PROBLEMAS DE PROGRAMAÇÃO LINEAR: UMA PROPOSTA DE RESOLUÇÃO GEOMÉTRICA PARA O ENSINO MÉDIO COM O USO DO GEOGEBRA

❯♥✐✈❡rs✐❞❛❞❡ ❊st❛❞✉❛❧ P❛✉❧✐st❛ ✏❏ú❧✐♦ ❞❡ ▼❡sq✉✐t❛ ❋✐❧❤♦✑ ■♥st✐t✉t♦ ❞❡ ●❡♦❝✐ê♥❝✐❛s ❡ ❈✐ê♥❝✐❛s ❊①❛t❛s

❈â♠♣✉s ❞❡ ❘✐♦ ❈❧❛r♦

Pr♦❜❧❡♠❛s ❞❡ Pr♦❣r❛♠❛çã♦ ▲✐♥❡❛r✿ ✉♠❛

♣r♦♣♦st❛ ❞❡ r❡s♦❧✉çã♦ ❣❡♦♠étr✐❝❛ ♣❛r❛ ♦

❊♥s✐♥♦ ▼é❞✐♦ ❝♦♠ ♦ ✉s♦ ❞♦ ●❡♦●❡❜r❛

❏✉❧✐❛♥❛ ▼❛❧❧✐❛ ❩❛❝❤✐

❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ Pr♦❣r❛♠❛ ❞❡ Pós✲ ●r❛❞✉❛çã♦ ✕ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡✲ ♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r✲ ❝✐❛❧ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ ❣r❛✉ ❞❡ ▼❡str❡

❖r✐❡♥t❛❞♦r❛

Pr♦❢❛✳ ❉r❛✳ ❆r✐❛♥❡ ▲✉③✐❛ ❞♦s ❙❛♥t♦s

✺✶✾✳✼✷ ❩✶✻♣

❩❛❝❤✐✱ ❏✉❧✐❛♥❛ ▼❛❧❧✐❛

Pr♦❜❧❡♠❛s ❞❡ Pr♦❣r❛♠❛çã♦ ▲✐♥❡❛r✿ ✉♠❛ ♣r♦♣♦st❛ ❞❡ r❡s♦❧✉çã♦ ❣❡♦♠étr✐❝❛ ♣❛r❛ ♦ ❊♥s✐♥♦ ▼é❞✐♦ ❝♦♠ ♦ ✉s♦ ❞♦ ●❡♦●❡❜r❛✴ ❏✉❧✐❛♥❛ ▼❛❧❧✐❛ ❩❛❝❤✐✲ ❘✐♦ ❈❧❛r♦✱ ✷✵✶✻✳

✶✶✺ ❢✳✱ ✐❧✳✱ ✜❣s✳✱ ❣rá❢s✳✱ t❛❜s✳

❉✐ss❡rt❛çã♦ ✭♠❡str❛❞♦✮ ✲ ❯♥✐✈❡rs✐❞❛❞❡ ❊st❛❞✉❛❧ P❛✉❧✐st❛✱ ■♥st✐✲ t✉t♦ ❞❡ ●❡♦❝✐ê♥❝✐❛s ❡ ❈✐ê♥❝✐❛s ❊①❛t❛s✳

❖r✐❡♥t❛❞♦r❛✿ ❆r✐❛♥❡ ▲✉③✐❛ ❞♦s ❙❛♥t♦s

✶✳ Pr♦❣r❛♠❛çã♦ ▲✐♥❡❛r✳ ✷✳ ➪❧❣❡❜r❛ ▲✐♥❡❛r✳ ✸✳ Pr♦❣r❛♠❛çã♦ ▲✐♥❡❛r ●❡♦♠étr✐❝❛✳ ■✳ ❚ít✉❧♦

❚❊❘▼❖ ❉❊ ❆P❘❖❱❆➬➹❖

❏✉❧✐❛♥❛ ▼❛❧❧✐❛ ❩❛❝❤✐

Pr♦❜❧❡♠❛s ❞❡ Pr♦❣r❛♠❛çã♦ ▲✐♥❡❛r✿ ✉♠❛ ♣r♦♣♦st❛ ❞❡

r❡s♦❧✉çã♦ ❣❡♦♠étr✐❝❛ ♣❛r❛ ♦ ❊♥s✐♥♦ ▼é❞✐♦ ❝♦♠ ♦ ✉s♦ ❞♦

●❡♦●❡❜r❛

❉✐ss❡rt❛çã♦ ❛♣r♦✈❛❞❛ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ ❣r❛✉ ❞❡ ▼❡str❡ ♥♦ ❈✉rs♦ ❞❡ Pós✲●r❛❞✉❛çã♦ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ ❞♦ ■♥st✐t✉t♦ ❞❡ ●❡♦❝✐ê♥❝✐❛s ❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❞❛ ❯♥✐✲ ✈❡rs✐❞❛❞❡ ❊st❛❞✉❛❧ P❛✉❧✐st❛ ✏❏ú❧✐♦ ❞❡ ▼❡sq✉✐t❛ ❋✐❧❤♦✑✱ ♣❡❧❛ s❡❣✉✐♥t❡ ❜❛♥❝❛ ❡①❛♠✐♥❛❞♦r❛✿

Pr♦❢❛✳ ❉r❛✳ ❆r✐❛♥❡ ▲✉③✐❛ ❞♦s ❙❛♥t♦s ❖r✐❡♥t❛❞♦r❛

Pr♦❢❛✳ ❉r❛✳ ❘❡♥❛t❛ ❩♦t✐♥ ●♦♠❡s ❞❡ ❖❧✐✈❡✐r❛ ❯♥❡s♣ ✲ ■●❈❊ ✲ ❘✐♦ ❈❧❛r♦

Pr♦❢❛✳ ❉r❛✳ ➱r✐❝❛ ❘❡❣✐♥❛ ❋✐❧❧❡tt✐ ◆❛s❝✐♠❡♥t♦ ❯♥❡s♣ ✲ ■◗ ✲ ❆r❛r❛q✉❛r❛

❆❣r❛❞❡❝✐♠❡♥t♦s

❆❣r❛❞❡ç♦ ❛ ❉❡✉s ♣♦r t❡r ♠❡ ❣✉✐❛❞♦ ❝♦♠ s❛❜❡❞♦r✐❛ ❞✉r❛♥t❡ t♦❞❛ ♠✐♥❤❛ ❥♦r♥❛❞❛✳

➚ ♠✐♥❤❛ ♦r✐❡♥t❛❞♦r❛ ❆r✐❛♥❡✱ ♣❡❧❛ ❛❝♦❧❤✐❞❛ ❡ ♦r✐❡♥t❛çã♦✱ ♣❡❧❛s ♣❛❧❛✈r❛s ❞❡ ✐♥❝❡♥✲ t✐✈♦ ❛ ❝❛❞❛ ❡t❛♣❛ ❝♦♥❝❧✉í❞❛ ❡ ♣❡❧❛s s✉❣❡stõ❡s ❞❡ ♠❡❧❤♦r✐❛s ❞✉r❛♥t❡ ❛ ❡❧❛❜♦r❛çã♦ ❞❡st❡ tr❛❜❛❧❤♦✳

❆♦ ♠❡✉ ♠❛r✐❞♦ ❈❧❛✉❞✐♥❡✐✱ ♣❡❧♦ ❛♣♦✐♦✱ ❝♦♠♣r❡❡♥sã♦ ❡ ♣❛❝✐ê♥❝✐❛✳

❆♦s ♠❡✉s ♣❛✐s✱ ▼❛r❝í❧✐♦ ❡ ❈❧❡✉③❛✱ ♣♦✐s ❢♦r❛♠ ♠❡✉s ♠❛✐♦r❡s ✐♥❝❡♥t✐✈❛❞♦r❡s✳ ◆♦s ♠♦♠❡♥t♦s ❞❡ ❝❛♥s❛ç♦✱ ❡r❛♠ q✉❡♠ ♠❡ ♠♦t✐✈❛✈❛♠ ❛ ♣r♦ss❡❣✉✐r✳

➚ ♠✐♥❤❛ ✐r♠ã ❏✉ss❛r❛✱ q✉❡ ♠❡ ♠♦str♦✉ ✉♠❛ r❡❛❧✐❞❛❞❡ ♣r♦✜ss✐♦♥❛❧ q✉❡ ❡✉ ❣♦st❛r✐❛ ❞❡ s❡❣✉✐r✳

❆ t♦❞♦s ♦s ♠❡✉s ❝♦❧❡❣❛s ❞❡ ♠❡str❛❞♦✱ ❡s♣❡❝✐❛❧♠❡♥t❡ ❛ ❆❧✐♥❡✱ ❈❛r✐♥❛ ❡ ❆♥❞ré✱ ♣♦✐s ❡st✐✈❡r❛♠ s❡♠♣r❡ ♣r❡s❡♥t❡s ♥♦s ♠♦♠❡♥t♦s ♠❛✐s ❞✐❢í❝❡✐s✱ ❡♠ q✉❡ ❛❧❣✉♠❛s ✈❡③❡s ♣❡♥s❡✐ ❡♠ ❞❡s✐st✐r✳

❆ t♦❞♦s ♦s ♣r♦❢❡ss♦r❡s ❞♦ P❘❖❋▼❆❚✱ ♣♦r ❛❝r❡❞✐t❛r❡♠ ♥♦ ♣r♦❣r❛♠❛ ❡ ❡♠ ♠✐♠✳

❆♦s ❢✉♥❝✐♦♥ár✐♦s ❞❛ ❙❡çã♦ ❞❡ Pós ●r❛❞✉❛çã♦✱ ❞❛ ❇✐❜❧✐♦t❡❝❛ ❡ t♦❞♦s q✉❡ ♣❛rt✐❝✐♣❛✲ r❛♠ ❞✐r❡t❛ ♦✉ ✐♥❞✐r❡t❛♠❡♥t❡ ♣❛r❛ ✈✐❛❜✐❧✐③❛r ❛ r❡❛❧✐③❛çã♦ ❞❡st❡ tr❛❜❛❧❤♦✳

❆ ●❡♦♠❡tr✐❛ ❡①✐st❡ ♣♦r t♦❞❛ ♣❛rt❡✳ ➱ ♣r❡❝✐s♦✱ ♣♦ré♠✱ ♦❧❤♦s ♣❛r❛ ✈ê✲❧❛✱ ✐♥t❡❧✐❣ê♥❝✐❛ ♣❛r❛ ❝♦♠♣r❡❡♥❞ê✲❧❛ ❡ ❛❧♠❛ ♣❛r❛ ❛❞♠✐rá✲❧❛✳

❘❡s✉♠♦

◆❡st❡ tr❛❜❛❧❤♦ sã♦ ❛♣r❡s❡♥t❛❞♦s ♦s ❢✉♥❞❛♠❡♥t♦s ❞❛ Pr♦❣r❛♠❛çã♦ ▲✐♥❡❛r✱ ❡♠ ❡s♣❡✲ ❝✐❛❧✱ ❞❛ Pr♦❣r❛♠❛çã♦ ▲✐♥❡❛r ●❡♦♠étr✐❝❛✱ ✐♥str✉♠❡♥t♦ ✐♠♣♦rt❛♥t❡ ❞❡ ♦t✐♠✐③❛çã♦ ♣❛r❛ ♣r♦❜❧❡♠❛s ❞❡ ❊❝♦♥♦♠✐❛✱ ❣❡stã♦ ❞❡ ❡♠♣r❡s❛s✱ ♣r♦❜❧❡♠❛s ❞❡ tr❛♥s♣♦rt❡s✱ ♦❜t❡♥çã♦ ❞❡ ♠✐st✉r❛s ót✐♠❛s✱ ❡♥tr❡ ♦✉tr♦s✳ ❆❧é♠ ❞✐ss♦✱ é ❛♣r❡s❡♥t❛❞❛ ✉♠❛ ♣r♦♣♦st❛ ❞✐❞át✐❝❛ ♣❛r❛ ♦s ♣r♦❢❡ss♦r❡s ❞❡ ❡❞✉❝❛çã♦ ❜ás✐❝❛ ❞❛ ❡s❝♦❧❛ ♣ú❜❧✐❝❛✱ ✉t✐❧✐③❛♥❞♦ ♦ s♦❢t✇❛r❡ ●❡♦●❡❜r❛ ❝♦♠♦ ✐♥str✉♠❡♥t♦ ♠♦t✐✈❛❞♦r ♣❛r❛ ♦ ❡st✉❞♦ ❞❡ ✉♠❛ s✐t✉❛çã♦ ❞❡ ❛♣r❡♥❞✐③❛❣❡♠ ♣r♦♣♦st❛ ♥♦ ♠❛t❡r✐❛❧ ❞❡ ❛♣♦✐♦ ✐❞❡❛❧✐③❛❞♦ ♣❡❧❛ ❙❡❝r❡t❛r✐❛ ❞❛ ❊❞✉❝❛çã♦ ❞♦ ❊st❛❞♦ ❞❡ ❙ã♦ P❛✉❧♦✱ ❛❜♦r❞❛❞❛ ♥♦ ❝❛❞❡r♥♦ ❞♦ ❛❧✉♥♦ ❞♦ ✸♦ _{❛♥♦ ❞♦ ❊♥s✐♥♦ ▼é❞✐♦✳}

