♦s t❛❧ q✉❡ lim
k→∞ρk =∞.
P❛r❛ t♦❞♦ k∈IN✱ ♦ ✐t❡r❛♥❞♦ xk é ✉♠❛ s♦❧✉çã♦ ❣❧♦❜❛❧ ❞♦ s✉❜♣r♦❜❧❡♠❛
▼✐♥✐♠✐③❛r f(x) + ρk
2 [kh(x)k
2+
kg(x)+k2] s✉❥❡✐t❛ ❛ x∈Ω. ✭✷✳✶✮
❖❜s❡r✈❡ q✉❡✱ ♥❛ ❞❡s❝r✐çã♦ ❞♦ ▼ét♦❞♦ ❞❡ P❡♥❛❧✐❞❛❞❡ ❊①t❡r♥❛ ❞❛❞❛ ♥♦ ❆❧❣♦✲ r✐t♠♦ ✷✳✶✱ ♥ã♦ ❡①✐st❡ ✉♠ ✏❝r✐tér✐♦ ❞❡ ♣❛r❛❞❛✑✳ ■st♦ s✐❣♥✐✜❝❛ q✉❡✱ t❡♦r✐❝❛♠❡♥t❡✱ ❛ s❡qüê♥❝✐❛ {xk
} t❡♠ ✐♥✜♥✐t♦s t❡r♠♦s✱ ♦♥❞❡ ♥ã♦ s❡ ❡①❝❧✉✐ q✉❡ ♠✉✐t♦s ❞❡❧❡s s❡✲
❥❛♠ ✐❣✉❛✐s✳ ■♠♣❧❡♠❡♥t❛çõ❡s ♣rát✐❝❛s✱ ♣♦r ♦✉tr♦ ❧❛❞♦✱ ❞❡✈❡♠ ❡st❛r ❡q✉✐♣❛❞❛s ❝♦♠ ❝r✐tér✐♦s ❞❡ ♣❛r❛❞❛✱ ❝♦♠♦ ✈❡r❡♠♦s ❡♠ ❝❛♣ít✉❧♦s ♣♦st❡r✐♦r❡s✳
◆♦ ▼ét♦❞♦ ❞❡ P❡♥❛❧✐❞❛❞❡ ❊①t❡r♥❛ s✉❜st✐t✉í♠♦s ♦ ♣r♦❜❧❡♠❛ ♦r✐❣✐♥❛❧ ♣♦r ✉♠❛ s❡qüê♥❝✐❛ ❞❡ ♣r♦❜❧❡♠❛s ♥♦s q✉❛✐s ♦ ❝♦♥❥✉♥t♦ ❞❡ r❡str✐çõ❡s ❢♦✐ r❡❞✉③✐❞♦ ❛
x∈Ω.
❊♠ ✐♠♣❧❡♠❡♥t❛çõ❡s ♣rát✐❝❛s✱ ♦ ❝♦♥❥✉♥t♦ Ω s❡rá✱ ❣❡r❛❧♠❡♥t❡✱ ✏s✐♠♣❧❡s✑✱ ❞❡ ♠❛✲
♥❡✐r❛ q✉❡ ❝❛❞❛ s✉❜♣r♦❜❧❡♠❛ ❞❡✜♥✐❞♦ ♣❡❧♦ ▼ét♦❞♦ ❞❡ P❡♥❛❧✐❞❛❞❡ ❊①t❡r♥❛ s❡❥❛ ♠❛✐s ❢á❝✐❧ q✉❡ ♦ ♣r♦❜❧❡♠❛ ♦r✐❣✐♥❛❧ P◆▲✳ ❊♥tr❡t❛♥t♦✱ ❞♦ ♣♦♥t♦ ❞❡ ✈✐st❛ t❡ór✐❝♦✱ ♥ã♦ ❤á r❡str✐çõ❡s s♦❜r❡ ❛ ❝♦♠♣❧❡①✐❞❛❞❡ ❞❡Ω✳
❖❜s❡r✈❛♠♦s q✉❡✱ ❢♦r♠❛❧♠❡♥t❡✱ ♦ ✐t❡r❛♥❞♦ xk+1 ❞♦ ♠ét♦❞♦ ♥ã♦ ❡stá ❝♦♥❡❝t❛❞♦
❡♠ ❛❜s♦❧✉t♦ ❝♦♠ ♦ ✐t❡r❛♥❞♦ xk✳ ◆❛ ♣rát✐❝❛✱ ❡♥tr❡t❛♥t♦✱ ❡st❛ ❝♦♥❡①ã♦ s❡♠♣r❡
❡①✐st✐rá✱ ♣♦✐sxk é ♦ ✏♣♦♥t♦ ✐♥✐❝✐❛❧✑ ♥❛t✉r❛❧ ♣❛r❛ r❡s♦❧✈❡r ♦ s✉❜♣r♦❜❧❡♠❛ ❞❡ ♠❛♥❡✐r❛
✐t❡r❛t✐✈❛✳
❆ ❡①✐❣ê♥❝✐❛ ❞❡ ♦❜t❡♥çã♦ ❞❡ ✉♠ ♠✐♥✐♠✐③❛❞♦r ❣❧♦❜❛❧ ❞♦ s✉❜♣r♦❜❧❡♠❛ é ✉♠ ❡♥♦r♠❡ ❡♠♣❡❝✐❧❤♦ ♣❛r❛ ❛ ✐♠♣❧❡♠❡♥t❛çã♦ ❞❡st❡ ♠ét♦❞♦✳ ◆❛ ♣rát✐❝❛✱ s❛❧✈♦ ❝❛s♦s ♠✉✐t♦ ❡s♣❡❝í✜❝♦s✱ ✐st♦ é ♠✉✐t♦ ❞✐❢í❝✐❧✳ ▼❡s♠♦ ❞♦ ♣♦♥t♦ ❞❡ ✈✐st❛ t❡ór✐❝♦✱ ❛ ❡①✐s✲ tê♥❝✐❛ ❞❡ ✉♠❛ s♦❧✉çã♦ ❣❧♦❜❛❧ ❞♦ s✉❜♣r♦❜❧❡♠❛ s♦♠❡♥t❡ é ❣❛r❛♥t✐❞❛ s✉♣♦♥❞♦✱ ♣♦r ❡①❡♠♣❧♦✱ ❛ ❝♦♠♣❛❝✐❞❛❞❡ ❞♦ ❝♦♥❥✉♥t♦ s✐♠♣❧❡sΩ✳ ◗✉❛♥❞♦xk❡①✐st❡ ♣❛r❛ t♦❞♦k ∈IN
❞✐③❡♠♦s q✉❡ ♦ ♠ét♦❞♦ ❡stá ❜❡♠ ❞❡✜♥✐❞♦✳
❆ ✐❞é✐❛ ❞❛ ♣❡♥❛❧✐❞❛❞❡ ❡①t❡r♥❛ é s✉❜st✐t✉ír ❛ ❢✉♥çã♦ ♦❜❥❡t✐✈♦ ♦r✐❣✐♥❛❧f ♣♦r ✉♠❛ ❢✉♥çã♦ ♦♥❞❡ s❡ ❝❛st✐❣❛ ❛ ♥ã♦✲❛❞♠✐ss✐❜✐❧✐❞❛❞❡ ❞❡ x✳ ◗✉❛♥t♦ ♠❡♥♦s ❛❞♠✐ssí✈❡❧ ❢♦r ♦ ♣♦♥t♦ x ❝♦♠ r❡s♣❡✐t♦ às r❡str✐çõ❡s h(x) = 0 ❡ g(x) ≤ 0✱ ♠❛✐s ❝❛st✐❣❛❞♦ ❡❧❡ s❡rá✱
♠❡❞✐❛♥t❡ ✉♠ ❛✉♠❡♥t♦ ❞❛ ❢✉♥çã♦ ♦❜❥❡t✐✈♦ ❞♦ s✉❜♣r♦❜❧❡♠❛✳ ❆ ♣❡♥❛❧✐❞❛❞❡ é ♠❛✐♦r q✉❛♥t♦ ♠❛✐♦r ❢♦rρk✳ P♦❞❡♠♦s ✐♥t❡r♣r❡t❛r q✉❡✱ q✉❛♥❞♦ρk é ♠✉✐t♦ ❣r❛♥❞❡✱ ❛ ❢✉♥çã♦
♦❜❥❡t✐✈♦ ❞♦ s✉❜♣r♦❜❧❡♠❛ ❝♦✐♥❝✐❞❡ ❝♦♠ f(x) s❡ x é ❛❞♠✐ssí✈❡❧ ❡ s❡ ❛♣r♦①✐♠❛ ❞❡ ✐♥✜♥✐t♦ s❡x ♥ã♦ é ❛❞♠✐ssí✈❡❧✳ ❆ ❞✐✜❝✉❧❞❛❞❡ ❞♦s s✉❜♣r♦❜❧❡♠❛s ❛✉♠❡♥t❛ ❥✉♥t♦ ❝♦♠ ♦ ✈❛❧♦r ❞❡ ρk✳ ❆♣❡s❛r ❞❡ q✉❡✱ ❡♠ ❝❛s♦s s✐♠♣❧❡s✱ ❝♦♠❡ç❛r ❝♦♠ ✈❛❧♦r❡s ❡♥♦r♠❡s ❞❡
ρk ❞á ❜♦♥s r❡s✉❧t❛❞♦s✱ ❡♠ s✐t✉❛çõ❡s ♠❛✐s ❝♦♠♣❧❡①❛s ✈❛❧♦r❡s ❡①tr❡♠♦s ❞❡ρk ❢❛③❡♠
❝♦♠ q✉❡ ♦ t❡r♠♦f(x)♥♦ s✉❜♣r♦❜❧❡♠❛ s❡❥❛ ✏❝❛♥❝❡❧❛❞♦✑ ♣❡❧♦ t❡r♠♦ ❞❡ ♣❡♥❛❧✐❞❛❞❡✱
♣❡❧♦ ♠❡♥♦s q✉❛♥❞♦ s❡ tr❛❜❛❧❤❛ ❝♦♠ ❛♣r♦①✐♠❛çõ❡s ♥♦ ❝♦♠♣✉t❛❞♦r✳
■♥❞❡♣❡♥❞❡♥t❡♠❡♥t❡ ❞✐ss♦✱ ♦ q✉❡ ♥♦s ✐♥t❡r❡ss❛ ♥❡st❡ ❝❛♣ít✉❧♦ sã♦ ❛s ♣r♦♣r✐❡❞❛❞❡s t❡ór✐❝❛s ❞♦ ♠ét♦❞♦ t❛❧ ❝♦♠♦ ❞❡✜♥✐❞♦ ❛❝✐♠❛✳ ❊①✐st❡♠ ❞✉❛s ♣❡r❣✉♥t❛s ❜ás✐❝❛s q✉❡✱ ❞❡ ❢❛t♦✱ sã♦ ❝♦♠✉♥s ❛ t♦❞♦s ♦s ❛❧❣♦r✐t♠♦s ❞❡ ♦t✐♠✐③❛çã♦ ❝♦♠ r❡str✐çõ❡s✿
• ❖ ♠ét♦❞♦ ❡♥❝♦♥tr❛ ✭❝♦♥✈❡r❣❡ ❛✮ ♣♦♥t♦s ❛❞♠✐ssí✈❡✐s❄
2.1. Penalidade Externa Pura 5 ❆ ♣r✐♠❡✐r❛ r❡s♣♦st❛ ♥ã♦ ♣♦❞❡ s❡r s❡♠♣r❡ ♣♦s✐t✐✈❛ ♣♦rq✉❡✱ às ✈❡③❡s✱ ♦ ♣r♦❜❧❡♠❛ ♦r✐❣✐♥❛❧ ♥ã♦ t❡♠ ♣♦♥t♦s ❛❞♠✐ssí✈❡✐s✳ ▲♦❣♦✱ ♥♦ss♦ ✐♥t❡r❡ss❡ s❡r✐❛ q✉❡ ♦ ♠ét♦❞♦ ❡♥❝♦♥tr❛ss❡ ♣♦♥t♦s ❛❞♠✐ssí✈❡✐s q✉❛♥❞♦ ❡st❡s ❡①✐st❡♠ ❡ q✉❡✱ ❡♠ ❝❛s♦ ❝♦♥trár✐♦✱ ❡♥✲ ❝♦♥tr❛ss❡ ❛❧❣♦ q✉❡ ♥♦s s✐r✈❛ ♣❛r❛ ❣❛r❛♥t✐r ❛ ♥ã♦✲❡①✐stê♥❝✐❛ ❞❡ t❛✐s ♣♦♥t♦s✳ ❖s t❡♦r❡♠❛s ✷✳✶ ❡ ✷✳✷ ♠♦str❛♠ q✉❡✱ ❢❡❧✐③♠❡♥t❡✱ ✐ss♦ é ♦ q✉❡ ❛❝♦♥t❡❝❡ ❝♦♠ ♦ ▼ét♦❞♦ ❞❡ P❡♥❛❧✐❞❛❞❡ ❊①t❡r♥❛✳
◆♦s ❚❡♦r❡♠❛s ✷✳✶✕✷✳✺ s✉♣♦r❡♠♦s q✉❡ ❛ s❡qüê♥❝✐❛ {xk} ❡stá ❜❡♠ ❞❡✜♥✐❞❛ ♣❡❧♦
❆❧❣♦r✐t♠♦ ✷✳✶✳
❚❡♦r❡♠❛ ✷✳✶✳ ❙✉♣♦♥❤❛♠♦s q✉❡ ❛ s❡qüê♥❝✐❛ {xk} ❛❞♠✐t❡ ✉♠ ♣♦♥t♦ ❧✐♠✐t❡ x∗✳
❊♥tã♦✱ x∗ é ✉♠❛ s♦❧✉çã♦ ❣❧♦❜❛❧ ❞♦ ♣r♦❜❧❡♠❛
▼✐♥✐♠✐③❛r kh(x)k2+kg(x)
+k2 s✉❥❡✐t❛ ❛ x∈Ω. ✭✷✳✷✮
Pr♦✈❛✳ ❙❡❥❛ K⊂
∞IN t❛❧ q✉❡
lim k∈Kx
k =x∗.
❈♦♠♦ xk → x∗ ♣❛r❛ k ∈ K✱ xk ∈ Ω ♣❛r❛ t♦❞♦ k✱ ❡ Ω é ❢❡❝❤❛❞♦ ✱ t❡♠♦s q✉❡
x∗ ∈Ω✳
❙✉♣♦♥❤❛♠♦s q✉❡x∗ ♥ã♦ s❡❥❛ ✉♠ ♠✐♥✐♠✐③❛❞♦r ❣❧♦❜❛❧ ❞❡ ✭✷✳✷✮✳ ■st♦ ✐♠♣❧✐❝❛ q✉❡
❡①✐st❡ z ∈Ω t❛❧ q✉❡
kh(x∗)k2+kg(x∗)
+k2 >kh(z)k2+kg(z)+k2.
❈♦♠♦h❡g sã♦ ❝♦♥tí♥✉❛s✱ ❡①✐st❡c >0t❛❧ q✉❡✱ ♣❛r❛k ∈K s✉✜❝✐❡♥t❡♠❡♥t❡ ❣r❛♥❞❡✱
kh(xk)
k2+kg(xk)
+k2 >kh(z)k2+kg(z)+k2+c.
P♦rt❛♥t♦✱ ♣❛r❛ ❡ss❡s í♥❞✐❝❡s k✱
f(xk)+ρk 2 [kh(x
k
)k2+kg(xk
)+k2]> f(z)+
ρk
2 [kh(z)k
2+kg(z) +k2]+
ρkc 2 +f(x
k
)−f(z).
❆❣♦r❛✱ ❝♦♠♦{xk}
k∈K é ✉♠ ❝♦♥❥✉♥t♦ ❧✐♠✐t❛❞♦✱f é ❝♦♥tí♥✉❛ ❡ ρk→ ∞✱ t❡♠♦s q✉❡✱
♣❛r❛ k ∈K s✉✜❝✐❡♥t❡♠❡♥t❡ ❣r❛♥❞❡✱ ρkc
2 +f(x k
)−f(z)>0.