❆❜str❛❝t

■♥ t❤✐s ✇♦r❦ ♣r❡s❡♥ts t❤❡ ❢✉♥❞❛♠❡♥t❛❧s ♦❢ ❧✐♥❡❛r ♣r♦❣r❛♠♠✐♥❣ ✐♥ ♣❛rt✐❝✉❧❛r✱ ♦❢ ❣❡✲ ♦♠❡tr✐❝ ❧✐♥❡❛r ♣r♦❣r❛♠♠✐♥❣✱ ✐♠♣♦rt❛♥t ✐♥str✉♠❡♥t ♦❢ ♦♣t✐♠✐③❛t✐♦♥ ❢♦r ❡❝♦♥♦♠✐❝ ♣r♦✲ ❜❧❡♠s✱ ❜✉s✐♥❡ss ♠❛♥❛❣❡♠❡♥t✱ tr❛♥s♣♦rt ♣r♦❜❧❡♠s✱ ♦❜t❛✐♥✐♥❣ ♦♣t✐♠❛❧ ♠✐①t✉r❡s✱ ❛♠♦♥❣ ♦t❤❡rs✳ ■♥ ❛❞❞✐t✐♦♥✱ ♣r❡s❡♥ts ❛ ❞✐❞❛❝t✐❝ ♣r♦♣♦s❛❧ ❢♦r t❡❛❝❤❡rs ♦❢ t❤❡ ❜❛s✐❝ ❡❞✉❝❛t✐♦♥ ♦❢ ♣✉❜❧✐❝ s❝❤♦♦❧✱ ✉s✐♥❣ t❤❡ ●❡♦●❡❜r❛ s♦❢t✇❛r❡ ❛s ❛ ♠♦t✐✈❛t✐♥❣ t♦♦❧ ❢♦r t❤❡ st✉❞② ♦❢ ❛ ❧❡✲ ❛r♥✐♥❣ s✐t✉❛t✐♦♥ ♣r♦♣♦s❡❞ ✐♥ t❤❡ s✉♣♣♦rt ♠❛t❡r✐❛❧ ❞❡s✐❣♥❡❞ ❜② t❤❡ ❊❞✉❝❛t✐♦♥ s❡❝r❡t❛r② ♦❢ t❤❡ ❙t❛t❡ ♦❢ ❙ã♦ P❛✉❧♦ ✐s ♣r❡s❡♥t❡❞✱ ❛❞r❡ss❡❞ ✐♥ t❤❡ st✉❞❡♥t ♥♦t❡❜♦♦❦ ♦❢ ✸r❞ ②❡❛r ♦❢ ❤✐❣❤ s❝❤♦♦❧✳

▲✐st❛ ❞❡ ❋✐❣✉r❛s

✷✳✶ ❊①❡♠♣❧♦s ❞❡ ❝♦♥❥✉♥t♦s ❝♦♥✈❡①♦s ❡ ♥ã♦ ❝♦♥✈❡①♦s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✷✳✷ ❆✱ ❇ ❡ ❈ sã♦ ✈ért✐❝❡s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✷✳✸ AB✱ BC✱ CA sã♦ ❛r❡st❛s✱ BD ♥ã♦ é ✉♠❛ ❛r❡st❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹

✸✳✶ ❘❡♣r❡s❡♥t❛çã♦ ❞❛ r❡❣✐ã♦ ✈✐á✈❡❧ ❞❡ ✉♠ PP▲ ❡ s✉❛s ❝✉r✈❛s ❞❡ ♥í✈❡❧✳ ✳ ✳ ✳ ✷✼ ✹✳✶ ❋❧✉①♦❣r❛♠❛ ❞❛ ❞❡s❝r✐çã♦ ❣❡r❛❧ ❞♦ ❛❧❣♦rít♠♦ ❙✐♠♣❧❡①✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹ ✺✳✶ ❙♦❧✉çã♦ ót✐♠❛ ❞❡ ✉♠❛ r❡❣✐ã♦ ✈✐á✈❡❧✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸ ✺✳✷ ❙♦❧✉çã♦ ót✐♠❛ r❡♣r❡s❡♥t❛❞❛ ♣♦r ✉♠ s❡❣♠❡♥t♦ ❞❡ r❡t❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸ ✺✳✸ ❙♦❧✉çã♦ ✐❧✐♠✐t❛❞❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹ ✺✳✹ ❙♦❧✉çã♦ ót✐♠❛ ❞❛❞❛ ♣♦r ✉♠❛ s❡♠✐✲r❡t❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹ ✺✳✺ ❈♦♥❥✉♥t♦ s♦❧✉çã♦ ✈❛③✐♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺ ✺✳✻ ❘❡❣✐ã♦ ✈✐á✈❡❧ ❞♦ ❡①❡♠♣❧♦ ✺✳✶✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✼ ✺✳✼ ❙❡♠✐♣❧❛♥♦s ✐♥❞✐❝❛❞♦s ♣❡❧❛s r❡str✐çõ❡s ❞♦ ❊①❡♠♣❧♦ ✺✳✷ ❡ ❛ r❡❣✐ã♦ ✈✐á✈❡❧✳ ✺✾ ✺✳✽ ❘❡❣✐ã♦ ✈✐á✈❡❧ ❞♦ ❡①❡♠♣❧♦ ✺✳✸✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✶ ✺✳✾ ❘❡❣✐ã♦ ✈✐á✈❡❧ ❞♦ ❡①❡♠♣❧♦ ✺✳✺✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✸ ✺✳✶✵ ❘❡❣✐ã♦ ✈✐á✈❡❧ ❞♦ ❡①❡♠♣❧♦ ✺✳✻✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✹ ✺✳✶✶ ❘❡❣✐ã♦ ✈✐á✈❡❧ ❞♦ ❡①❡♠♣❧♦ ✺✳✽✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✺ ✻✳✶ ❈♦❧❡çõ❡s ❞♦ P◆▲❉ ✷✵✶✷ ✲ ❋♦♥t❡✿ ▼❛rt✐♥s✭♣✳✺✸✱ ✷✵✶✷✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✼ ✻✳✷ ❘❡❣✐ã♦ ✈✐á✈❡❧ ❞♦ ❡①❡♠♣❧♦ ✻✳✶✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✾ ✻✳✸ ❊sq✉❡♠❛ ❞❡ ❣❛st♦s ❞❡ tr❛♥s♣♦rt❡ ♣♦r ✉♥✐❞❛❞❡ ❞❡ ♠❡r❝❛❞♦r✐❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✵ ✻✳✹ ❘❡❣✐ã♦ ✈✐á✈❡❧ ❞♦ ❡①❡♠♣❧♦ ✻✳✷✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✶ ✻✳✺ ❘❡❣✐ã♦ ✈✐á✈❡❧ ❞♦ ❡①❡♠♣❧♦ ✻✳✸✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✷ ✻✳✻ ❈♦♥t❡ú❞♦s ❡ t❡♠❛s✱ ❝♦♠♣❡tê♥❝✐❛s ❡ ❤❛❜✐❧✐❞❛❞❡s ❡ s✉❣❡stã♦ ❞❡ ❡str❛té❣✐❛

✻✳✶✸ ❆t✐✈✐❞❛❞❡ ✷ ❞❛ s✐t✉❛çã♦ ❞❡ ❛♣r❡♥❞✐③❛❣❡♠ ✸ ❞♦ ❝❛❞❡r♥♦ ❞♦ ❛❧✉♥♦✳ ✳ ✳ ✳ ✳ ✼✼ ✻✳✶✹ ●rá✜❝♦ ✐♥t❡rs❡❝çã♦ ❞❛s ✐♥❡q✉❛çõ❡s ✶✳✷① ✰ ✵✳✶✺② ≥6✱ x≥0❡ y≥0✳ ✳ ✳ ✼✽

✻✳✶✺ ❆t✐✈✐❞❛❞❡ ✹ ❞❛ s✐t✉❛çã♦ ❞❡ ❛♣r❡♥❞✐③❛❣❡♠ ✸ ❞♦ ❝❛❞❡r♥♦ ❞♦ ❛❧✉♥♦✳ ✳ ✳ ✳ ✳ ✼✾ ✻✳✶✻ ●rá✜❝♦ ❞❛ ❡q✉❛çã♦ ✺① ✰ ✷② ❂ ✹✵✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✾ ✻✳✶✼ ●rá✜❝♦ ❞❛s ❡q✉❛çõ❡s ✺① ✰ ✷② ❂ ✹✵✱ ✺① ✰ ✷② ❂ ✻✵ ❡ ✺① ✰ ✷② ❂ ✽✵✳ ✳ ✳ ✽✵ ✻✳✶✽ ❆t✐✈✐❞❛❞❡ ✺ ❞❛ s✐t✉❛çã♦ ❞❡ ❛♣r❡♥❞✐③❛❣❡♠ ✸ ❞♦ ❝❛❞❡r♥♦ ❞♦ ❛❧✉♥♦✳ ✳ ✳ ✳ ✳ ✽✶ ✻✳✶✾ ●rá✜❝♦ ❞❛ ✐♥❡q✉❛çã♦ ① ✰ ② ≤ ✽✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✷ ✻✳✷✵ ●rá✜❝♦ ❞❛ ✐♥❡q✉❛çã♦ ✷① ✰ ② ≤ ✶✷✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✷ ✻✳✷✶ ●rá✜❝♦ ❞❛ ✐♥t❡rs❡❝çã♦ ❞❛s ✐♥❡q✉❛çõ❡s ✷① ✰ ② ≤ ✶✷✱ ✷① ✰ ② ≤ ✶✷✱ ①≥ ✵

❡ ② ≥ ✵✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✸ ✻✳✷✷ ❘❡t❛s 4x+ 3y= 15 ❡ 4x+ 3y= 24 ✐♥s❡r✐❞❛s ♥♦ ❣rá✜❝♦ ❞♦ ✐t❡♠ ✺❞✳ ✳ ✳ ✳ ✽✸

✻✳✷✸ P♦♥t♦s ❡①tr❡♠♦s ❞❛ r❡❣✐ã♦ ✈✐á✈❡❧✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✹ ✻✳✷✹ ❉❡s❛✜♦ ❞❛ s✐t✉❛çã♦ ❞❡ ❛♣r❡♥❞✐③❛❣❡♠ ✸ ❞♦ ❝❛❞❡r♥♦ ❞♦ ❛❧✉♥♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✹ ❆✳✶ Pá❣✐♥❛ ✐♥✐❝✐❛❧ ❞♦ s✐t❡ ●❡♦●❡❜r❛ ❝♦♠ ✐♥❞✐❝❛çã♦ ❞❡ ♦♥❞❡ r❡❛❧✐③❛r ♦ ❞♦✇♥✲

❧♦❛❞ ❞♦ s♦❢t✇❛r❡✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✵ ❆✳✷ Pá❣✐♥❛ ❞♦ ●❡♦●❡❜r❛ ✐♥❞✐❝❛♥❞♦ ❝♦♠♦ ❡s❝♦❧❤❡r ♦ s✐st❡♠❛ ♦♣❡r❛❝✐♦♥❛❧ ❡ ♦

t✐♣♦ ❞♦ ❛♣❛r❡❧❤♦ ❛ s❡r ✉t✐❧✐③❛❞♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✵ ❆✳✸ ❮❝♦♥❡ ❛ s❡r ❡①✐❜✐❞♦ ♣❡❧♦ ●❡♦●❡❜r❛ ❛♣ós s✉❛ ✐♥st❛❧❛çã♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✶ ❆✳✹ ❏❛♥❡❧❛ ♣r✐♥❝✐♣❛❧ ❞♦ ●❡♦●❡❜r❛ ❡ ❛♠❜✐❡♥t❡ ❞❡ tr❛❜❛❧❤♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✶ ❆✳✺ ❏❛♥❡❧❛ ♣r✐♥❝✐♣❛❧ ❞♦ ●❡♦●❡♦❜r❛ ✐♥❞✐❝❛♥❞♦ ♦ ❧♦❝❛❧ ♣❛r❛ ❛ ❞✐❣✐t❛çã♦ ❞❛s

❡q✉❛çõ❡s✱ ✐♥❡q✉❛çõ❡s✱ ♣♦♥t♦s✱ ❡♥tr❡ ♦✉tr♦s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✷ ❆✳✻ ❏❛♥❡❧❛ ♣r✐♥❝✐♣❛❧ ❞♦ ●❡♦●❡❜r❛ ❛♣ós ❛ ❞✐❣✐t❛çã♦ ❞❡ t♦❞❛s ❛s ❡q✉❛çõ❡s✳ ✳ ✾✸ ❆✳✼ ❏❛♥❡❧❛ ♣r✐♥❝✐♣❛❧ ❞♦ ●❡♦●❡❜r❛ ❛♣ós ❛ ❞✐❣✐t❛çã♦ ❞❡ t♦❞❛s ❛s ✐♥❡q✉❛çõ❡s✳ ✾✸ ❆✳✽ ❏❛♥❡❧❛ ♣r✐♥❝✐♣❛❧ ❞♦ ●❡♦●❡❜r❛ ♠♦str❛♥❞♦ ❛ ✐♥t❡rs❡❝çã♦ ❞❛s r❡❣✐õ❡s✳ ✳ ✳ ✾✹ ❆✳✾ ❏❛♥❡❧❛ ♣r✐♥❝✐♣❛❧ ❞♦ ●❡♦●❡❜r❛ ♠♦str❛♥❞♦ ❛ ✈✐s✉❛❧✐③❛çã♦ ❞❛ ✐♥t❡rs❡❝çã♦