▲♦❣♦✱
f(xk) + ρk 2 [kh(x
k)
k2+kg(xk)
+k2]> f(z) +
ρk
2[kh(z)k
2+
♦ q✉❡ ❝♦♥tr❛❞✐③ ♦ ❢❛t♦ ❞❡ q✉❡xké ✉♠❛ s♦❧✉çã♦ ❣❧♦❜❛❧ ❞♦ s✉❜♣r♦❜❧❡♠❛ ❞❡✜♥✐❞♦ ♣♦r
ρk✳ ❊st❛ ❝♦♥tr❛❞✐çã♦ ❞❡❝♦rr❡ ❞❡ s✉♣♦r q✉❡ x∗ ♥ã♦ ❡r❛ ✉♠ ♠✐♥✐♠✐③❛❞♦r ❣❧♦❜❛❧ ❞❡
✭✷✳✷✮✳ ◗❊❉
❚❡♦r❡♠❛ ✷✳✷✳ ❙✉♣♦♥❤❛♠♦s q✉❡ ❛ r❡❣✐ã♦ ❛❞♠✐ssí✈❡❧ ❞♦ ♣r♦❜❧❡♠❛ P◆▲ é ♥ã♦✲✈❛③✐❛✳ ❙✉♣♦♥❤❛♠♦s✱ ❛❧é♠ ❞✐ss♦✱ q✉❡ ❛ s❡qüê♥❝✐❛ {xk
}❛❞♠✐t❡ ✉♠ ♣♦♥t♦ ❧✐♠✐t❡ x∗✳ ❊♥tã♦✱
x∗ é ❛❞♠✐ssí✈❡❧✳
Pr♦✈❛✳ P❡❧♦ ❚❡♦r❡♠❛ ✷✳✶✱ x∗ é ♠✐♥✐♠✐③❛❞♦r ❣❧♦❜❛❧ ❞❡ ✭✷✳✷✮✳ ▼❛s✱ ❝♦♠♦ ❡①✐st❡
♣❡❧♦ ♠❡♥♦s ✉♠ ♣♦♥t♦ ❛❞♠✐ssí✈❡❧ z✱ ♥❡st❡ ♣♦♥t♦ ❛ ❢✉♥çã♦ ♦❜❥❡t✐✈♦ ❞❡ ✭✷✳✷✮ ✈❛❧❡ ③❡r♦✳ P♦rt❛♥t♦✱ ❡ss❛ ❢✉♥çã♦ t❛♠❜é♠ ❞❡✈❡ s❡ ❛♥✉❧❛r ❡♠ x∗✱ ♦ q✉❡ ✐♠♣❧✐❝❛ q✉❡ x∗ é
❛❞♠✐ssí✈❡❧✳ ◗❊❉
P❡❧♦ ❚❡♦r❡♠❛ ✷✳✷✱ s❡ ❡①✐st❡♠ ♣♦♥t♦s ❛❞♠✐ssí✈❡✐s ❡ ❛ s❡qüê♥❝✐❛ ❣❡r❛❞❛ ♣❡❧♦ ❛❧❣♦r✐t♠♦ é ❧✐♠✐t❛❞❛✱ ✉♠ ❞♦s ♣♦♥t♦s ❛❞♠✐ssí✈❡✐s s❡rá ❢❛t❛❧♠❡♥t❡ ❡♥❝♦♥tr❛❞♦ ♣❡❧♦ ❛❧❣♦r✐t♠♦✳ P♦r ♦✉tr♦ ❧❛❞♦✱ ♦ ❚❡♦r❡♠❛ ✷✳✶ ❣❛r❛♥t❡ q✉❡✱ s❡ ❛ r❡❣✐ã♦ ❛❞♠✐ssí✈❡❧ ❢♦r ✈❛③✐❛✱ ✉♠ ♣♦♥t♦ q✉❡ ♠✐♥✐♠✐③❛ ❛ s♦♠❛ ❞❡ q✉❛❞r❛❞♦s ❞❛s ✐♥❛❞♠✐ss✐❜✐❧✐❞❛❞❡s ✭♦✉ s❡❥❛✱ ♦ ✏♠❡♥♦s ✐♥❛❞♠✐ssí✈❡❧✑✮ s❡rá ❛❝❤❛❞♦ ✳
◆♦ ❚❡♦r❡♠❛ ✷✳✹ ✈❡r❡♠♦s q✉❡✱ q✉❛♥❞♦ ❛ r❡❣✐ã♦ ❛❞♠✐ssí✈❡❧ é ♥ã♦✲✈❛③✐❛✱ ♦ ❆❧❣♦✲ r✐t♠♦ ✷✳✶ ❡♥❝♦♥tr❛ ✉♠ ♠✐♥✐♠✐③❛❞♦r ❣❧♦❜❛❧ ❞♦ ♣r♦❜❧❡♠❛ P◆▲✳ ❊st❡ r❡s✉❧t❛❞♦ s❡rá ❝♦♥s❡qüê♥❝✐❛ ❞❡ ✉♠❛ ♣r♦♣r✐❡❞❛❞❡ ♠❛✐s ❢♦rt❡✱ ❞❛❞❛ ♥♦ ❚❡♦r❡♠❛ ✷✳✸✳ ◆❡❧❡ ✈❡r❡♠♦s q✉❡ ♦s ♣♦♥t♦s ❧✐♠✐t❡ ❞♦ ▼ét♦❞♦ ❞❡ P❡♥❛❧✐❞❛❞❡ ❊①t❡r♥❛ ♠✐♥✐♠✐③❛♠ f(x) ♥♦ ❝♦♥✲
❥✉♥t♦ ❞♦s ♣♦♥t♦s ♠❡♥♦s ✐♥❛❞♠✐ssí✈❡✐s✳
❚❡♦r❡♠❛ ✷✳✸✳ ❙❡❥❛ z ∈ Ω ✉♠ ♠✐♥✐♠✐③❛❞♦r ❣❧♦❜❛❧ ❞❡ ✭✷✳✷✮✳ ❙✉♣♦♥❤❛♠♦s q✉❡ ❛
s❡qüê♥❝✐❛ {xk} ❛❞♠✐t❡ ✉♠ ♣♦♥t♦ ❧✐♠✐t❡ x∗✳ ❊♥tã♦ x∗ t❛♠❜é♠ é ✉♠ ♠✐♥✐♠✐③❛❞♦r
❣❧♦❜❛❧ ❞❡ ✭✷✳✷✮ ❡✱ ❛❧é♠ ❞✐ss♦✱ f(x∗)≤f(z)✳
Pr♦✈❛✳ ❖ ❢❛t♦ ❞❡ q✉❡ x∗ é ♠✐♥✐♠✐③❛❞♦r ❣❧♦❜❛❧ ❞❡ ✭✷✳✷✮ ❢♦✐ ♣r♦✈❛❞♦ ♥♦ ❚❡♦r❡♠❛ ✷✳✶✳
❙❡❥❛ K⊂
∞IN t❛❧ q✉❡ limk∈Kx
k =x∗✳
P❡❧❛ ❞❡✜♥✐çã♦ ❞❡ xk✱ t❡♠♦s q✉❡✿
f(xk) + ρk 2 [kh(x
k
)k2+kg(xk
)+k2]≤f(z) +
ρk
2 [kh(z)k
2+kg(z) +k2],
♣❛r❛ t♦❞♦k ∈K✳
▼❛s✱ ❝♦♠♦ z é ✉♠ ♠✐♥✐♠✐③❛❞♦r ❣❧♦❜❛❧ ❞❡ ✭✷✳✷✮✱ ρk
2[kh(x k)
k2+kg(xk)
+k2]≥
ρk
2[kh(z)k
2+kg(z) +k2].
P♦rt❛♥t♦✱
f(xk)
2.2. Penalidade Externa com Controle de Admissibilidade 7 ❚♦♠❛♥❞♦ ❧✐♠✐t❡s ♥❡st❛ ❞❡s✐❣✉❛❧❞❛❞❡✱ ♣❡❧❛ ❝♦♥t✐♥✉✐❞❛❞❡ ❞❡f ❡ ♦ ❢❛t♦ ❞❡ q✉❡xk →x∗
♣❛r❛ k ∈K✱ ♦❜t❡♠♦s q✉❡ f(x∗)≤f(z)✳ ◗❊❉
❚❡♦r❡♠❛ ✷✳✹✳ ❙✉♣♦♥❤❛♠♦s q✉❡ ❛ r❡❣✐ã♦ ❛❞♠✐ssí✈❡❧ ❞♦ ♣r♦❜❧❡♠❛ P◆▲ é ♥ã♦✲✈❛③✐❛ ❡ ❛ s❡qüê♥❝✐❛{xk} ❛❞♠✐t❡ ✉♠ ♣♦♥t♦ ❧✐♠✐t❡x∗✳ ❊♥tã♦✱ x∗ é ✉♠ ♠✐♥✐♠✐③❛❞♦r ❣❧♦❜❛❧
❞❡ P◆▲✳
Pr♦✈❛✳ P❡❧♦ ❚❡♦r❡♠❛ ✷✳✸✱x∗ é ❛❞♠✐ssí✈❡❧ ❡f(x∗)≤f(z)♣❛r❛ q✉❛❧q✉❡r ♦✉tr♦ ♣♦♥t♦
❛❞♠✐ssí✈❡❧ z ∈Ω✳ P♦rt❛♥t♦✱ x∗ é ♠✐♥✐♠✐③❛❞♦r ❣❧♦❜❛❧ ❞❡ P◆▲✳ ◗❊❉
❯s✉❛❧♠❡♥t❡✱ ♦s ✐t❡r❛♥❞♦s xk ❞♦ ▼ét♦❞♦ ❞❡ P❡♥❛❧✐❞❛❞❡ ❊①t❡r♥❛ ♥ã♦ sã♦ ♣♦♥t♦s
❛❞♠✐ssí✈❡✐s✳ ❖✉ s❡❥❛✱ ❛❧❣✉♠❛ r❡str✐çã♦ ❞♦ t✐♣♦ gi(xk) ≤ 0 ♦✉ hi(xk) = 0 q✉❛s❡
s❡♠♣r❡ é ✈✐♦❧❛❞❛✳ ◗✉❛♥❞♦ ✐st♦ ♥ã♦ é ❛ss✐♠✱ ♦✉ s❡❥❛✱ q✉❛♥❞♦ t♦❞❛s ❛s r❡str✐çõ❡s sã♦ s❛t✐s❢❡✐t❛s ❡♠ ✉♠ ✐t❡r❛♥❞♦ xk✱ ❛❧❣♦ ❜❛st❛♥t❡ s✉r♣r❡❡♥❞❡♥t❡ ❛❝♦♥t❡❝❡✿ xk ❞❡✈❡ s❡r
s♦❧✉çã♦ ❞♦ ♣r♦❜❧❡♠❛ P◆▲✳ Pr♦✈❛♠♦s ❡st❛ ♣r♦♣r✐❡❞❛❞❡ ♥♦ ❚❡♦r❡♠❛ ✷✳✺✳
❚❡♦r❡♠❛ ✷✳✺✳ ❙✉♣♦♥❤❛♠♦s q✉❡ ❡①✐st❡ k ∈ IN t❛❧ q✉❡ xk é ❛❞♠✐ssí✈❡❧✳ ❊♥tã♦✱ xk
é ♠✐♥✐♠✐③❛❞♦r ❣❧♦❜❛❧ ❞❡ P◆▲✳
Pr♦✈❛✳ ❙❡❥❛ z ∈Ω ✉♠ ♣♦♥t♦ ❛❞♠✐ssí✈❡❧✳ P❡❧❛ ❞❡✜♥✐çã♦ ❞❡ xk t❡♠♦s✿
f(xk) + ρk 2 [kh(x
k
)k2+kg(xk
)+k2]≤f(z) +
ρk
2 [kh(z)k
2+kg(z) +k2].
▼❛s✱ ❝♦♠♦xk ❡ z sã♦ ❛❞♠✐ssí✈❡✐s✱
kh(xk)k2+kg(xk
)+k2 =kh(z)k2+kg(z)+k2 = 0.
P♦rt❛♥t♦✿
f(xk)
≤f(z).
❈♦♠♦ z é ✉♠ ♣♦♥t♦ ❛❞♠✐ssí✈❡❧ ❛r❜✐trár✐♦✱ ❞❡❞✉③✐♠♦s q✉❡ xk é ✉♠ ♠✐♥✐♠✐③❛❞♦r
❣❧♦❜❛❧ ❞❡ P◆▲✳ ◗❊❉
2.2
Penalidade Externa com Controle de Admissibilidade
❉❛❞♦ q✉❡ tr❛❜❛❧❤❛r ❝♦♠ ♣❛râ♠❡tr♦s ❞❡ ♣❡♥❛❧✐❞❛❞❡ ♠✉✐t♦ ❣r❛♥❞❡s é✱ ✈✐❛ ❞❡ r❡❣r❛✱ ✐♥❝♦♥✈❡♥✐❡♥t❡✱ ♣r♦❝✉r❛r❡♠♦s ❛✉♠❡♥t❛r ♦ ♣❛râ♠❡tr♦ ❞❡ ♣❡♥❛❧✐❞❛❞❡ só s❡ ❢♦r ❡str✐✲ t❛♠❡♥t❡ ♥❡❝❡ssár✐♦✳ ❏✉❧❣❛r❡♠♦s q✉❡ ❛✉♠❡♥t❛r ρk é ❞❡s♥❡❝❡ssár✐♦ s❡ ❛ ♠❡❞✐❞❛ ❞❡
✐♥❛❞♠✐ss✐❜✐❧✐❞❛❞❡ ❞❡xké s✉✜❝✐❡♥t❡♠❡♥t❡ ♠❡♥♦r q✉❡ ❛ ♠❡❞✐❞❛ ❞❡ ✐♥❛❞♠✐ss✐❜✐❧✐❞❛❞❡
❆❧❣♦r✐t♠♦ ✷✳✷✳
❙❡❥❛ x0 ∈ Ω ✉♠ ♣♦♥t♦ ✐♥✐❝✐❛❧ ❛r❜✐trár✐♦✱ ρ
1 >0 ✉♠ ♣❛râ♠❡tr♦ ✐♥✐❝✐❛❧ ❞❡ ♣❡♥❛✲
❧✐❞❛❞❡✱ τ ∈[0,1)❡ γ >1✳ ■♥✐❝✐❛❧✐③❛r ♦ ✏❝♦♥t❛❞♦r✑ k ←1✳
P❛ss♦ ✶✳ ❊♥❝♦♥tr❛r xk ∈Ω s♦❧✉çã♦ ❣❧♦❜❛❧ ❞♦ s✉❜♣r♦❜❧❡♠❛ ✭✷✳✶✮✳
P❛ss♦ ✷✳ ❙❡
max{kh(xk)k∞,kg(xk)+k∞} ≤τmax{kh(xk−1)k∞,kg(xk−1)+k∞},
❞❡✜♥✐r ρk+1 =ρk✳ ❊♠ ❝❛s♦ ❝♦♥trár✐♦✱ ❞❡✜♥✐rρk+1 =γρk✳
P❛ss♦ ✸✳ ❆t✉❛❧✐③❛r k ←k+ 1 ❡ ✈♦❧t❛r ❛♦ P❛ss♦ ✶✳
◆♦ t❡st❡ ❞♦ P❛ss♦ ✷ ♥ã♦ é ♦❜r✐❣❛tór✐♦ ✉s❛r k · k∞✱ ❡♠❜♦r❛ ❡st❛ ♥♦r♠❛ s❡❥❛ ❛
♠❛✐s ❢r❡qü❡♥t❡ ♥❛s ❛♣❧✐❝❛çõ❡s ♣rát✐❝❛s✳
❖ ❧❡✐t♦r ❝ét✐❝♦ ♣♦❞❡ ❛r❣✉♠❡♥t❛r q✉❡ ♥ã♦ ❤á ♥❡♥❤✉♠❛ ❞✐❢❡r❡♥ç❛ ❡ss❡♥❝✐❛❧ ❡♥tr❡ ♦ ❆❧❣♦r✐t♠♦ ✷✳✷ ❡ ♦ ❆❧❣♦r✐t♠♦ ✷✳✶✳ ❉❡ ❢❛t♦✱ s❡ ❡♠ ❞❡t❡r♠✐♥❛❞❛ ✐t❡r❛çã♦ ♦ ♣❛râ♠❡tr♦ ❞❡ ♣❡♥❛❧✐❞❛❞❡ ♥ã♦ é ❛✉♠❡♥t❛❞♦✱ ♦ s✉❜♣r♦❜❧❡♠❛ ♥❛ ✐t❡r❛çã♦ s❡❣✉✐♥t❡ s❡rá ♦ ♠❡s♠♦ ❡✱ ♣♦rt❛♥t♦✱ ♦ ♠❡s♠♦ ♣♦♥t♦ ♣♦❞❡ s❡r ❡s❝♦❧❤✐❞♦ ❝♦♠♦ s♦❧✉çã♦✳ ◆♦ ♥♦✈♦ t❡st❡✱ ❛ ♠❡♥♦s q✉❡ xk s❡❥❛ ❛❞♠✐ssí✈❡❧ ✭❡✱ ♣♦rt❛♥t♦✱ ót✐♠♦✮ ❛ r❡❞✉çã♦ ❞❛ ✐♥❛❞♠✐ss✐❜✐❧✐❞❛❞❡
♥ã♦ ♣♦❞❡rá ♦❝♦rr❡r ✭♣♦✐sxk=xk−1✮ ❡✱ ❡♠ ❝♦♥s❡qüê♥❝✐❛✱ ♦ ♣❛râ♠❡tr♦ ❞❡ ♣❡♥❛❧✐❞❛❞❡
s❡rá ❛✉♠❡♥t❛❞♦✳ ❆ss✐♠✱ ❛ s❡qüê♥❝✐❛ρkt❡♥❞❡rá ❛ ✐♥✜♥✐t♦ ❡ ❡st❛r❡♠♦s ✐♥t❡✐r❛♠❡♥t❡
s♦❜ ❛s ❤✐♣ót❡s❡s ❞♦ ❆❧❣♦r✐t♠♦ ✷✳✶✳
❊♥tr❡t❛♥t♦✱ ❞❡ ❛❝♦r❞♦ ❝♦♠ ♦ ❆❧❣♦r✐t♠♦ ✷✳✷✱ s❡ ♦ t❡st❡ ❞❡ ♠❡❧❤♦r❛ ❞❡ ❛❞♠✐ss✐✲ ❜✐❧✐❞❛❞❡ ❢♦r ❜❡♠ s✉❝❡❞✐❞♦✱ ♦ ♣♦♥t♦ xk+1 ♣♦❞❡ ♠❛s ♥ã♦ ♣r❡❝✐s❛ s❡r ❡s❝♦❧❤✐❞♦ ✐❣✉❛❧ ❛
xk✳ ❖✉ s❡❥❛✱ ♦ s✉❜♣r♦❜❧❡♠❛ ✭✷✳✶✮ ❡♠❜♦r❛ s❡♥❞♦ ♦ ♠❡s♠♦ ♥❛s ❞✉❛s ✐t❡r❛çõ❡s✱ ♣♦❞❡
t❡r ♠❛✐s ❞❡ ✉♠❛ s♦❧✉çã♦ ❣❧♦❜❛❧ ❡ ♥❛❞❛ ✐♠♣❡❞❡ q✉❡ ♥❛ ✐t❡r❛çã♦k+ 1s❡❥❛ ❡s❝♦❧❤✐❞❛
✉♠❛ s♦❧✉çã♦ ❞✐❢❡r❡♥t❡ ❞❛q✉❡❧❛ ❡s❝♦❧❤✐❞❛ ♥❛ ✐t❡r❛çã♦ k✳ ❈♦♥s✐❞❡r❡♠♦s ♦ ❡①❡♠♣❧♦ ❡♠ ✉♠❛ ✈❛r✐á✈❡❧ ❞❡✜♥✐❞♦ ♣♦r✿
f(x) = −x2, m= 1, p= 0, h1(x) = x,Ω =IR.