❞❡ ❞✉❛s r❡t❛s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✺ ❆✳✶✵ ❏❛♥❡❧❛ ♣r✐♥❝✐♣❛❧ ❞♦ ●❡♦●❡❜r❛ ✐♥❞✐❝❛♥❞♦ ♦ ♣♦♥t♦ ❞❡ ✐♥t❡rs❡❝çã♦ ❞❡ ❞✉❛s

▲✐st❛ ❞❡ ❚❛❜❡❧❛s

❙✉♠ár✐♦

✶ ■◆❚❘❖❉❯➬➹❖ ✶✾

✷ ❈❖◆❏❯◆❚❖❙ ❈❖◆❱❊❳❖❙ ✷✸

✸ P❘❖●❘❆▼❆➬➹❖ ▲■◆❊❆❘ ✷✺

✸✳✶ ❉❡✜♥✐çõ❡s ❡ r❡s✉❧t❛❞♦s ♣r❡❧✐♠✐♥❛r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✸✳✷ ❋♦r♠✉❧❛çã♦ ❞❡ ✉♠ Pr♦❜❧❡♠❛ ❞❡ Pr♦❣r❛♠❛çã♦ ▲✐♥❡❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✸✳✸ ▼♦❞❡❧♦ ❞❡ Pr♦❣r❛♠❛çã♦ ▲✐♥❡❛r ♥❛s ❢♦r♠❛s ♣❛❞rã♦ ❡ ❝❛♥ô♥✐❝❛ ✳ ✳ ✳ ✳ ✳ ✷✾ ✸✳✸✳✶ ▼♦❞❡❧♦ ❞❡ Pr♦❣r❛♠❛çã♦ ▲✐♥❡❛r ♥❛ ❢♦r♠❛ ♣❛❞rã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✸✳✸✳✷ ▼♦❞❡❧♦ ❞❡ Pr♦❣r❛♠❛çã♦ ▲✐♥❡❛r ♥❛ ❢♦r♠❛ ❝❛♥ô♥✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✸✳✸✳✸ ❚r❛♥s❢♦r♠❛çõ❡s ♣❛r❛ ❛ ❢♦r♠❛ ♣❛❞rã♦ ♦✉ ❝❛♥ô♥✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✸✳✹ ❊①❡♠♣❧♦s ❞❡ ♠♦❞❡❧❛❣❡♠ ❞❡ ✉♠ Pr♦❜❧❡♠❛ ❞❡ Pr♦❣r❛♠❛çã♦ ▲✐♥❡❛r ✳ ✳ ✳ ✸✷ ✹ ▼➱❚❖❉❖❙ ❉❊ ❘❊❙❖▲❯➬➹❖ ❉❊ ❯▼ P❘❖❇▲❊▼❆ ❉❊ P❘❖●❘❆✲

▼❆➬➹❖ ▲■◆❊❆❘ ✸✾

✹✳✶ ▼ét♦❞♦ ❛♥❛❧ít✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ✹✳✷ ❖ ▼ét♦❞♦ ❙✐♠♣❧❡① ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸

✺ P❘❖●❘❆▼❆➬➹❖ ▲■◆❊❆❘ ●❊❖▼➱❚❘■❈❆ ✺✶

✺✳✶ ❯♠❛ s♦❧✉çã♦ ❣❡♦♠étr✐❝❛ ♣❛r❛ Pr♦❜❧❡♠❛s ❞❡ Pr♦❣r❛♠❛çã♦ ▲✐♥❡❛r ✳ ✳ ✳ ✳ ✺✶ ✺✳✷ ❊sq✉❡♠❛t✐③❛çã♦ ❣rá✜❝❛ ❞❡ s♦❧✉çõ❡s ❞❡ Pr♦❜❧❡♠❛s ❞❡ Pr♦❣r❛♠❛çã♦ ▲✐✲

♥❡❛r ❝♦♠ ❞✉❛s ✈❛r✐á✈❡✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷ ✺✳✸ ❊①❡♠♣❧♦s ❞❡ r❡s♦❧✉çã♦ ❞❡ PP▲ ♣❡❧♦ ♠ét♦❞♦ ❣❡♦♠étr✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺

✻ ❆P▲■❈❆➬➹❖ ◆❖ ❊◆❙■◆❖ ▼➱❉■❖ ✻✼

✻✳✶ ❙❡q✉ê♥❝✐❛ ❞✐❞át✐❝❛ ♣❛r❛ ❛♣❧✐❝❛çã♦ ♥♦ ❊♥s✐♥♦ ▼é❞✐♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✸

✼ ❈❖◆❙■❉❊❘❆➬Õ❊❙ ❋■◆❆■❙ ✽✺

❘❡❢❡rê♥❝✐❛s ✽✼

❆ ❚❯❚❖❘■❆▲ ●❊❖●❊❇❘❆ P❆❘❆ ❯❚■▲■❩❆➬➹❖ ◆❆ ❙❊◗❯✃◆❈■❆

❇ ❘❖❚❊■❘❖ P❆❘❆ ❆P▲■❈❆➬➹❖ ❉❆ ❙■❚❯❆➬➹❖ ❉❊ ❆P❘❊◆❉■❩❆✲

✶ ■◆❚❘❖❉❯➬➹❖

❆ ♦r✐❣❡♠ ❞♦s ❡st✉❞♦s s♦❜r❡ ♣r♦❜❧❡♠❛s ❞❡ ♦t✐♠✐③❛r ✉♠❛ ❢✉♥çã♦ ❧✐♥❡❛r s✉❥❡✐t❛ ❛ r❡str✐çõ❡s r❡♠❡t❡♠ ❛ ❋♦✉rr✐❡r✱ ❡♠ ✶✽✷✻✱ ❝♦♠ s❡✉s ❡st✉❞♦s s♦❜r❡ s✐st❡♠❛s ❧✐♥❡❛r❡s ❞❡ ✐♥❡q✉❛çõ❡s✳

❊♠ ✶✾✸✾✱ ♣❡r❝❡❜❡✉✲s❡ ❛ ✐♠♣♦rtâ♥❝✐❛ ♣rát✐❝❛ ❞❡ss❡s ♣r♦❜❧❡♠❛s✱ q✉❛♥❞♦ ♦ r✉ss♦ ▲❡♦✲ ♥✐❞ ❑❛♥t♦r♦✈✐❝❤ ❞❡s❡♥✈♦❧✈❡✉ ♦s ♣r✐♠❡✐r♦s Pr♦❜❧❡♠❛s ❞❡ Pr♦❣r❛♠❛çã♦ ▲✐♥❡❛r ♣❛r❛ ✉s♦ ❞✉r❛♥t❡ ❛ ❙❡❣✉♥❞❛ ●✉❡rr❛ ▼✉♥❞✐❛❧✱ ♣❛r❛ ♣❧❛♥❡❥❛r ❣❛st♦s ❡ r❡t♦r♥♦s✱ ❛ ✜♠ ❞❡ r❡❞✉③✐r ♦s ❝✉st♦s ♣❛r❛ ♦ ❡①ér❝✐t♦ ❡ ❛✉♠❡♥t❛r ❛s ♣❡r❞❛s ♣❛r❛ ♦ ✐♥✐♠✐❣♦✳ ❑❛♥t♦r♦✈✐❝❤✱ ❡♠ ✶✾✼✺✱ ❣❛♥❤♦✉ ♦ Prê♠✐♦ ◆♦❜❡❧ ❞❛ ❊❝♦♥♦♠✐❛✱ ♥♦ ❡♥t❛♥t♦✱ s❡✉ tr❛❜❛❧❤♦ ✜❝♦✉ ❞❡s❝♦♥❤❡❝✐❞♦ ❞✉r❛♥t❡ ✈✐♥t❡ ❛♥♦s✳

P♦r ✈♦❧t❛ ❞❡ ✶✾✹✵✱ ♦s ❡st✉❞♦s s♦❜r❡ ❛ Pr♦❣r❛♠❛çã♦ ▲✐♥❡❛r ❛t✐♥❣✐r✐❛♠ s❡✉ ❝❧í♠❛①✱ ❝♦♠ ♦s ❡st✉❞♦s ❞❡ ●❡♦r❣❡ ❉❛♥t③✐❣ ❡ ❝♦♠ ♦ ♣r❡♠✐♦ ◆♦❜❡❧ ❞❛ ❊❝♦♥♦♠✐❛ ❡♥tr❡❣✉❡ ❛ ●❡♦r❣❡ ❙t✐❣❧❡r✳

❊♠ ✶✾✹✼✱ ❉❛♥t③✐❣✱ ❛❧é♠ ❞❡ ❢♦r♠✉❧❛r ♦ Pr♦❜❧❡♠❛ ❞❡ Pr♦❣r❛♠❛çã♦ ▲✐♥❡❛r t❛♠❜é♠ ❝r✐❛ ♦ ❆❧❣♦rít♠♦ ❙✐♠♣❧❡① ♣❛r❛ ❛ s✉❛ s♦❧✉çã♦✳ ❆♣ós ❛ ❣✉❡rr❛✱ ♠✉✐t❛s ❡♠♣r❡s❛s ❡♥❝♦♥✲ tr❛r❛♠ ♦ ✉s♦ ❡♠ s❡✉ ♣❧❛♥❡❥❛♠❡♥t♦ ❞✐ár✐♦✳

P❛r❛ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞♦ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦✱ ❢♦r❛♠ r❡❛❧✐③❛❞❛s ❞✐✈❡rs❛s ❧❡✐t✉r❛s✱ ♥❛s q✉❛✐s ❜✉s❝♦✉✲s❡ ❡♠❜❛s❛♠❡♥t♦ t❡ór✐❝♦ ❥✉♥t♦ ❛ ❞✐✈❡rs♦s ❛✉t♦r❡s✳ ❆s ❢♦♥t❡s ❝♦♥s✉❧t❛❞❛s ❢♦r❛♠ ❛s ♠❛✐s ❞✐✈❡rs❛s ♣♦ssí✈❡✐s✱ ❞❡s❞❡ ❧✐✈r♦s✱ ❛rt✐❣♦s✱ ❞✐ss❡rt❛çõ❡s ❞❡ ♠❡str❛❞♦ ❡ ❞♦✉✲ t♦r❛❞♦✱ ❜❡♠ ❝♦♠♦ ❡①♣❡r✐ê♥❝✐❛s ✈✐✈❡♥❝✐❛❞❛s ♥❛ ♣rát✐❝❛✳ ❆tr❛✈és ❞❡ ❝♦♥✈❡rs❛s r❡❛❧✐③❛❞❛s ❝♦♠ ❛♥t✐❣♦s ❡ ♥♦✈♦s ❞♦❝❡♥t❡s ❞❛ ❞✐s❝✐♣❧✐♥❛ ❞❡ ♠❛t❡♠át✐❝❛✱ ✜❝♦✉ ❡✈✐❞❡♥t❡ ❛ ♥❡❝❡ss✐❞❛❞❡ ❞❡ ♥♦✈❛s ♠❡t♦❞♦❧♦❣✐❛s ❡ ♣rát✐❝❛s ❞❡ ❝♦♠♦ ❡♥s✐♥❛r ❡ ❛♣r❡♥❞❡r ♠❛t❡♠át✐❝❛✱ ❜✉s❝❛♥❞♦ ✉♠❛ ♠❛✐♦r ♠♦t✐✈❛çã♦ ❡ ✐♥t❡r❡ss❡ ♣♦r ♣❛rt❡ ❞♦s ❞✐s❝❡♥t❡s✳