❈♦♠❡ç❛♥❞♦ ❝♦♠ρ1 = 2❡x0 = 100✱ ❛ ❢✉♥çã♦ ♦❜❥❡t✐✈♦ ❞♦ s✉❜♣r♦❜❧❡♠❛ é ♥✉❧❛✳ ▲♦❣♦
♣♦❞❡♠♦s ❡s❝♦❧❤❡r ♦ ♠✐♥✐♠✐③❛❞♦r ❣❧♦❜❛❧ x1 = 10✳ ❖ t❡st❡ ❞❡ ♠❡❧❤♦r❛ ❞❡ ❛❞♠✐ss✐✲
❜✐❧✐❞❛❞❡ ❞á r❡s✉❧t❛❞♦ ♣♦s✐t✐✈♦ ❡✱ ♣♦rt❛♥t♦✱ ρ2 = ρ1 = 2✳ ❈♦♠♦ ❛ ❢✉♥çã♦ ♦❜❥❡t✐✈♦
❞♦ s✉❜♣r♦❜❧❡♠❛ ❝♦♥t✐♥✉❛ s❡♥❞♦ ❛ ❢✉♥çã♦ ♥✉❧❛✱ ♣♦❞❡♠♦s ❡s❝♦❧❤❡r ❛❣♦r❛ ❛ s♦❧✉çã♦ xk = 1 q✉❛♥❞♦ k = 2✳ ❉❡ ♥♦✈♦✱ ♦ t❡st❡ é ♣♦s✐t✐✈♦ ❡ ❛ss✐♠ ♣♦r ❞✐❛♥t❡✳ ◆❡st❡ ❝❛s♦✱
❛ s❡qüê♥❝✐❛ ❣❡r❛❞❛ ♣❡❧♦ ❆❧❣♦r✐t♠♦ ✷✳✷ ♣♦❞❡r✐❛ s❡r {100,10,1,0.1,0.01,0.001, . . .} ❡ ♦ ♣❛râ♠❡tr♦ ❞❡ ♣❡♥❛❧✐❞❛❞❡ ♥ã♦ ❛✉♠❡♥t❛r✐❛ ♥✉♥❝❛✳
2.2. Penalidade Externa com Controle de Admissibilidade 9 ❡♠ ❢✉♥çã♦ ❞❛ ❢✉t✉r❛ ❞❡✜♥✐çã♦ ❞❡ ♦✉tr♦s ♠ét♦❞♦s ♦♥❞❡ ♦ t❡st❡ ❞❡ ❛❞♠✐ss✐❜✐❧✐❞❛❞❡ ❞❡s❡♠♣❡♥❤❛ ✉♠ ♣❛♣❡❧ ♠❛✐s ✐♠♣♦rt❛♥t❡✳
❙❡✱ ♥♦ ❆❧❣♦r✐t♠♦ ✷✳✷✱ ♦ ♣❛râ♠❡tr♦ ❞❡ ♣❡♥❛❧✐❞❛❞❡ ♣r❡❝✐s❛ s❡r ❛✉♠❡♥t❛❞♦ ✐♥✜♥✐t❛s ✈❡③❡s✱ ♦ ❛❧❣♦r✐t♠♦ s❡ r❡❞✉③ ❛♦ ❆❧❣♦r✐t♠♦ ✷✳✶✳ P♦rt❛♥t♦✱ s✉❛s ♣r♦♣r✐❡❞❛❞❡s✱ ❞❛❞❛s ♣❡❧♦s ❚❡♦r❡♠❛s ✷✳✶✕✷✳✺✱ sã♦ ❛s ♠❡s♠❛s✳ ▲♦❣♦✱ ❛ ú♥✐❝❛ ❛♥á❧✐s❡ ❛❞✐❝✐♦♥❛❧ q✉❡ ♠❡r❡❝❡ ♦ ♥♦✈♦ ❛❧❣♦r✐t♠♦ ❝♦rr❡s♣♦♥❞❡ ❛♦ ❝❛s♦ ❡♠ q✉❡✱ ❛ ♣❛rt✐r ❞❡ ❝❡rt♦ k0✱ ♦ ♣❛râ♠❡tr♦
❞❡ ♣❡♥❛❧✐❞❛❞❡ ♥ã♦ ❛✉♠❡♥t❛ ♠❛✐s✳ ❖ ❚❡♦r❡♠❛ ✷✳✻ ❡①♣❧✐❝❛ ♦ q✉❡ ❛❝♦♥t❡❝❡ ♥❡ss❡ ❝❛s♦✳
❚❡♦r❡♠❛ ✷✳✻✳ ❙✉♣♦♥❤❛♠♦s q✉❡ ❛ s❡qüê♥❝✐❛ {xk
} ❡stá ❜❡♠ ❞❡✜♥✐❞❛ ♣❡❧♦ ❆❧❣♦✲
r✐t♠♦ ✷✳✷ ❡ q✉❡✱ ♣❛r❛ t♦❞♦ k ≥ k0✱ t❡♠♦s q✉❡ ρk =ρk0✳ ❙✉♣♦♥❤❛♠♦s q✉❡ x
∗ é ✉♠
♣♦♥t♦ ❧✐♠✐t❡ ❞❛ s❡qüê♥❝✐❛✳ ❊♥tã♦✱ x∗ é ✉♠ ♠✐♥✐♠✐③❛❞♦r ❣❧♦❜❛❧ ❞❡ P◆▲✳
Pr♦✈❛✳ P❛r❛ t♦❞♦ k ≥ k0 ♦ t❡st❡ ❞❡ ♠❡❧❤♦r❛ ❞❛ ❛❞♠✐ss✐❜✐❧✐❞❛❞❡ ❞❡✈❡ t❡r ❞❛❞♦
r❡s✉❧t❛❞♦ ♣♦s✐t✐✈♦✱ ♣♦rt❛♥t♦✿
max{kh(xk)k∞,kg(xk)+k∞} ≤τ max{kh(xk−1)k∞,kg(xk−1)+k∞}
♣❛r❛ t♦❞♦k ≥k0✳ P♦rt❛♥t♦✱ ❝♦♠♦ τ <1✱
lim
k→∞max{kh(x
k)
k∞,kg(xk)+k∞}= 0.
▲♦❣♦✱ s❡ limk∈Kxk=x∗✱
lim
k∈Kmax{kh(x k)
k∞,kg(xk)+k∞}= 0.
P❡❧❛ ❝♦♥t✐♥✉✐❞❛❞❡ ❞❡ h ❡ g✱ ✐st♦ ✐♠♣❧✐❝❛ q✉❡ kh(x∗)k=kg(x∗)
+k= 0✳ ❖✉ s❡❥❛✱ x∗
é ❛❞♠✐ssí✈❡❧✳
❙❡❥❛ z ∈Ω♦✉tr♦ ♣♦♥t♦ ❛❞♠✐ssí✈❡❧ ❛r❜✐trár✐♦✳ P❡❧❛ ❞❡✜♥✐çã♦ ❞♦ ❛❧❣♦r✐t♠♦✱
f(xk)
≤f(xk) +ρ
k0[kh(x
k)
k2+kg(xk)
+k2] =f(xk) +ρk[kh(xk)k2+kg(xk)+k2]
≤f(z) +ρk[kh(z)k2+kg(z)
+k2] =f(z).
▲♦❣♦✱ f(xk)
≤ f(z) ♣❛r❛ t♦❞♦ k ≥ k0✳ ❚♦♠❛♥❞♦ ❧✐♠✐t❡s ♥❡st❛ ❞❡s✐❣✉❛❧❞❛❞❡✱ ♣❡❧❛
❊❳❊❘❈❮❈■❖❙
✶✳ ▲❡✐❛ ♦s r❡s✉❧t❛❞♦s ❡①✐❜✐❞♦s ♥❡st❡ ❝❛♣ít✉❧♦✳ ❖ q✉❡ ♣♦❞❡ ❢❛❧❤❛r❄ Pr♦❝✉r❡ tr❛❞✉✲ ③✐r ❝❛❞❛ r❡s✉❧t❛❞♦ ❡♠ t❡r♠♦s ❞❡ ❢✉♥❝✐♦♥❛♠❡♥t♦ ♣rát✐❝♦ ❞♦ ♠ét♦❞♦ ❛♥❛❧✐s❛❞♦✳ ✷✳ ❈♦♥s✐❞❡r❡✱ ♥♦s r❡s✉❧t❛❞♦s ❞❡st❡ ❝❛♣ít✉❧♦✱ ❛ ♣♦ss✐❜✐❧✐❞❛❞❡ ❞❡ s✉❜st✐t✉ír ❛s ♥♦r✲ ♠❛s k · k ❡ k · k∞ ♣♦r ♥♦r♠❛s ❛r❜✐trár✐❛s✳ ❊①✐st❡ ❛❧❣✉♠ ✐♠♣❡❞✐♠❡♥t♦ ♣❛r❛
♣r♦❝❡❞❡r ❛ss✐♠❄
✸✳ ❊s❝r❡✈❛ ✉♠ ♣r♦❣r❛♠❛ q✉❡ ✐♠♣❧❡♠❡♥t❡✱ ❞❡ ♠❛♥❡✐r❛ tã♦ ✜❡❧ q✉❛♥t♦ ❢♦r ♣♦ssí✈❡❧✱ ♦s ♠ét♦❞♦s ❞❡ ♣❡♥❛❧✐❞❛❞❡ ❡①t❡r♥❛✳ Pr♦❝✉r❡✱ ❝♦♠ ❛❥✉❞❛ ❞❡ss❡ ♣r♦❣r❛♠❛✱ ❡♥❝♦♥tr❛r ❡①❡♠♣❧♦s ❞❡ s✉❝❡ss♦ ❡ ❢r❛❝❛ss♦ ❞♦ ♠ét♦❞♦✳ ❊s❝r❡✈❛ ✉♠❛ ♠♦♥♦❣r❛✜❛ ❡①♣❧✐❝❛♥❞♦ ❛s ❢♦r♠❛s q✉❡ ♦ ❢r❛❝❛ss♦ t♦♠❛ ♥❛ ♣rát✐❝❛✱ ❡ s❡✉ ❝♦rr❡❧❛t♦ t❡ór✐❝♦✳ ✹✳ ❈♦♥s✐❞❡r❡ ♦ ♣r♦❜❧❡♠❛ ✏❞❡ ❞♦✐s ♥í✈❡✐s✑
▼✐♥✐♠✐③❛r f(x)
s✉❥❡✐t❛ à ❝♦♥❞✐çã♦ ❞❡ q✉❡ x é ♠✐♥✐♠✐③❛❞♦r ❣❧♦❜❛❧ ❞❡ f1(x)✳ ❆♥❛❧✐s❡ s✉❛
♣♦ssí✈❡❧ r❡s♦❧✉çã♦ ❛tr❛✈és ❞❡ s✉❜♣r♦❜❧❡♠❛s ❞♦ t✐♣♦ ▼✐♥✐♠✐③❛rf(x) +ρf1(x).
❙✉❣❡stã♦✿ s✉♣♦♥❤❛ q✉❡ c é ✉♠ ❧✐♠✐t❛♥t❡ ✐♥❢❡r✐♦r ❞❡ f1(x) ❡ ❝♦♥s✐❞❡r❡ ❛ r❡s✲
tr✐çã♦ h1(x) = 0✱ ♦♥❞❡ h1(x) =
q
f1(x)−c✳
✺✳ ❈♦♥s✐❞❡r❡ ♦ ♣r♦❜❧❡♠❛ ❞❡ ❞♦✐s ♥í✈❡✐s ✏❝♦♠ r❡str✐çõ❡s ♥♦ ♥í✈❡❧ ✐♥❢❡r✐♦r✑✿ ▼✐♥✐♠✐③❛rx,uf(x, u)
s✉❥❡✐t❛ ❛ q✉❡ x é ♠✐♥✐♠✐③❛❞♦r ✭❝♦♠ r❡s♣❡✐t♦ à ♣r✐♠❡✐r❛ ✈❛r✐á✈❡❧✮ ❞❡ f1(x, u)
❝♦♠ ❞❡t❡r♠✐♥❛❞❛s r❡str✐çõ❡s✳ ❯s❛♥❞♦ ♦ ❡①❡r❝í❝✐♦ ❛♥t❡r✐♦r✱ ❞❡✜♥❛ ✉♠ ♠é✲ t♦❞♦ ❞❡ ♣❡♥❛❧✐❞❛❞❡ ❡①t❡r♥❛ ❞✉♣❧❛ ♣❛r❛ r❡s♦❧✈❡r ❡st❡ ♣r♦❜❧❡♠❛ ❡ ❛♥❛❧✐s❡ s✉❛s ♣r♦♣r✐❡❞❛❞❡s✳
✻✳ ❆❝❡ss❡ ❛ ♣á❣✐♥❛ ❞♦ ♣r♦❥❡t♦ ❚❆◆●❖ ✇✇✇✳✐♠❡✳✉s♣✳❜r✴∼❡❣❜✐r❣✐♥✴t❛♥❣♦✱
❝♦♣✐❡ ♦ ♣r♦❣r❛♠❛ ❆❧❣❡♥❝❛♥✱ ❡st✉❞❡ ♦s ❝♦♠❡♥tár✐♦s ❡ ♦ ♣ró♣r✐♦ ♣r♦❣r❛♠❛ t❛♥t♦ q✉❛♥t♦ ❢♦r ♣♦ssí✈❡❧ ❡ ✉s❡ ♦ ♣r♦❣r❛♠❛ ♣❛r❛ ✏✈❡r✐✜❝❛r✑ ♦s r❡s✉❧t❛❞♦s t❡ór✐❝♦s ❡♥✉♥❝✐❛❞♦s ♥❡st❡ ❝❛♣ít✉❧♦✱ ✐♠♣❧❡♠❡♥t❛♥❞♦ ❡①❡♠♣❧♦s ❛❞❡q✉❛❞♦s ❞❡ s✉❛ ✐♥✈❡♥çã♦✳
✼✳ Pr♦✈❡ q✉❡✱ ♥♦ ▼ét♦❞♦ ❞❡ P❡♥❛❧✐❞❛❞❡ ❊①t❡r♥❛✱ f(xk+1) ≥ f(xk) ♣❛r❛ t♦❞♦
Cap´ıtulo 3
M´
etodo de Penalidade
Deslocada
❈♦♥s✐❞❡r❡♠♦s ♦ ♣r♦❜❧❡♠❛ P◆▲✱ ❞❡✜♥✐❞♦ ♥♦ ❈❛♣ít✉❧♦ ✷ ❝♦♠ ❛s ♠❡s♠❛s ❤✐♣ót❡s❡s ❞❡ ❝♦♥t✐♥✉✐❞❛❞❡ s♦❜r❡ f✱ h ❡ g ❡♥✉♥❝✐❛❞❛s ♥❡ss❡ ❝❛♣ít✉❧♦✳ ❱✐♠♦s ♥♦ ❚❡♦r❡♠❛ ✷✳✺ q✉❡✱ s❡ ✉♠ ✐t❡r❛♥❞♦ xk ❢♦r ❛❞♠✐ssí✈❡❧✱ ❡♥tã♦ ❡ss❡ ✐t❡r❛♥❞♦ é ❛ s♦❧✉çã♦ ❞❡ P◆▲✳ ■♥✲
❢❡❧✐③♠❡♥t❡✱ ♦ ❝❛s♦ ❡♠ q✉❡xk é ❛❞♠✐ssí✈❡❧ é ♣♦✉❝♦ ❢r❡qü❡♥t❡✳ ❆ r❛③ã♦ ❞❡ss❛ ❢❛❧t❛ ❞❡
❢r❡qüê♥❝✐❛ é ❢á❝✐❧ ❞❡ ❡♥t❡♥❞❡r✿ ❛ ❢✉♥çã♦ q✉❡ ✏❝❛st✐❣❛✑ ❛ ♥ã♦✲❛❞♠✐ss✐❜✐❧✐❞❛❞❡ ❞❡ ✉♠ ♣♦♥t♦xé✱ ♥♦ ▼ét♦❞♦ ❞❡ P❡♥❛❧✐❞❛❞❡ ❊①t❡r♥❛✱ ρk
2 [kh(x)k2+kg(x)+k2]✳ ❖ ♣❛râ♠❡tr♦
❞❡ ♣❡♥❛❧✐❞❛❞❡ρk ♣♦❞❡ s❡r ❣r❛♥❞❡✱ ♠❛s ❛s ♠❡❞✐❞❛s ❞❡ ♥ã♦✲❛❞♠✐ss✐❜✐❧✐❞❛❞❡kh(x)k❡ kg(x)+k❛♣❛r❡❝❡♠ ❡❧❡✈❛❞❛s ❛♦ q✉❛❞r❛❞♦ ♥❡st❛ ❢✉♥çã♦✳ ■st♦ s✐❣♥✐✜❝❛ q✉❡✱ ♣❛r❛ ♣♦♥✲
t♦s x ❧❡✈❡♠❡♥t❡ ♥ã♦✲❛❞♠✐ssí✈❡✐s✱ ♦ ❝❛st✐❣♦ é ❜r❛♥❞♦ ✭ kh(x)k<1 ⇒ kh(x)k2 ≪ 1
❡kg(x)+k<1⇒ kg(x)+k2 ≪1✮✳ P♦r ♦✉tr♦ ❧❛❞♦✱ é ♣r♦✈á✈❡❧ q✉❡ ❛ ❢✉♥çã♦ ♦❜❥❡t✐✈♦
f(x)t❡♥❤❛✱ ♥♦ ♣♦♥t♦ ♥ã♦✲❛❞♠✐ssí✈❡❧ x✱ ✉♠ ✈❛❧♦r ❢♦rt❡♠❡♥t❡ ♠❡♥♦r q✉❡ ♥♦s ♣♦♥t♦s
❛❞♠✐ssí✈❡✐s ♠❛✐s ♣ró①✐♠♦s ✭s❡ ♥ã♦ ❢♦ss❡ ❛ss✐♠✱ ❛s r❡str✐çõ❡s ♥ã♦ s❡r✐❛♠ ♥❡❝❡ssá✲ r✐❛s✦✮✳ ❊♠ t❛✐s ❝✐r❝✉♥stâ♥❝✐❛s✱ ♣❛r❛ q✉❛❧q✉❡r ✈❛❧♦r ✜♥✐t♦ ❞❡ ρk ♦ ❜❛✐①♦ ✈❛❧♦r ❞❡
f(x) ❝♦♠♣❡♥s❛rá ♦ ❝❛st✐❣♦ ♣❡❧❛ ♥ã♦✲❛❞♠✐ss✐❜✐❧✐❞❛❞❡✳ ▲♦❣♦✱ é ♣♦ssí✈❡❧ ❛✜r♠❛r q✉❡✱
q✉❛s❡ s❡♠♣r❡✱ ♦ ▼ét♦❞♦ ❞❡ P❡♥❛❧✐❞❛❞❡ ❊①t❡r♥❛ ❛♣r♦①✐♠❛ ❛ s♦❧✉çã♦ ❞♦ ♣r♦❜❧❡♠❛ P◆▲ ♣♦r ♠❡✐♦ ❞❡ ✉♠❛ s❡qüê♥❝✐❛ ❞❡ ♣♦♥t♦s ♥ã♦✲❛❞♠✐ssí✈❡✐s✳ ❖ ❧❡✐t♦r ✜❝❛rá ♠❛✐s ❝♦♥✈❡♥❝✐❞♦ ❞❡st❛ ♣r♦♣r✐❡❞❛❞❡ ❝♦♠♣✉t❛♥❞♦ ♦s ✐t❡r❛♥❞♦s ❡♠ ✉♠ ❡①❡♠♣❧♦ ♥✉♠ér✐❝♦ s✐♠♣❧❡s✿ ▼✐♥✐♠✐③❛r x✱ s✉❥❡✐t❛ ❛−x≤0✳