P❛r❛ ▼❡❧♦✭✷✵✶✷✮✱ ♦ ♣r♦❝❡ss♦ ❞❡ ❡♥s✐♥♦ ❡ ❛♣r❡♥❞✐③❛❣❡♠ ❞❛ ♠❛t❡♠át✐❝❛ é ♠✉✐t♦ ❝♦♠✲ ♣❧❡①♦ ❡ ❝❤❡✐♦ ❞❡ ♦❜stá❝✉❧♦s✱ ♦ q✉❡ ♣♦❞❡ s❡r✈✐r ❞❡ ❡stí♠✉❧♦ ♣❛r❛ ❛ ❜✉s❝❛ ❞❡ ♥♦✈❛s té❝♥✐✲ ❝❛s ❡ ♠❡t♦❞♦❧♦❣✐❛s ♣❛r❛ ❛ ❞✐♠✐♥✉✐çã♦ ❣r❛❞❛t✐✈❛ ❞❡ t❛✐s ❛❞✈❡rs✐❞❛❞❡s✳ ❆✐♥❞❛✱ s❡❣✉♥❞♦ ▼❡❧♦✭✷✵✶✷✱ ♣á❣✳ ✸✮✱ ❤á ✧❛ ✐♠❡♥s❛ ♥❡❝❡ss✐❞❛❞❡ ❞❡ ❜✉s❝❛r♠♦s ✉♠❛ ♠❛✐♦r ♠♦t✐✈❛çã♦ ♣❛r❛ ♦s ❞✐s❝❡♥t❡s✱ ❛tr❛✈és ❞❡ ✉♠❛ ♠❛t❡♠át✐❝❛ ♠❛✐s ✐♥t❡r❡ss❛♥t❡ ❡ r❡❧❛❝✐♦♥❛❞❛ ❛♦s ❛s♣❡❝t♦s s♦❝✐♦❝✉❧t✉r❛✐s ❞♦s ❛❧✉♥♦s✳✧ ❊♠ s✉❛ ❞✐ss❡rt❛çã♦ ❞❡ ♠❡str❛❞♦✱ ♦ ❛✉t♦r ❝✐t❛❞♦ ❛♣r❡s❡♥t❛ ✉♠❛ ♣r♦♣♦st❛ ❞❡ ❡♥s✐♥♦ ❞♦ ❛ss✉♥t♦ Pr♦❣r❛♠❛çã♦ ▲✐♥❡❛r✱ ❥✉♥t❛♠❡♥t❡ ❝♦♠ ♦ ❝♦♥t❡ú❞♦ ❞❡ ▼❛tr✐③❡s ❡ s✐st❡♠❛s ❧✐♥❡❛r❡s ♣❛r❛ ✉♠ ❣r✉♣♦ ❞❡ ❛❧✉♥♦s ❞♦ ❊♥s✐♥♦ ▼é❞✐♦✱ ❡♠ ✉♠❛ ❡s❝♦❧❛ ❞❛ ❝✐❞❛❞❡ ❞❡ P♦rt♦ ❆❧❡❣r❡✳ ❆ ♣r♦♣♦st❛ ❢♦✐ ❛ ❛♣❧✐❝❛çã♦ ❞❡ s✐t✉❛çõ❡s✲♣r♦❜❧❡♠❛

✷✵ ■◆❚❘❖❉❯➬➹❖

❝♦♠ ♣❛rt❡ ❞❛ s♦❧✉çã♦ ❢❛③❡♥❞♦ ✉s♦ ❞♦ ❙♦❢t✇❛r❡ ●r❛♣❤♠❛t✐❝❛✱ ❝♦♠♦ ❢❛❝✐❧✐t❛❞♦r ♥❛ s♦✲ ❧✉çã♦ ❣❡♦♠étr✐❝❛ ❞♦s s✐st❡♠❛s ❞❡ ✐♥❡q✉❛çã♦ ❝♦♠ ❞✉❛s ✈❛r✐á✈❡✐s✳ ❖ ✉s♦ ❞♦ s♦❢t✇❛r❡ ❢♦✐ ❜❛st❛♥t❡ ❡❧✉❝✐❞❛t✐✈♦ ❡ ❡s❝❧❛r❡❝❡❞♦r✳ ◗✉❛♥❞♦ ❢♦r❛♠ ❛♣r❡s❡♥t❛❞♦s ♣r♦❜❧❡♠❛s ❝♦♠ ♠❛✐s ❞❡ ❞✉❛s ✈❛r✐á✈❡✐s✱ ❤♦✉✈❡ ✉♠❛ ❝❡rt❛ ❢r✉str❛çã♦ ♣♦r ♥ã♦ s❡r ♣♦ssí✈❡❧ ❛ ✉t✐❧✐③❛çã♦ ❞♦ s♦❢t✇❛r❡ ♣❛r❛ r❡s♦❧✈ê✲❧♦s✳ ◆❡st❡ ♠♦♠❡♥t♦✱ ❢♦✐ ❝♦♠❡♥t❛❞♦ s♦❜r❡ ♦ ♠ét♦❞♦ s✐♠♣❧❡① ❡ ❛ ❢✉♥çã♦ ❙♦❧✈❡r ❞♦ ❊①❝❡❧✳

❖s r❡s✉❧t❛❞♦s ♦❜t✐❞♦s ❢♦r❛♠ ♠✉✐t♦ ♣♦s✐t✐✈♦s✱ ♣♦✐s ♦s ❛❧✉♥♦s ❛♣r❡s❡♥t❛r❛♠ ót✐♠♦s ❡♥❝❛♠✐♥❤❛♠❡♥t♦s ♣❛r❛ ❛s s♦❧✉çõ❡s ❞♦s ♣r♦❜❧❡♠❛s✱ ♦♥❞❡ ♠✉✐t♦s ❝❤❡❣❛r❛♠ ♣r❛t✐❝❛♠❡♥t❡ s♦③✐♥❤♦s✳ ❙❡♥t✐✉✲s❡ ✉♠❛ s❛t✐s❢❛çã♦ ❣❡r❛❧ ❡ ✐♥t❡r❡ss❡ ❡♠ ❝♦♥t✐♥✉❛r ❛ ❡st✉❞❛r ♦ ❛ss✉♥t♦✱ ♣♦✐s ♠✉✐t♦s ❝♦♥t❡ú❞♦s✱ ❝♦♠♦ ✐♥❡q✉❛çã♦ ❞♦ ✶♦ _{❣r❛✉✱ ♣❛ss❛r❛♠ ❛ ❢❛③❡r s❡♥t✐❞♦✳}

▼❛rt✐♥s✭✷✵✶✸✮ ❥✉st✐✜❝❛ s❡✉ tr❛❜❛❧❤♦ ♣❡❧❛ ♥❡❝❡ss✐❞❛❞❡ ❞❡ r❡♣❡♥s❛r ♦ ❝✉rrí❝✉❧♦✱ ❜✉s✲ ❝❛♥❞♦ ❡str❛té❣✐❛s ❡ ❛❧t❡r♥❛t✐✈❛s ♣❛r❛ r❡s♣♦♥❞❡r ❛♦s q✉❡st✐♦♥❛♠❡♥t♦s ❞♦s ❛❧✉♥♦s✱ ❢❛✲ ③❡♥❞♦ ❛ss✐♠✱ ✉♠❛ ❝♦♥❡①ã♦ ❞❛ ♠❛t❡♠át✐❝❛ ❝♦♠ ♦ ❝♦t✐❞✐❛♥♦✳ ❊♠ s✉❛ ❞✐ss❡rt❛çã♦ ❞❡ ♠❡str❛❞♦✱ ♦ ❛✉t♦r ♦❢❡r❡❝❡✉ ❛♦s ❡st✉❞❛♥t❡s ❞♦ ❊♥s✐♥♦ ▼é❞✐♦ ❞❡ ✉♠❛ ✐♥st✐t✉✐çã♦ ♣r✐✈❛❞❛ ❞❡ P♦rt♦ ❆❧❡❣r❡✱ ❛t✐✈✐❞❛❞❡s ♥❛ ♠♦❞❛❧✐❞❛❞❡ ❞❡ ♦✜❝✐♥❛✱ t♦t❛❧✐③❛♥❞♦ ✷✵ ❤♦r❛s ❞❡ ❛✉❧❛s ❡♠ ✽ ❡♥❝♦♥tr♦s✳ ❈♦♠♦ ♦ ❝✉♠♣r✐♠❡♥t♦ ❞❛s ♦✜❝✐♥❛s ♥ã♦ ❡r❛ ♦❜r✐❣❛tór✐♦✱ ♦ ♥ú♠❡r♦ ❞❡ ♣❛r✲ t✐❝✐♣❛♥t❡s ✈❛r✐♦✉ ❞✉r❛♥t❡ ❛ ❛♣❧✐❝❛çã♦ ❞❛ s❡q✉ê♥❝✐❛✳ ❆♣❡♥❛s ♥♦✈❡ ❛❧✉♥♦s ♣❛rt✐❝✐♣❛r❛♠ ❞❡ t♦❞❛s ❛s ♦✜❝✐♥❛s✳

❆ s❡q✉ê♥❝✐❛ ❞✐❞át✐❝❛ ❢♦✐ ❛♣❧✐❝❛❞❛ ❡♠ ❣r✉♣♦s✱ ❛tr❛✈és ❞❡ r❡s♦❧✉çã♦ ❞❡ ♣r♦❜❧❡♠❛s✱ ♣❡r♠✐t✐♥❞♦ ❛♦s ❛❧✉♥♦s ❡❧❛❜♦r❛r ❝♦♥❝❧✉sõ❡s✳ ❆s ❛t✐✈✐❞❛❞❡s ❢♦r❛♠ ❛♣❧✐❝❛❞❛s s❡♠ ❡①♣❧❛✲ ♥❛çã♦ ❞❡ ❝♦♥❝❡✐t♦s t❡ór✐❝♦s ♣ré✈✐♦s✱ ❛ ✜♠ ❞❡ ❞❡s♣❡rt❛r ❛ ♣❡r❝❡♣çã♦ ❞❡ ✉♠ ♠ét♦❞♦ ♣❛r❛ ❛ r❡s♦❧✉çã♦ ❞❡ ♣r♦❜❧❡♠❛s✳ ❆ t❡♦r✐❛ ❢♦✐ ❞❡s❡♥✈♦❧✈✐❞❛ ♥♦ ❞❡❝♦rr❡r ❞❛s ♦✜❝✐♥❛s✱ ♣❛r❛❧❡❧❛♠❡♥t❡ ❝♦♠ ❛ r❡✈✐sã♦ ❞❡ ❛❧❣✉♥s ❝♦♥t❡ú❞♦s✳ ❋❡③✲s❡ ✉s♦ ❞♦ s♦❢t✇❛r❡ ●❡♦●❡❜r❛✳

P❛r❛ ▼❛rt✐♥s✭✷✵✶✸✱ ♣á❣✳ ✻✵✮ ✧❛s ❛t✐✈✐❞❛❞❡s ♦♣♦rt✉♥✐③❛r❛♠ q✉❡ ♦s ❛❧✉♥♦s ❝r✐❛ss❡♠ s❡✉ ♣ró♣r✐♦ ❛❧❣♦r✐t♠♦ ♣❛r❛ ❛ r❡s♦❧✉çã♦ ❞❡ Pr♦❜❧❡♠❛s ❞❡ Pr♦❣r❛♠❛çã♦ ▲✐♥❡❛r✧✱ ❛❝r❡✲ ❞✐t❛♥❞♦ ♥♦ ❢❛✈♦r❡❝✐♠❡♥t♦ ❞♦ ❛♣r❡♥❞✐③❛❞♦ ♣♦r ♠❡✐♦s ♥ã♦ ♠❡❝❛♥✐③❛❞♦s ❞❡ ♠❡♠♦r✐③❛çã♦ ❞❡ ♣r♦❝❡❞✐♠❡♥t♦s ❡ ❛❧❣♦r✐t♠♦s ♣r♦♥t♦s✳

❆♦ ✜♥❛❧ ❞❛s ♦✜❝✐♥❛s✱ ▼❛rt✐♥s✭✷✵✶✸✮✱ ❝♦♥st❛t♦✉✱ ❞❡ ♠♦❞♦ ❣❡r❛❧✱ ✉♠ ✐♥t❡r❡ss❡ ♥❡ss❡ tó♣✐❝♦✱ ❛té ❡♥tã♦ ❞❡s❝♦♥❤❡❝✐❞♦✱ ❝♦♥❝❧✉✐♥❞♦ q✉❡ ❤♦✉✈❡ ❡♥t❡♥❞✐♠❡♥t♦ ❞❡ q✉❡✱ s♦♠❡♥t❡ t❛❜❡❧❛s ♥ã♦ r❡s♦❧✈❡♠ t♦❞♦s ♦s Pr♦❜❧❡♠❛s ❞❡ Pr♦❣r❛♠❛çã♦ ▲✐♥❡❛r✱ s✉r❣✐♥❞♦ ❛ ♥❡❝❡ss✐✲ ❞❛❞❡ ❞❡ ✉♠ ♠ét♦❞♦ ♠❛✐s ❡✜❝❛③ ❡ ❞♦ ✉s♦ ❞❡ ❡q✉❛çõ❡s ❡ ✐♥❡q✉❛çõ❡s✳ ❖ ❡st✉❞♦ ❝♦♥❡❝t♦✉ ❝♦♥❤❡❝✐♠❡♥t♦s ♣ré✈✐♦s ❡ s❡♠ s❡♥t✐❞♦ ❛té ♦ ♠♦♠❡♥t♦✱ ❥á q✉❡ ♥ã♦ ❤♦✉✈❡ ❛ ❛❜♦r❞❛❣❡♠ ❞❡ ❝♦♥❝❡✐t♦s ♥♦✈♦s✳ ❖ ✉s♦ ❞♦ s♦❢t✇❛r❡ ❝♦♠♣✉t❛❝✐♦♥❛❧ ❢♦✐ ✉♠ ✐♥str✉♠❡♥t♦ ❢❛❝✐❧✐t❛❞♦r✱ ♣♦✐s ♣❡r♠✐t✐✉ ❛ ♠❛♥✐♣✉❧❛çã♦ ❞❡ ♣❛râ♠❡tr♦s✳

✷✶

❘✐❜❛s ♣r♦♣õ❡ ❛t✐✈✐❞❛❞❡s ♣❛r❛ ✽ ❛✉❧❛s✱ ♦✉ ♣r♦❥❡t♦ ♥♦ ❝♦♥tr❛✲t✉r♥♦✱ s❡♥❞♦ ❛s ♣r✐♥✲ ❝✐♣❛✐s ❛t✐✈✐❞❛❞❡s✿ ❞✐s❝✉ssã♦ ❞♦s ♣r♦❜❧❡♠❛s ❛♣r❡s❡♥t❛❞♦s✱ ❛❜♦r❞❛❣❡♠ ❞♦s ❝♦♥❝❡✐t♦s ❞❡ Pr♦❣r❛♠❛çã♦ ▲✐♥❡❛r✱ ❝♦♥str✉çõ❡s ♥♦ s♦❢t✇❛r❡ ●❡♦●❡❜r❛✱ ✉s♦ ❞♦ s♦❢t✇❛r❡ ❊①❝❡❧❧ ♦✉ ▲✐♥✉① ❈❛❧❝ ♣❛r❛ r❡s♦❧✉çã♦ ❞♦s ♣r♦❜❧❡♠❛s✳ ◆ã♦ ❤♦✉✈❡ ❛✈❛❧✐❛çã♦ ❞♦s r❡s✉❧t❛❞♦s ♦❜t✐❞♦s✱ ♣♦✐s ❛s ❛t✐✈✐❞❛❞❡s ♥ã♦ ❢♦r❛♠ ❛♣❧✐❝❛❞❛s ❛♦s s❡✉s ❛❧✉♥♦s✳

❘✐❣✉❡tt♦✭✷✵✶✺✮ ❡s❝♦❧❤❡✉ ❡st❡ t❡♠❛ ♣♦r ❡st❛r ♣r❡s❡♥t❡ ♥♦ ▼❛t❡r✐❛❧ ❞❡ ❆♣♦✐♦ ❛♦ ❈✉r✲ rí❝✉❧♦ ❞♦ ❊st❛❞♦ ❞❡ ❙ã♦ P❛✉❧♦✱ ❈❛❞❡r♥♦ ❞♦ ❆❧✉♥♦ ❞❛ ✸❛ _{sér✐❡ ❞♦ ❊♥s✐♥♦ ▼é❞✐♦ ✈♦❧✉♠❡}

✶✱ ❛♣r❡s❡♥t❛♥❞♦ ✈ár✐♦s Pr♦❜❧❡♠❛s ❞❡ Pr♦❣r❛♠❛çã♦ ▲✐♥❡❛r✱ ♣r♦♣✐❝✐❛♥❞♦ ✐♥tr♦❞✉③✐r ❡ss❡ ❛ss✉♥t♦ ♥♦ ❊♥s✐♥♦ ▼é❞✐♦ ❞❡ ♠❛♥❡✐r❛ s✐♠♣❧❡s✳ ❋❛③✲s❡ ♥❡❝❡ssár✐♦ ❛❜♦r❞❛r ❛ ▼❛t❡♠át✐❝❛ ❞❡ ❢♦r♠❛ q✉❡ ♦ ❛❧✉♥♦ ❝♦♠♣r❡❡♥❞❛ ❛ ✐♠♣♦rtâ♥❝✐❛ ❡♠ ❛♣r❡♥❞ê✲❧❛✱ ❡ ♥ã♦ s✐♠♣❧❡s♠❡♥t❡✱ ❛ ♦❜r✐❣❛çã♦✳ P❛r❛ ✐ss♦✱ ❛ ❛✉t♦r❛ s✉❣❡r❡ ✉♠❛ s❡q✉ê♥❝✐❛ ❞✐❞át✐❝❛ ❝♦♠ ✶✵ ♣r♦❜❧❡♠❛s✱ ❡♠ ❣r❛✉ ❝r❡s❝❡♥t❡ ❞❡ ❞✐✜❝✉❧❞❛❞❡✱ ❛ s❡r❡♠ r❡s♦❧✈✐❞♦s ❡♠ ❞✉♣❧❛s✱ ✉t✐❧✐③❛♥❞♦ ❛ ♠❡t♦❞♦❧♦❣✐❛ ❞❡ r❡s♦❧✉çã♦ ❞❡ ♣r♦❜❧❡♠❛s✱ ❝♦♥❝❡♣çã♦ ❡st❛ ♣r❡❝♦♥✐③❛❞❛ ♥♦ P❛râ♠❡tr♦s ❈✉rr✐❝✉❧❛r❡s ◆❛❝✐♦♥❛✐s✳

❘✐❣✉❡tt♦✭✷✵✶✺✮ é ♣r♦❢❡ss♦r❛ ♥❛ ♠❡s♠❛ ✉♥✐❞❛❞❡ ❡s❝♦❧❛r ❞❡s❞❡ ✶✾✽✸✱ s❡♥❞♦ ❡st❛ ♣ú✲ ❜❧✐❝❛✱ ♥❛ ❝✐❞❛❞❡ ❞❡ ■❧❤❛ ❙♦❧t❡✐r❛✳ ❆ s❡q✉ê♥❝✐❛ ❢♦✐ ❛♣❧✐❝❛❞❛ ❛ ✉♠❛ t✉r♠❛ ❞❡ ✸♦ _❛♥♦✱

✐♥✐❝✐❛❧♠❡♥t❡ ♥❛ ❢♦r♠❛ ❞❡ ♣ré✲t❡st❡✱ ❡♥t❡♥❞❡♥❞♦ q✉❡✱ ❞❡st❛ ❢♦r♠❛✱ ♦s ❛❧✉♥♦s s❡ ❡♠♣❡✲ ♥❤❛r✐❛♠ ♥❛ r❡s♦❧✉çã♦ ❞♦s ♣r♦❜❧❡♠❛s✳

❖s ♣r♦❜❧❡♠❛s ♠❛✐s s✐♠♣❧❡s ❢♦r❛♠ r❡s♦❧✈✐❞♦s s❡♠ ❞✐✜❝✉❧❞❛❞❡s ♣❡❧❛ ♠❛✐♦r✐❛✱ ❥á ♦ ♣r♦❜❧❡♠❛ q✉❡ ❡♥✈♦❧✈❡✉ t♦❞♦s ♦s ❝♦♥❝❡✐t♦s ❢♦✐ r❡s♦❧✈✐❞♦ ♣♦r ❛♣❡♥❛s ✉♠❛ ❞✉♣❧❛✱ ❛♣❡s❛r ❞❛ ✐♥t❡r✈❡♥çã♦ ❞❛ ♣r♦❢❡ss♦r❛✳

❋✐♥❛❧✐③❛♥❞♦✱ ❘✐❣✉❡tt♦✭✷✵✶✺✮ ❞❡✜♥❡ s✉❛ ♣r♦♣♦st❛ ❝♦♠♦ ✉♠❛ ❢♦r♠❛ ❞❡ ❛✉①✐❧✐❛r ♦ ♣r♦❢❡ss♦r ♥❛ ✐♥tr♦❞✉çã♦ ❞❡st❡ ❛ss✉♥t♦✱ ♣♦✐s ♦❜s❡r✈♦✉ ✉♠❛ ❡s❝❛ss❡③ ❞❡ ♠❛t❡r✐❛❧ ✈♦❧t❛❞♦ ❛♦ ❊♥s✐♥♦ ▼é❞✐♦✳

❈♦♠♦ ✈✐st♦ ♥♦s ❡st✉❞♦ r❡❛❧✐③❛❞♦s✱ ♦ t❡♠❛ Pr♦❣r❛♠❛çã♦ ▲✐♥❡❛r é ❞❡ ❢á❝✐❧ ❛♣❧✐❝❛çã♦ ❡ ❝♦♥s❡❣✉❡ r❡s♣♦♥❞❡r ❞❡ ♠❛♥❡✐r❛ ❝❧❛r❛ ❛♦s q✉❡st✐♦♥❛♠❡♥t♦s s♦❜r❡ ♦ ✉s♦ ❞❛ ♠❛t❡♠át✐❝❛ ♥♦ ❞✐❛ ❛ ❞✐❛✳ ▼❡s♠♦ s❡♥❞♦ ✉♠ ❛ss✉♥t♦ ✐♥t❡r❡ss❛♥t❡✱ é ♠✉✐t♦ ♣♦✉❝♦ ❡①♣❧♦r❛❞♦ ♥♦ ❊♥s✐♥♦ ▼é❞✐♦✱ s❡♥❞♦ ❛❜♦r❞❛❞♦ ❡♠ ♣♦✉❝♦s ❧✐✈r♦s ❞✐❞át✐❝♦s✳ ❖ ❝❛❞❡r♥♦ ❞❡ ❛♣♦✐♦ ❞♦ ❛❧✉♥♦ ❞♦ ✸♦ _{❛♥♦ ❞♦ ❊♥s✐♥♦ ▼é❞✐♦ ❞♦ ❊st❛❞♦ ❞❡ ❙ã♦ P❛✉❧♦ ✲ ✈♦❧✉♠❡ ✶✱ tr❛③ ✉♠❛ s✐t✉❛çã♦ ❞❡}

❛♣r❡♥❞✐③❛❣❡♠ s♦❜r❡ Pr♦❜❧❡♠❛s ❞❡ Pr♦❣r❛♠❛çã♦ ▲✐♥❡❛r✱❝♦♠♦ ♠♦t✐✈❛çã♦ ♣❛r❛ ✉s♦ ❞❛s ❡q✉❛çõ❡s ❡ ✐♥❡q✉❛çõ❡s ❛ss♦❝✐❛❞❛s ❛ r❡t❛s ❡ r❡❣✐õ❡s ❞♦ ♣❧❛♥♦✳ P♦r ❡st❡ ♠♦t✐✈♦ ♦♣t❡✐ ♣♦r ❡①♣❧♦r❛r ❡st❡ t❡♠❛✱ ❝♦♠ ♦ ✉s♦ ❞❡ s♦❢t✇❛r❡ ♠❛t❡♠át✐❝♦ ●❡♦●❡❜r❛ ❝♦♠♦ ❢❛❝✐❧✐t❛❞♦r✱ tr❛③❡♥❞♦ ❛ss✐♠ ❛ ♣♦ss✐❜✐❧✐❞❛❞❡ ❞❡ ✐♥tr♦❞✉③✐r ❛ss✉♥t♦s ♠❛✐s ❝♦♥❝r❡t♦s✱ t❡♥t❛♥❞♦ ❞❡✐①❛r ❛ ♠❛t❡♠át✐❝❛ ♠❛✐s ✧♣❛❧♣á✈❡❧✧ ❛♦s ♦❧❤♦s ❞♦s ❞✐s❝❡♥t❡s✳

◗✉❛♥t♦ à ♦r❣❛♥✐③❛çã♦ ❞❡st❡ tr❛❜❛❧❤♦✱ ♥♦ s❡❣✉♥❞♦ ❝❛♣ít✉❧♦ é ❢❡✐t♦ ♦ ❡st✉❞♦ s♦❜r❡ ❝♦♥❥✉♥t♦s ❝♦♥✈❡①♦s✳

❖ t❡r❝❡✐r♦ ❝❛♣ít✉❧♦ ❛♣r❡s❡♥t❛ ✉♠ ❡st✉❞♦ s♦❜r❡ Pr♦❣r❛♠❛çã♦ ▲✐♥❡❛r✱ s✉❛s ♠♦t✐✈❛✲ çõ❡s✱ ❢♦r♠✉❧❛çã♦ ❞❡ Pr♦❜❧❡♠❛s ❞❡ Pr♦❣r❛♠❛çã♦ ▲✐♥❡❛r✱ ♠♦❞❡❧♦s ♥❛s ❢♦r♠❛s ♣❛❞rã♦ ❡ ❝❛♥ô♥✐❝❛✱ ❛❧é♠ ❞❡ ❡①❡♠♣❧♦s ❞❡ ♠♦❞❡❧❛❣❡♠ ❞❡ Pr♦❜❧❡♠❛s ❞❡ Pr♦❣r❛♠çã♦ ▲✐♥❡❛r✳

✷✷ ■◆❚❘❖❉❯➬➹❖

❞❡ r❡s♦❧✉çã♦ ❞❡ Pr♦❜❧❡♠❛s ❞❡ Pr♦❣r❛♠❛çã♦ ▲✐♥❡❛r✱ ❡s♣❡❝✐✜❝❛♥❞♦ q✉❛❧ s❡rá ♠❛✐s ✈✐á✈❡❧ ♣❛r❛ ❝❛❞❛ t✐♣♦ ❞❡ ♣r♦❜❧❡♠❛✳

❖ ❝❛♣ít✉❧♦ ❝✐♥❝♦ ❢❛③ ✉♠ ❡st✉❞♦ s♦❜r❡ Pr♦❣r❛♠❛çã♦ ▲✐♥❡❛r ●❡♦♠étr✐❝❛✱ ❛❜♦r❞❛♥❞♦ ❞❡✜♥✐çõ❡s ❡ r❡s✉❧t❛❞♦s ✐♠♣♦rt❛♥t❡s✱ ❡sq✉❡♠❛s ❞❡ s♦❧✉çõ❡s ♣❛r❛ ♣r♦❜❧❡♠❛s ❞❡ ❞✉❛s ✈❛r✐á✈❡✐s✱ ❛❧é♠ ❞❛ r❡s♦❧✉çã♦ ❞❡ ❞✐✈❡rs♦s ❡①❡♠♣❧♦s q✉❡ ❛❜♦r❞❛♠ ❛s ✈❛r✐❛çõ❡s ♥❛ ♥❛t✉r❡③❛ ❞❛s s♦❧✉çõ❡s q✉❡ ♣♦❞❡♠ ♦❝♦rr❡r✳

◆♦ ❝❛♣ít✉❧♦ s❡✐s sã♦ ❛❜♦r❞❛❞❛s ❛s ❛♣❧✐❝❛çõ❡s ♥♦ ❊♥s✐♥♦ ▼é❞✐♦✱ ❝♦♠ ❛❧❣✉♥s ❡①❡♠♣❧♦s ❞❡ ❛♣❧✐❝❛çõ❡s ♣r♦♣♦st❛s ❡♠ ❧✐✈r♦s ❞✐❞át✐❝♦s✱ ❡ ❛ ❛♣r❡s❡♥t❛çã♦ ❞❡ ✉♠❛ ♣r♦♣♦st❛ ❞✐❞át✐❝❛ ♣❛r❛ ♠♦t✐✈❛r ✉♠❛ s✐t✉❛çã♦ ❞❡ ❛♣r❡♥❞✐③❛❣❡♠ ❛♣r❡s❡♥t❛❞❛ ♥♦ ♠❛t❡r✐❛❧ ❞❡ ❛♣♦✐♦ ❞♦ ❝✉rrí❝✉❧♦ ❞♦ ❊st❛❞♦ ❞❡ ❙ã♦ P❛✉❧♦✱ ♣r♦♣♦st❛ ♥♦ ❝❛❞❡r♥♦ ❞♦ ❛❧✉♥♦ ❞♦ ✸♦ _{❛♥♦ ❞♦ ❊♥s✐♥♦}

▼é❞✐♦✳

❋✐♥❛❧♠❡♥t❡✱ ♥♦s ❛♣ê♥❞✐❝❡s✱ ✐♥❝❧✉í♠♦s ✉♠ ♠❛♥✉❛❧ ♣❛r❛ ✉s♦ ❞♦ s♦❢t✇❛r❡ ●❡♦●❡❜r❛ ❡ ♦ r♦t❡✐r♦ ❞❡ ❛♣❧✐❝❛çã♦ ❞❡s❝r✐t♦ ♥♦ ♠❛t❡r✐❛❧ ❞✐r✐❣✐❞♦ ❞❡s❝r✐t♦ ♥♦ ❝❛❞❡r♥♦ ❞♦ ♣r♦❢❡ss♦r ❞♦ ❝✉rrí❝✉❧♦ ❞♦ ❊st❛❞♦ ❞❡ ❙ã♦ P❛✉❧♦✱ q✉❡ ♠♦t✐✈♦✉ ❛ s✐t✉❛çã♦ ❞❡ ❛♣r❡♥❞✐③❛❣❡♠ ✉t✐❧✐③❛❞❛ ♥❛ ♣r♦♣♦st❛ ❞✐❞át✐❝❛ ❞❡st❡ tr❛❜❛❧❤♦✳

✷ ❈❖◆❏❯◆❚❖❙ ❈❖◆❱❊❳❖❙

◆❡st❡ ❝❛♣ít✉❧♦ ❢❛r❡♠♦s ✉♠ ❜r❡✈❡ ❡st✉❞♦ s♦❜r❡ ❝♦♥❥✉♥t♦s ❝♦♥✈❡①♦s✱ ♣♦✐s ✈❡r❡♠♦s ♥♦ ❝❛♣ít✉❧♦ ✸ q✉❡ ♦ ❝♦♥❥✉♥t♦ ❞♦s ♣♦♥t♦s ✈✐á✈❡✐s ♣❛r❛ ✉♠ Pr♦❜❧❡♠❛ ❞❡ Pr♦❣r❛♠❛çã♦ ▲✐♥❡❛r é ✉♠ ❝♦♥❥✉♥t♦ ❝♦♥✈❡①♦ ❝✉❥♦s ✈ért✐❝❡s ❝♦rr❡s♣♦♥❞❡♠ às s♦❧✉çõ❡s✳

❉❡✜♥✐çã♦ ✷✳✶✳ ❙❡❥❛ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ♣♦♥t♦s xi ∈X✳ ❉✐③✲s❡ q✉❡ ❳ é ✉♠❛ ❝♦♠❜✐♥❛çã♦

❧✐♥❡❛r ❝♦♥✈❡①❛ ❞♦s ♣♦♥t♦s xi s❡

Pn

i=1λixi✱ ❡♠ q✉❡ λi sã♦ ♦s ❝♦❡✜❝✐❡♥t❡s ❡s❝❛❧❛r❡s q✉❡

t❡rã♦ q✉❡ ❛ss✉♠✐r ♦s s❡❣✉✐♥t❡s ✈❛❧♦r❡s✿ (

0≤λi ≤1

Pn

i=1λi = 1

❊♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱ ❞♦✐s ♣♦♥t♦s ❞❡✜♥❡♠ ✉♠ s❡❣♠❡♥t♦ ❞❡ r❡t❛ ❡ ✉♠❛ ❝♦♠❜✐♥❛çã♦ ❧✐♥❡❛r ❝♦♥✈❡①❛✱ q✉❡ é ✐❣✉❛❧ ❛♦ s❡❣♠❡♥t♦ q✉❡ ♦s ✉♥❡✳

❉❡✜♥✐çã♦ ✷✳✷✳ ❯♠ ❝♦♥❥✉♥t♦ ❑ é ❝♦♥✈❡①♦ q✉❛♥❞♦ t♦❞♦s ♦s s❡❣♠❡♥t♦s ❞❡ r❡t❛ q✉❡ ✉♥❡♠ ❞♦✐s ♣♦♥t♦s q✉❛✐sq✉❡r ❞❡ ❑ ❡stã♦ ❝♦♥t✐❞♦s ❡♠ ❑✳ ❯♠ ❝♦♥❥✉♥t♦ é ❢❡❝❤❛❞♦ s❡ ❡❧❡ ❝♦♠♣r❡❡♥❞❡ ❛ s✉❛ ❢r♦♥t❡✐r❛✳

❆ r❡♣r❡s❡♥t❛çã♦ ❣rá✜❝❛ ❞❡ ❝♦♥❥✉♥t♦s ❝♦♥✈❡①♦s✱ ♣♦r ♠❡✐♦ ❞❡ ✉♠ ❡①❡♠♣❧♦ ✐❧✉str❛t✐✈♦✱ ❡stá ♥❛ ❋✐❣✉r❛ ✷

❋✐❣✉r❛ ✷✳✶✿ ❊①❡♠♣❧♦s ❞❡ ❝♦♥❥✉♥t♦s ❝♦♥✈❡①♦s ❡ ♥ã♦ ❝♦♥✈❡①♦s✳

❉❡✜♥✐çã♦ ✷✳✸✳ ❯♠ ✈ért✐❝❡ é ✉♠ ♣♦♥t♦ ♣❡rt❡♥❝❡♥t❡ ❛ ✉♠ ❝♦♥❥✉♥t♦ ❝♦♥✈❡①♦ q✉❡ ♥ã♦ ♣♦❞❡ s❡r ♦❜t✐❞♦ ♣♦r ♠❡✐♦ ❞❡ ❝♦♠❜✐♥❛çã♦ ❝♦♥✈❡①❛ ❞♦s ♦✉tr♦s ♣♦♥t♦s ❞♦ ❝♦♥❥✉♥t♦✳ P♦❞❡♠♦s ❞✐③❡r q✉❡ ✉♠ ✈ért✐❝❡ é ✉♠ ♣♦♥t♦ ❡①tr❡♠♦ ❞❡ ✉♠ ❝♦♥❥✉♥t♦ ❝♦♥✈❡①♦✳ ❱❡r ❋✐❣✉r❛ ✷✳✷✳

✷✹ ❈❖◆❏❯◆❚❖❙ ❈❖◆❱❊❳❖❙

❋✐❣✉r❛ ✷✳✷✿ ❆✱ ❇ ❡ ❈ sã♦ ✈ért✐❝❡s✳

❉❡✜♥✐çã♦ ✷✳✹✳ ❆ ❝♦♠❜✐♥❛çã♦ ❝♦♥✈❡①❛ ❞❡ ❞♦✐s ✈ért✐❝❡s ❞❡ ❞♦✐s ✈ért✐❝❡s ❆ ❡ ❇ é ❝♦♥✲ s✐❞❡r❛❞❛ ✉♠❛ ❛r❡st❛ s❡ ♥❡♥❤✉♠ ♣♦♥t♦ ❞❡ AB ♣♦❞❡ s❡r ♦❜t✐❞♦ ♣❡❧❛ ❝♦♠❜✐♥❛çã♦ ❝♦♥✈❡①❛

❞❡ ♣♦♥t♦s ♥ã♦ ♣❡rt❡♥❝❡♥t❡s ❛ AB✳ ❱❡r ❋✐❣✉r❛ ✷✳✸✳

❋✐❣✉r❛ ✷✳✸✿ AB✱ BC✱CA sã♦ ❛r❡st❛s✱ BD ♥ã♦ é ✉♠❛ ❛r❡st❛✳

✸ P❘❖●❘❆▼❆➬➹❖ ▲■◆❊❆❘

❆ Pr♦❣r❛♠❛çã♦ ▲✐♥❡❛r ✭P▲✮ é ✉♠ ❞♦s r❛♠♦s ❞❛ ♣r♦❣r❛♠❛çã♦ ♠❛t❡♠át✐❝❛✱ ❡ t❛♠✲ ❜é♠ ✉♠❛ ❞❛s ♣r✐♥❝✐♣❛✐s ❢❡rr❛♠❡♥t❛s ❞❛ P❡sq✉✐s❛ ❖♣❡r❛❝✐♦♥❛❧ ✭P❖✮✳ ❆ P❡sq✉✐s❛ ❖♣❡✲ r❛❝✐♦♥❛❧ t❡✈❡ s✉❛s ♣r✐♠❡✐r❛s ❛t✐✈✐❞❛❞❡s ❞✉r❛♥t❡ ❛ ❙❡❣✉♥❞❛ ●✉❡rr❛ ▼✉♥❞✐❛❧✱ ♦♥❞❡ s❡✉s ❡st✉❞♦s ❢♦r❛♠ ✉t✐❧✐③❛❞♦s ♥❛s t♦♠❛❞❛s ❞❡ ❞❡❝✐sõ❡s s♦❜r❡ ♦ ✉s♦ ❞❡ ♠❛t❡r✐❛✐s ❞❡ ❣✉❡rr❛✳ ❆♣ós ❛ ❙❡❣✉♥❞❛ ●✉❡rr❛ ▼✉♥❞✐❛❧✱ s✉❛ ❛♣❧✐❝❛çã♦ ♣♦❞❡ s❡r ✈✐st❛ ❡♠ ❞✐✈❡rs♦s s❡❣♠❡♥t♦s ✐♥❞✉str✐❛✐s ❡ ❝♦♠❡r❝✐❛✐s ✭❡str❛té❣✐❛✱ ♠❛r❦❡t✐♥❣✱ ✜♥❛♥ç❛s✱ ♦♣❡r❛çõ❡s ❡ ❧♦❣íst✐❝❛✱ r❡❝✉rs♦s ❤✉♠❛♥♦s✱ ❡♥tr❡ ♦✉tr♦s✮✳

❙❡❣✉♥❞♦ ❊✐s❡❧t✭✷✵✵✼✱ ♣á❣✳ ✹✽✮✱ ✧♣r♦❜❧❡♠❛s ❞❡ ♣r♦❣r❛♠❛çã♦ ♠❛t❡♠át✐❝❛ sã♦ ♠♦❞❡✲ ❧♦s ♠❛t❡♠át✐❝♦s q✉❡ t❡♥t❛♠ ♠♦❞❡❧❛r ✉♠❛ s✐t✉❛çã♦ ❞❛ ✈✐❞❛ r❡❛❧✧✱ ❛tr❛✈és ❞❡ ✈❛r✐á✈❡✐s ❡ ♣❛râ♠❡tr♦s✱ ❛♠❜♦s r❡♣r❡s❡♥t❛❞♦s ♣♦r ♥ú♠❡r♦s✳ ❖s ♣❛râ♠❡tr♦s sã♦ ♥ú♠❡r♦s ❝♦♥❤❡❝✐❞♦s ❡ ❞❡✈❡♠ s❡r t♦♠❛❞♦s ❝♦♠♦ ✉♠ ❞❛❞♦ ✜①♦✱ ❡♥q✉❛♥t♦ ❛s ✈❛r✐á✈❡✐s sã♦ ♥ú♠❡r♦s ❝✉❥♦s ✈❛❧♦✲ r❡s s❡rã♦ ❞❡t❡r♠✐♥❛❞♦s ♥♦ ♣r♦❝❡ss♦✳ ❊♠ ❣❡r❛❧✱ ♦s ♣❛râ♠❡tr♦s ♥ã♦ sã♦ ❞❛ ❝♦♠♣❡tê♥❝✐❛ ❞♦ t♦♠❛❞♦r ❞❡ ❞❡❝✐sã♦✱ ❡♥q✉❛♥t♦ q✉❡ ❛s ✈❛r✐á✈❡✐s sã♦✳ ❖ t❡r♠♦ ♣r♦❣r❛♠❛çã♦ é ✉t✐❧✐③❛❞♦ ❛q✉✐ ❝♦♠♦ s✐♥ô♥✐♠♦ ❞❡ ♦t✐♠✐③❛çã♦✱ ❢❛③❡♥❞♦ r❡❢❡rê♥❝✐❛ ❛♦ t❡r♠♦ ✧♣r♦❣r❛♠❛✧✱ ✉t✐❧✐③❛❞♦ ♣❡❧♦s ♠✐❧✐t❛r❡s ♥♦rt❡✲❛♠❡r✐❝❛♥♦s✱ ❛♦ s❡ r❡❢❡r✐r à ❛❣❡♥❞❛ ❞❡ ❤♦rár✐♦s ♣❛r❛ tr❡✐♥❛♠❡♥t♦s ❡ ❛çõ❡s ❧♦❣íst✐❝❛s✱ ❡ ♥ã♦ à ♣r♦❣r❛♠❛çã♦ ❞❡ ❝♦♠♣✉t❛❞♦r❡s✳

❖ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❛ Pr♦❣r❛♠❛çã♦ ▲✐♥❡❛r ❢♦✐ ✐♥s♣✐r❛❞♦ ❡♠ três t✐♣♦s ❞❡ ♣r♦❜❧❡♠❛s✿ • ❚r❛♥s♣♦rt❡✿ ♦t✐♠✐③❛çã♦ ❞❡ s✐st❡♠❛s✭♣r♦❣r❛♠❛s✮ ❞❡ ❞✐str✐❜✉✐çã♦✱ ❝♦♥❤❡❝❡♥❞♦✲

s❡ ♦s ❝✉st♦s ❞❡ tr❛♥s♣♦rt❡✱ ❛ ♣r♦❝✉r❛ ♣r❡✈✐st❛ ♣❛r❛ ❝❛❞❛ ❧♦❥❛ ❡ ❛s ❝❛♣❛❝✐❞❛❞❡s ♠á①✐♠❛s ❞❡ ♣r♦❞✉çã♦ ❞❡ ❝❛❞❛ ❢á❜r✐❝❛✳

• ❈♦♠♣♦s✐çã♦✿ ♦t✐♠✐③❛çã♦ ❞❛ ❝♦♠♣♦s✐çã♦ ❞❡ ✉♠❛ ❞✐❡t❛✱ ♠✐♥✐♠✐③❛♥❞♦ ♦ s❡✉ ❝✉st♦ ❡ s❛t✐s❢❛③❡♥❞♦ ♦s ♥í✈❡✐s ♠í♥✐♠♦s ❞❡ ❝❛❧♦r✐❛s ❡ ✈✐t❛♠✐♥❛s ♥❡❝❡ssár✐♦s ♥❛ ❛❧✐♠❡♥t❛çã♦✳

• ❋♦r♠❛çã♦ ❡ ♣r♦❞✉çã♦✿ ♦t✐♠✐③❛çã♦ ❞❡ ♣r♦❣r❛♠❛s ❞❡ ❝♦♥tr❛t❛çã♦ ❡ ❢♦r✲ ♠❛çã♦ ❞❡ ♣❡ss♦❛❧✱ ❛ss✐♠ ❝♦♠♦ ❞❡ ♣r♦❞✉çã♦ ❡ ❛r♠❛③❡♥❛♠❡♥t♦✱ ❞❡ ❢♦r♠❛ ❛ ♠✐♥✐♠✐③❛r ♦s ❝✉st♦s ❡ ♠❛①✐♠✐③❛r ♦s ❧✉❝r♦s✳✭❈❛r✈❛❧❤♦✱ ✷✵✶✹✱ ♣á❣✳ ✶✶✮ P❛r❛ ▼❛❝✉❧❛♥✭♣✳✶✶✱ ✶✾✽✵✮✱ ✧♦ ♣r♦❜❧❡♠❛ ❞❡ t♦♠❛❞❛ ❞❡ ❞❡❝✐sã♦ ❧❡✈❛ ❡♠ ❝♦♥t❛ ✈❛r✐á✲ ✈❡✐s ❡ ❛s ❝♦♥❞✐çõ❡s q✉❡ ✧♣r❡♥❞❡♠✧ ❡st❛s ✈❛r✐á✈❡✐s✱ às q✉❛✐s ❞❡♥♦♠✐♥❛r❡♠♦s r❡str✐çõ❡s✧✱ ♣♦❞❡♥❞♦ ❤❛✈❡r ♠✐❧❤❛r❡s ❞❡ r❡str✐çõ❡s ❡ ✈❛r✐á✈❡✐s✳ ●❡r❛❧♠❡♥t❡✱ ❛ ❞❡❝✐sã♦ ❡stá ❧✐❣❛❞❛

✷✻ P❘❖●❘❆▼❆➬➹❖ ▲■◆❊❆❘

❛ ❝❡rt♦ ♦❜❥❡t✐✈♦✿ ♠✐♥✐♠✐③❛r ♦s ❝✉st♦s ❞❡ ♣r♦❞✉çã♦✱ ♠❛①✐♠✐③❛r ♦s ❧✉❝r♦s✱ ♠❡❧❤♦r❛r ❛s ❝♦♥❞✐çõ❡s ❞❡ ✈✐❞❛ ❞❡ ✉♠❛ ♣♦♣✉❧❛çã♦✱ ❡♥tr❡ ♦✉tr♦s✳

❖s ♣r♦❜❧❡♠❛s ❞❡ t♦♠❛❞❛ ❞❡ ❞❡❝✐sã♦✱ ❡♠ ❣❡r❛❧✱ tr❛t❛♠ ❞❡ r❡❝✉rs♦s ❧✐♠✐t❛❞♦s ❡ ❛ ✐❞❡✐❛ é ♠❡❧❤♦r❛r s❡✉ r❡♥❞✐♠❡♥t♦ ♦✉ ♣r♦❞✉t✐✈✐❞❛❞❡✳

✸✳✶ ❉❡✜♥✐çõ❡s ❡ r❡s✉❧t❛❞♦s ♣r❡❧✐♠✐♥❛r❡s

❙❡❣✉❡♠ ❛❧❣✉♠❛s ❞❡✜♥✐çõ❡s ❡ r❡s✉❧t❛❞♦s ✐♠♣♦rt❛♥t❡s ♣❛r❛ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡st❡ tr❛❜❛❧❤♦✿

❉❡✜♥✐çã♦ ✸✳✶✳ ❯♠❛ ❢✉♥çã♦ é ❧✐♥❡❛r q✉❛♥❞♦ ❡♥✈♦❧✈❡ ❛♣❡♥❛s ❝♦♥st❛♥t❡s ❡ t❡r♠♦s ❝♦♠ ✈❛r✐á✈❡✐s ❞❡ ♣r✐♠❡✐r❛ ♦r❞❡♠✳

❉❡✜♥✐çã♦ ✸✳✷✳ ❆s ✈❛r✐á✈❡✐s sã♦ ❝♦♥tí♥✉❛s q✉❛♥❞♦ ♣♦❞❡♠ ❛ss✉♠✐r q✉❛✐sq✉❡r ✈❛❧♦r❡s ❡♠ ✉♠ ✐♥t❡r✈❛❧♦ ❞❡ ♥ú♠❡r♦s r❡❛✐s✳

❉❡✜♥✐çã♦ ✸✳✸✳ ❙❡❥❛ ❳ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ♣♦♥t♦sX ⊂_Rnt❛❧ q✉❡ ♣❛r❛ t♦❞♦_{x= (x}

1, x2, ..., xn)∈

X✱ ❡①✐st❡♠ ai ❡ ❜✱ ❝♦♠ ♣❡❧♦ ♠❡♥♦s ✉♠ ai ♥ã♦ ♥✉❧♦✳

a₁x₁+a₂x₂+...+anxn=b

é ✉♠ ❤✐♣❡r♣❧❛♥♦ ❡♠ Rn✳

❊①❡♠♣❧♦ ✸✳✶✳ P♦♥t♦s sã♦ ❤✐♣❡r♣❧❛♥♦s ❡♠R✱ r❡t❛s sã♦ ❤✐♣❡r♣❧❛♥♦s ❡♠R2 ❡ ♣❧❛♥♦s sã♦ ❤✐♣❡r♣❧❛♥♦s ❡♠R3

❉❡✜♥✐çã♦ ✸✳✹✳ ❊♠ Rn✱ ✉♠ s❡♠✐❡s♣❛ç♦ é ❛ r❡❣✐ã♦ ❞❡ ✉♠ ❞♦s ❧❛❞♦s ❞❡ ✉♠ ❤✐♣❡r♣❧❛♥♦✳ ❊♠ ♦✉tr❛s ♣❛❧❛✈r❛s✱ sã♦ ♦s ♣♦♥t♦s ① t❛✐s q✉❡

a1x1+a2x2+...+anxn ≥b ♦✉ a1x1+a2x2+...+anxn≤b

♣❛r❛ ❞❡t❡r♠✐♥❛❞♦s a_i ❡ ❜ ∈R✳

❉❡✜♥✐çã♦ ✸✳✺✳ ❆s ✈❛r✐á✈❡✐s ❞❡ ❞❡❝✐sã♦ sã♦ ❛s ✐♥❝ó❣♥✐t❛s✱ ♦✉ ✈❛❧♦r❡s ❞❡s❝♦♥❤❡❝✐❞♦s✱ q✉❡ s❡rã♦ ❞❡t❡r♠✐♥❛❞♦s ♣❡❧❛ s♦❧✉çã♦ ❞♦ ♠♦❞❡❧♦✳ P♦❞❡♠ s❡r ❝❧❛ss✐✜❝❛❞❛s ❞❡ ❛❝♦r❞♦ ❝♦♠ ❛s s❡❣✉✐♥t❡s ❡s❝❛❧❛s ❞❡ ♠❡♥s✉r❛çã♦✿ ✈❛r✐á✈❡✐s ❝♦♥tí♥✉❛s✱ ❞✐s❝r❡t❛s ♦✉ ❜✐♥ár✐❛s✳ ❆s ✈❛r✐á✈❡✐s ❞❡ ❞❡❝✐sã♦ ❞❡✈❡♠ ❛ss✉♠✐r ✈❛❧♦r❡s ♥ã♦ ♥❡❣❛t✐✈♦s✳

❉❡✜♥✐çõ❡s ❡ r❡s✉❧t❛❞♦s ♣r❡❧✐♠✐♥❛r❡s ✷✼

❉❡✜♥✐çã♦ ✸✳✽✳ ❙♦❧✉çã♦ ✈✐á✈❡❧ ♦✉ ❢❛❝tí✈❡❧ é ❛q✉❡❧❛ q✉❡ s❛t✐s❢❛③ t♦❞❛s ❛s r❡str✐çõ❡s ❞♦ ♠♦❞❡❧♦✱ ✐♥❝❧✉s✐✈❡ ❛s ❞❡ ♥ã♦ ♥❡❣❛t✐✈✐❞❛❞❡✳

❆ ✐♠♣♦rtâ♥❝✐❛ ❞♦s ♣♦♥t♦s ❡①tr❡♠♦s ❞❡ ✉♠❛ r❡❣✐ã♦ ✈✐á✈❡❧ é ♠♦str❛❞❛ ♣❡❧♦ s❡❣✉✐♥t❡ t❡♦r❡♠❛✳

❚❡♦r❡♠❛ ✸✳✶✳ ❙❡ ❛ r❡❣✐ã♦ ✈✐á✈❡❧ ❞❡ ✉♠ Pr♦❜❧❡♠❛ ❞❡ Pr♦❣r❛♠❛çã♦ ▲✐♥❡❛r é ♥ã♦✲✈❛③✐❛ ❡ ❧✐♠✐t❛❞❛✱ ❡♥tã♦ ❛ ❢✉♥çã♦✲♦❜❥❡t✐✈♦ ❛t✐♥❣❡ t❛♥t♦ ✉♠ ✈❛❧♦r ♠á①✐♠♦ q✉❛♥t♦ ✉♠ ✈❛❧♦r ♠í♥✐♠♦ ❡ ❡st❡s ♦❝♦rr❡♠ ❡♠ ♣♦♥t♦s ❡①tr❡♠♦s ❞❛ r❡❣✐ã♦ ✈✐á✈❡❧✳ ❙❡ ❛ r❡❣✐ã♦ ✈✐á✈❡❧ é ✐❧✐♠✐t❛❞❛✱ ❡♥tã♦ ❛ ❢✉♥çã♦✲♦❜❥❡t✐✈♦ ♣♦❞❡ ♦✉ ♥ã♦ ❛t✐♥❣✐r ✈❛❧♦r❡s ♠á①✐♠♦ ♦✉ ♠í♥✐♠♦❀ ❝♦♥t✉❞♦✱ s❡ ❛t✐♥❣✐r ✉♠ ♠á①✐♠♦ ♦✉ ✉♠ ♠í♥✐♠♦✱ ❡st❡ ♦❝♦rr❡rá ❡♠ ♣♦♥t♦s ❡①tr❡♠♦s✳

❆ ❋✐❣✉r❛ ✸✳✶ s✉❣❡r❡ ❛ ✐❞❡✐❛ ♣♦r trás ❞❛ ♣r♦✈❛ ❞♦ t❡♦r❡♠❛ ✸✳✶✳ ❈♦♠♦ ❛ ❢✉♥çã♦ ♦❜❥❡t✐✈♦ z =c₁x₁+c₂x₂ ❞❡ ✉♠ PP▲ é ✉♠❛ ❢✉♥çã♦ ❧✐♥❡❛r ❞❡ ①₁ ❡ ❞❡ ①₂✱ ❛s s✉❛s ❝✉r✈❛s

❞❡ ♥í✈❡❧✭❛s ❝✉r✈❛s ❛♦ ❧♦♥❣♦ ❞❛s q✉❛✐s ③ t❡♠ ✈❛❧♦r ❝♦♥st❛♥t❡✮ sã♦ r❡t❛s✳ ➚ ♠❡❞✐❞❛ q✉❡ ❞❡s❧♦❝❛♠♦s ♣❡r♣❡♥❞✐❝✉❧❛r♠❡♥t❡ ❡st❛s r❡t❛s✱ ❛ ❢✉♥çã♦✲♦❜❥❡t✐✈♦ ♦✉ ❝r❡s❝❡ ♦✉ ❞❡❝r❡s❝❡ ♠♦♥♦t♦♥❛♠❡♥t❡✳ ❉❡♥tr♦ ❞❡ ✉♠❛ r❡❣✐ã♦ ✈✐á✈❡❧ ❧✐♠✐t❛❞❛✱ ♦s ✈❛❧♦r❡s ♠á①✐♠♦s ❡ ♠í♥✐♠♦s ❞❡ ③ ❞❡✈❡♠ ♦❝♦rr❡r✱ ♣♦rt❛♥t♦✱ ♥♦s ♣♦♥t♦s ❡①tr❡♠♦s✱ ❝♦♠♦ ✐♥❞✐❝❛ ❛ ❋✐❣✉r❛ ✸✳✶✳

❋✐❣✉r❛ ✸✳✶✿ ❘❡♣r❡s❡♥t❛çã♦ ❞❛ r❡❣✐ã♦ ✈✐á✈❡❧ ❞❡ ✉♠ PP▲ ❡ s✉❛s ❝✉r✈❛s ❞❡ ♥í✈❡❧✳

❉❡✜♥✐çã♦ ✸✳✾✳ ❙♦❧✉çã♦ Ót✐♠❛ é ❛ s♦❧✉çã♦ ❢❛❝tí✈❡❧ q✉❡ ❛♣r❡s❡♥t❡ ♦ ♠❡❧❤♦r ✈❛❧♦r ❞❛ ❢✉♥çã♦ ♦❜❥❡t✐✈♦✳

❚❡♦r❡♠❛ ✸✳✷✳ ❖ ❝♦♥❥✉♥t♦ ❙ ❞❡ s♦❧✉çõ❡s ✈✐á✈❡✐s ♣❛r❛ ✉♠ Pr♦❜❧❡♠❛ ❞❡ Pr♦❣r❛♠❛çã♦ ▲✐♥❡❛r é ❢❡❝❤❛❞♦✱ ❝♦♥✈❡①♦ ❡ ❧✐♠✐t❛❞♦ ♣♦r ❜❛✐①♦✳

Pr♦✈❛✿ P❡❧❛ r❡str✐çã♦ x ≥ 0✱ ❙ é ❧✐♠✐t❛❞♦ ♣♦r ❜❛✐①♦✳ ❆❧é♠ ❞✐ss♦✱ ❙ é ✐♥t❡rs❡❝çã♦

✷✽ P❘❖●❘❆▼❆➬➹❖ ▲■◆❊❆❘

❚❡♦r❡♠❛ ✸✳✸✳ ❙❡❥❛ ❙ ♦ ❝♦♥❥✉♥t♦ ❞❡ s♦❧✉çõ❡s ✈✐á✈❡✐s ♣❛r❛ ✉♠ Pr♦❜❧❡♠❛ ❞❡ Pr♦❣r❛♠❛çã♦ ▲✐♥❡❛r✳ ❊♥tã♦✱ s❡ ❡①✐st❡ s♦❧✉çã♦ ót✐♠❛ ♣❛r❛ ♦ Pr♦❜❧❡♠❛ ❞❡ Pr♦❣r❛♠❛çã♦ ▲✐♥❡❛r✱ ❡①✐st❡ ✉♠ ♣♦♥t♦ ❡①tr❡♠♦ ❡♠ ❙ ❝♦♠ ✈❛❧♦r ót✐♠♦✳

Pr♦✈❛✿ ❙❡❥❛ c = (c₁, c₂, ..., c_n) ❛ ♠❛tr✐③ ❞♦s ❝♦❡✜❝✐❡♥t❡s ❞❛ ❢✉♥çã♦ ♦❜❥❡t✐✈♦ ❡ x = (x₁, x₂, ..., x_n) ❛ ♠❛tr✐③ ❞❛s ✈❛r✐á✈❡✐s ❞❡ ❞❡❝✐sã♦ ❞❛ ❢✉♥çã♦ ♦❜❥❡t✐✈♦✳ ❉❡st❛ ❢♦r♠❛✱ ❛

❢✉♥çã♦ ♦❜❥❡t✐✈♦ ♣♦❞❡ s❡r r❡♣r❡s❡♥t❛❞❛ ♣♦r✿ z =cT_x✳

❙ t❡♠ ✉♠ ♥ú♠❡r♦ ✜♥✐t♦ ❞❡ ♣♦♥t♦s ❡①tr❡♠♦s q✉❡ ❞❡♥♦t❛♠♦s ♣♦r x∗

1, x

∗

2, ..., x

∗

p✳ ❙❡❥❛

x₀ ✉♠ ♣♦♥t♦ ✈✐á✈❡❧ ♠❛①✐♠✐③❛♥❞♦ cT_x ❡♠ ❙✿

∀x∈S✱ cT_x

0 ≥cTx

❙✉♣♦♥❤❛ q✉❡ x₀ ♥ã♦ é ♣♦♥t♦ ❡①tr❡♠♦✳ ❊♥tã♦ x₀ ♣♦❞❡ s❡r ❞❡s❝r✐t♦ ❝♦♠♦ ❝♦♠❜✐♥❛çã♦

❝♦♥✈❡①❛ ❞♦s ♣♦♥t♦s ❡①tr❡♠♦s ❞❡ ❙✳

x₀ =

p

P

i=1

λix∗i✱ λi ≥0✱ Pλi = 1

❙❡❥❛ ❡♥tã♦ x∗

r ♦ ♣♦♥t♦ ❡①tr❡♠♦ ❝♦♠ ♠❛✐♦r ✈❛❧♦r ♦❜❥❡t✐✈♦✳ ❊♥tã♦✱

cT_x

0 =cT(

p

P

i=1

λix∗i)

=

p

P

i=1

λi(cTx∗i)

≤

p

P

i=1

λi(cTx∗r)

=cT_x∗

r p

P

i=1

λi

cT_x∗

r✳

❚❡♠♦s ❡♥tã♦cT_x

0 ≥cTx∗r✳ ❈♦♠♦x0 é ót✐♠♦✱ cTx0 =cTx∗r✱ ❡ ❡①✐st❡ ♦ ♣♦♥t♦ ❡①tr❡♠♦

x∗

r ♦♥❞❡ ♦ ✈❛❧♦r ❞♦ ♦❜❥❡t✐✈♦ é ót✐♠♦✳

❚❡♦r❡♠❛ ✸✳✹✳ ❖ ❝♦♥❥✉♥t♦ ❞❡ s♦❧✉çõ❡s ót✐♠❛s ♣❛r❛ ✉♠ Pr♦❜❧❡♠❛ ❞❡ Pr♦❣r❛♠❛çã♦ ▲✐✲ ♥❡❛r é ✉♠ ❝♦♥❥✉♥t♦ ❝♦♥✈❡①♦✳

Pr♦✈❛✿ ❙❡❥❛ ❑ ♦ ❝♦♥❥✉♥t♦ ❞❡ s♦❧✉çõ❡s ót✐♠❛s✱ x∗

1, x

∗

1 ∈ K✱ ❡ z

∗ ♦ ✈❛❧♦r ót✐♠♦ ❞❛

❢✉♥çã♦ ♦❜❥❡t✐✈♦✳ ❊♥tã♦✱

cT_x∗

1 =cTx

∗

2 =z

∗✳

❈♦♠♦ sã♦ ✈✐á✈❡✐s✱ x∗

1, x

∗

1 ∈S ❡ ❙ é ❝♦♥✈❡①♦✱ ♣♦rt❛♥t♦

λx∗

1+ (1−λ)x

∗

2 ∈S✳

0≤λ≤1.

❚❡♠♦s ❡♥tã♦

cT_(x∗

1, x

∗

2) =λcTx

∗

1+ (1−λ)cTx

∗

2

=λz∗

+ (1−λ)z∗

=z∗✳

❆ss✐♠✱ λx∗

1+ (1−λ)x

∗

❋♦r♠✉❧❛çã♦ ❞❡ ✉♠ Pr♦❜❧❡♠❛ ❞❡ Pr♦❣r❛♠❛çã♦ ▲✐♥❡❛r ✷✾

✸✳✷ ❋♦r♠✉❧❛çã♦ ❞❡ ✉♠ Pr♦❜❧❡♠❛ ❞❡ Pr♦❣r❛♠❛çã♦ ▲✐✲

♥❡❛r

❆ r❡s♦❧✉çã♦ ❞❡ ✉♠ Pr♦❜❧❡♠❛ ❞❡ Pr♦❣r❛♠❛çã♦ ▲✐♥❡❛r ✭PP▲✮ ❝♦♥s✐st❡ ❡♠ ♠❛①✐♠✐③❛r ♦✉ ♠✐♥✐♠✐③❛r ✉♠❛ ❢✉♥çã♦ ❧✐♥❡❛r ❝♦♠ ✈❛r✐á✈❡✐s ❞❡ ❞❡❝✐sã♦ ❝♦♥tí♥✉❛s ❞❡♥♦♠✐♥❛❞❛ ❝♦♠♦ ❢✉♥çã♦ ♦❜❥❡t✐✈♦✱ ❧❡✈❛♥❞♦ ❡♠ ❝♦♥s✐❞❡r❛çã♦ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ r❡str✐çõ❡s✱ ❢♦r♠❛❞♦ ♣♦r ✉♠ s✐st❡♠❛ ❧✐♥❡❛r ❞❡ ✐❣✉❛❧❞❛❞❡s ♦✉ ❞❡s✐❣✉❛❧❞❛❞❡s✳

❆ ❢♦r♠✉❧❛çã♦ ❞❡ ✉♠ ♠♦❞❡❧♦ ❣❡r❛❧ ❞❡ Pr♦❣r❛♠❛çã♦ ▲✐♥❡❛r ♣♦❞❡ s❡r r❡♣r❡s❡♥t❛❞❛ ❝♦♠♦✿

♠✐♥✐♠✐③❛r ♦✉ ♠❛①✐♠✐③❛r z =c1x1+c2x2 +...+cnxn ✱ s❛t✐s❢❛③❡♥❞♦ ❛s r❡str✐çõ❡s

a₁₁x₁+a₁₂x₂+...+a₁nxn ≥ b1

a₂₁x₁+a₂₂x₂+...+a₂nxn ≥ b2

✳ ✳ ✳

✳ ✳ ✳

✳ ✳ ✳

am1x1+am2x2+...+amnxn ≥ bm

❡♠ q✉❡ ③ é ❛ ❢✉♥çã♦ ♦❜❥❡t✐✈♦❀

❆s r❡str✐çõ❡s ♣♦❞❡♠ ✈✐r ❛❝♦♠♣❛♥❤❛❞❛s t❛♠❜é♠ ❞♦s s✐♥❛✐s ≤ ♦✉ ❂✳

x_j sã♦ ❛s ✈❛r✐á✈❡✐s ❞❡ ❞❡❝✐sã♦✱ ♣r✐♥❝✐♣❛✐s ♦✉ ❝♦♥tr♦❧á✈❡✐s✱ x_j ≥0✱ ❥❂✶✱ ✷✱ ✳✳✳✱ ♥❀

aij é ❛ ❝♦♥st❛♥t❡ ♦✉ ❝♦❡✜❝✐❡♥t❡ ❞❛s ✐✲és✐♠❛ r❡str✐çã♦ ❞❛ ❥ ✲és✐♠❛ ✈❛r✐á✈❡❧✱ ✐❂✶✱ ✷✱

✳✳✳✱ ♠✱ ❥❂✶✱ ✷✱ ✳✳✳✱ ♥❀

bi é ♦ t❡r♠♦ ✐♥❞❡♣❡♥❞❡♥t❡ ♦✉ q✉❛♥t✐❞❛❞❡ ❞❡ r❡❝✉rs♦s ❞✐s♣♦♥í✈❡✐s ❞❛ ✐✲és✐♠❛ r❡str✐✲

çã♦✱ ✐❂✶✱ ✷✱ ✳✳✳