➱ ✉♠ ♣♦✉❝♦ ❢r✉str❛♥t❡ ♦ ❢❛t♦ ❞❡ q✉❡ t♦❞♦s ♦s ✐t❡r❛♥❞♦s ❞❡ ✉♠ ♠ét♦❞♦ s❡❥❛♠ ♥ã♦✲❛❞♠✐ssí✈❡✐s✱ ❡♠❜♦r❛ ♦ ♠ét♦❞♦ ❝♦♥✈✐r❥❛ ❛ ✉♠❛ s♦❧✉çã♦✳ ■st♦ ♣♦rq✉❡✱ ♥❛ ♣rát✐❝❛✱ s❡♠♣r❡ ♣❛r❛♠♦s ❛ ❡①❡❝✉çã♦ ❞❡ ✉♠ ❛❧❣♦r✐t♠♦ ❡♠ ✉♠❛ ✐t❡r❛çã♦ k✱ ♥ã♦ ♥♦ ✐♥✜♥✐t♦✳ ❊♠ ♠✉✐t♦s ♣r♦❜❧❡♠❛s r❡❛✐s✱ ♣♦♥t♦s ❛❞♠✐ssí✈❡✐s ❧❡✈❡♠❡♥t❡ ♥ã♦✲ót✐♠♦s sã♦ t♦❧❡rá✲ ✈❡✐s ♠❛s ♣♦♥t♦s ♥ã♦✲❛❞♠✐ssí✈❡✐s ♥ã♦ tê♠ ✉t✐❧✐❞❛❞❡ ❛❧❣✉♠❛✳ P♦r ✐ss♦✱ ♣r♦❝✉r❛♠♦s ❝♦rr✐❣✐r ♦ ▼ét♦❞♦ ❞❡ P❡♥❛❧✐❞❛❞❡ ❊①t❡r♥❛ ♣❛r❛ ❛✉♠❡♥t❛r ❛s ❝❤❛♥❝❡s ❞❡ q✉❡ ✉♠ ✐t❡r❛♥❞♦ xk s❡❥❛ s♦❧✉çã♦ ♦✉✱ ♣❡❧♦ ♠❡♥♦s✱ ♣❛r❛ q✉❡ xk s❡❥❛ ❛❞♠✐ssí✈❡❧✳ P❛r❛ ✜①❛r
✐❞é✐❛s✱ s✉♣♦♥❤❛♠♦s q✉❡ m = 0✱ p = 1✳ ✭❘❡❝♦♠❡♥❞❛♠♦s ❛❝♦♠♣❛♥❤❛r ❡st❛ ❧❡✐t✉r❛
❝♦♠ s❡✉ ❞❡s❡♥❤♦ ❢❛✈♦r✐t♦✳✮ ❙❡ f(x) ❞❡s❝❡ ❢♦rt❡♠❡♥t❡ q✉❛♥❞♦ ♣❛ss❛♠♦s ❞♦ ♣♦♥t♦
ót✐♠♦ ❛ ♣♦♥t♦s ♥ã♦✲❛❞♠✐ssí✈❡✐s ♦ ❝❛st✐❣♦ q✉❛❞rát✐❝♦ ♥ã♦ s❡rá s✉✜❝✐❡♥t❡ ♣❛r❛ ❢❛③❡r
❛❞♠✐ssí✈❡❧ ♥❡♥❤✉♠ ❞♦s ✐t❡r❛♥❞♦sxk✳ ❊♥tr❡t❛♥t♦✱ ✐st♦ ❛❝♦♥t❡❝❡ ♣♦rq✉❡ ♥♦ ▼ét♦❞♦
❞❡ P❡♥❛❧✐❞❛❞❡ ❊①t❡r♥❛ ❡st❛♠♦s ❝❛st✐❣❛♥❞♦ ♦ ❞❡s✈✐♦ ❞❡ g(x)+ ❝♦♠ r❡s♣❡✐t♦ ❛♦ ✈❡✲
t♦r 0✳ ❙❡✱ ❡♠ ✈❡③ ❞✐ss♦✱ ❛♣❧✐❝áss❡♠♦s ✉♠❛ ♣❡♥❛❧✐❞❛❞❡✱ ♥ã♦ ❛♦ ❞❡s✈✐♦ ❡♠ r❡❧❛çã♦
❛♦ 0✱ ♠❛s ❛♦ ❞❡s✈✐♦ ❡♠ r❡❧❛çã♦ ❛ ✉♠ ✈❡t♦r ♥❡❣❛t✐✈♦ w✱ ♦ t❡r♠♦ ❞❡ ♣❡♥❛❧✐❞❛❞❡
❝♦rr❡s♣♦♥❞❡♥t❡ s❡r✐❛ k(g(x)−w)+k2 ❡ ♦ ♠✐♥✐♠✐③❛❞♦r ❞♦ s✉❜♣r♦❜❧❡♠❛ ✜❝❛r✐❛ ❧❡✲
✈❡♠❡♥t❡ ❞❡s❧♦❝❛❞♦ ❡ ♠❛✐s ♣❡rt♦ ❞❛ ❛❞♠✐ss✐❜✐❧✐❞❛❞❡✳ ❆ss✐♠✱ ❛s ❝❤❛♥❝❡s ❞❡ q✉❡ xk
❢♦ss❡ ❛❞♠✐ssí✈❡❧ ❡✱ t❛❧✈❡③✱ q✉❛s❡ ót✐♠♦✱ ❛✉♠❡♥t❛r✐❛♠✳ ❋❛③❡♥❞♦ ♦ ♠❡s♠♦ r❛❝✐♦❝í♥✐♦ ❡♠ r❡❧❛çã♦ ❛ h(x) ♣❛r❡❝❡ r❡❝♦♠❡♥❞á✈❡❧ q✉❡ ❛ ❢✉♥çã♦ ❞❡ ❝❛st✐❣♦ ❜ás✐❝❛✱ ❡♠ ✈❡③ ❞❡ kh(x)k2+kg(x)
+k2✱ s❡❥❛
kh(x)−vk
k2+k(g(x)−wk)
+k2
♣❛r❛ ✈❡t♦r❡s vk∈IRm ❡wk∈IRp
− q✉❡✱ ♣♦r ❡♥q✉❛♥t♦✱ ♥ã♦ s❛❜❡♠♦s ❝♦♠♦ ❝❛❧❝✉❧❛r✳
◆♦ ❝❛s♦ vk = 0, wk = 0 t❡r❡♠♦s ♦ ▼ét♦❞♦ ❞❡ P❡♥❛❧✐❞❛❞❡ ❊①t❡r♥❛ ❞❡✜♥✐❞♦ ♥♦
❈❛♣ít✉❧♦ ✷✳ ❆❣♦r❛✱ q✉❛♥❞♦ρké ♠✉✐t♦ ❣r❛♥❞❡ ❛ s♦❧✉çã♦ ❞♦ s✉❜♣r♦❜❧❡♠❛ ♥♦ ▼ét♦❞♦
❞❡ P❡♥❛❧✐❞❛❞❡ ❊①t❡r♥❛ ❛♣r♦①✐♠❛ ♠✉✐t♦ ❜❡♠ ❛ s♦❧✉çã♦ ❞♦ P◆▲✳ ◆❡ss❡ ❝❛s♦✱ ♥ã♦ ♣❛r❡❝❡ ❝♦♥✈❡♥✐❡♥t❡ ❛❢❛st❛r✲s❡ ❞❡♠❛s✐❛❞♦ ❞❡ v = 0, w = 0✳ ❊st❡ r❡q✉✐s✐t♦ ♣♦❞❡ s❡r
❢♦rç❛❞♦ ❞❡✜♥✐♥❞♦✿
vk=−¯λ k
ρk
, wk =−µ¯ k
ρk
❡ ✐♠♣♦♥❞♦ ❛ ❝♦♥❞✐çã♦ ❞❡ q✉❡ ¯λk ∈ IRm ❡ µ¯k ∈ IRp
+ ❡st❡❥❛♠ ❡♠ ✉♠ ❝♦♥❥✉♥t♦
❧✐♠✐t❛❞♦✱ ✐♥❞❡♣❡♥❞❡♥t❡♠❡♥t❡ ❞❡ k✳
❈♦♠ ✐st♦✱ ❡ ❛♣❡s❛r ❞❡ q✉❡ ♥ã♦ s❛❜❡♠♦s ❝❛❧❝✉❧❛r λ¯k ♥❡♠ µ¯k✱ ♣♦❞❡♠♦s ❞❡✜♥✐r ♦
▼ét♦❞♦ ❞❡ P❡♥❛❧✐❞❛❞❡ ❉❡s❧♦❝❛❞❛ ❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛✿
❆❧❣♦r✐t♠♦ ✸✳✶✳
❙❡❥❛ x0 ∈ Ω ✉♠ ♣♦♥t♦ ✐♥✐❝✐❛❧ ❛r❜✐trár✐♦✳ ❙❡❥❛♠ λ
min ≤ λmax ❡ µmax > 0 ♦s
♣❛râ♠❡tr♦s q✉❡ ❞❡✜♥❡♠ ❛ ❧✐♠✐t❛çã♦ ❞❡ λ¯k ❡µ¯k✳ ❙✉♣♦♥❤❛♠♦s q✉❡
¯
λ1i ∈[λmin, λmax]∀i= 1, . . . , m
❡
¯
µ1i ∈[0, µmax]∀i= 1, . . . , p.
❙❡❥❛ {ρk} ✉♠❛ s❡qüê♥❝✐❛ ❞❡ ♥ú♠❡r♦s ♣♦s✐t✐✈♦s q✉❡ t❡♥❞❡ ❛ ✐♥✜♥✐t♦✳ ■♥✐❝✐❛❧✐③❛r ♦
✏❝♦♥t❛❞♦r✑ k ←1✳
P❛ss♦ ✶✳ ❊♥❝♦♥tr❛r xk ∈Ω s♦❧✉çã♦ ❣❧♦❜❛❧ ❞♦ s✉❜♣r♦❜❧❡♠❛
▼✐♥✐♠✐③❛r f(x) + ρk 2
h(x) +
¯
λk
ρk
2
+
g(x) + µ¯ k
ρk
+
2
13 P❛ss♦ ✷✳ ❈❛❧❝✉❧❛r ¯λk+1
i ∈ [λmin, λmax] ♣❛r❛ t♦❞♦ i = 1, . . . , m ❡ µ¯ki+1 ∈ [0, µmax]
♣❛r❛ t♦❞♦i= 1, . . . , p✳ ❆t✉❛❧✐③❛r k←k+ 1 ❡ ✈♦❧t❛r ❛♦ P❛ss♦ ✶✳
❖❜s❡r✈❡ q✉❡✱ ❛♣❡s❛r ❞❡ q✉❡ ♣❛r❛ ❞❡✜♥✐r ♦ ❆❧❣♦r✐t♠♦ ✸✳✶ ✉s❛♠♦s ♦ ♠♦❞❡❧♦ ❞♦ ❆❧❣♦r✐t♠♦ ✷✳✷✱ ♥ã♦ ♦✉s❛♠♦s✱ ♣♦r ❡♥q✉❛♥t♦✱ r❡♣r♦❞✉③✐r ♦ P❛ss♦ ✷ ❞❛q✉❡❧❡ ❛❧❣♦r✐t♠♦✳ ❉❡✐①❛r❡♠♦s ❛ ♣♦ss✐❜✐❧✐❞❛❞❡ ❞❡ ♠❛♥t❡r s❡♠ ♠♦❞✐✜❝❛çã♦ ♦ ♣❛râ♠❡tr♦ ❞❡ ♣❡♥❛❧✐❞❛❞❡ ♣❛r❛ ♠❛✐s ❛❞✐❛♥t❡ ❡✱ ♣♦r ❡♥q✉❛♥t♦✱ ♥♦s ❝♦♥t❡♥t❛r❡♠♦s ❡♠ ❡st✉❞❛r ❛s ♣r♦♣r✐❡❞❛❞❡s t❡ór✐❝❛s ❞♦ ❆❧❣♦r✐t♠♦ ✸✳✶✳ ❊st❛s ♣r♦♣r✐❡❞❛❞❡s s❡rã♦ ❣❡♥❡r❛❧✐③❛çõ❡s ❞♦s t❡♦r❡♠❛s ♣r♦✈❛❞♦s ♥♦ ❈❛♣ít✉❧♦ ✷✱ ♦ q✉❡ ♥ã♦ é s✉r♣r❡❡♥❞❡♥t❡ ♣♦✐s✱ ❞❡ ❢❛t♦✱ ♦ ❆❧❣♦r✐t♠♦ ✷✳✶ é ♦ ❝❛s♦ ♣❛rt✐❝✉❧❛r ❞♦ ❆❧❣♦r✐t♠♦ ✸✳✶ ❡♠ q✉❡λ¯k= 0,µ¯k = 0 ♣❛r❛ t♦❞♦ k✳
❈♦♠❡❝❡♠♦s ♣❡❧♦s t❡♦r❡♠❛s r❡❧❛t✐✈♦s à ❛❞♠✐ss✐❜✐❧✐❞❛❞❡ ❞♦s ♣♦♥t♦s ❧✐♠✐t❡s ❞❡ s❡qüê♥❝✐❛s ❣❡r❛❞❛s ♣❡❧♦ ❆❧❣♦r✐t♠♦ ✸✳✶✳
◆♦s ❚❡♦r❡♠❛s ✸✳✶✕✸✳✺ s✉♣♦r❡♠♦s q✉❡ ❛ s❡qüê♥❝✐❛ {xk} ❡stá ❜❡♠ ❞❡✜♥✐❞❛ ♣❡❧♦
❆❧❣♦r✐t♠♦ ✸✳✶✳
❚❡♦r❡♠❛ ✸✳✶✳ ❙✉♣♦♥❤❛♠♦s q✉❡ ❛ s❡qüê♥❝✐❛ {xk} ❛❞♠✐t❡ ✉♠ ♣♦♥t♦ ❧✐♠✐t❡ x∗✳
❊♥tã♦✱ x∗ é ✉♠❛ s♦❧✉çã♦ ❣❧♦❜❛❧ ❞♦ ♣r♦❜❧❡♠❛
▼✐♥✐♠✐③❛r kh(x)k2+kg(x)+k2 s✉❥❡✐t❛ ❛ x∈Ω. ✭✸✳✷✮
Pr♦✈❛✳ ❊st❛ ♣r♦✈❛ ❣❡♥❡r❛❧✐③❛ ❛ ❞♦ ❚❡♦r❡♠❛ ✷✳✶✳ ❙❡❥❛ K⊂∞IN t❛❧ q✉❡
lim k∈Kx
k =x∗.
❈♦♠♦ xk → x∗ ♣❛r❛ k ∈ K✱ xk ∈ Ω ♣❛r❛ t♦❞♦ k✱ ❡ Ω é ❢❡❝❤❛❞♦ ✱ t❡♠♦s q✉❡
x∗ ∈Ω✳
❙✉♣♦♥❤❛♠♦s q✉❡ x∗ ♥ã♦ é ✉♠ ♠✐♥✐♠✐③❛❞♦r ❣❧♦❜❛❧ ❞❡ ✭✸✳✷✮✳ P♦rt❛♥t♦✱ ❡①✐st❡ z ∈Ωt❛❧ q✉❡
kh(x∗)k2+kg(x∗)
+k2 >kh(z)k2+kg(z)+k2.
❈♦♠♦ h ❡ g sã♦ ❝♦♥tí♥✉❛s ❡✱ ❛❧é♠ ❞✐ss♦✱ λ¯k ❡ µ¯k sã♦ ❧✐♠✐t❛❞❛s ❡ ρ
k → ∞✱ ❡①✐st❡
c >0 t❛❧ q✉❡ ♣❛r❛ k ∈K s✉✜❝✐❡♥t❡♠❡♥t❡ ❣r❛♥❞❡✿
h(xk) +
¯ λk ρk 2 +
g(xk) + µ¯ k ρk + 2 >
h(z) +
¯ λk ρk 2 +
g(z) + µ¯ k ρk + 2 +c.
P♦rt❛♥t♦✱ ♣❛r❛ ❡ss❡s í♥❞✐❝❡s k✱
f(xk) + ρk 2
h(xk) +
¯ λk ρk 2 +
g(xk) + µ¯ k ρk + 2
> f(z) + ρk 2
h(z) +
¯ λk ρk 2 +
g(z) + µ¯ k ρk + 2
+ ρkc 2 +f(x
k
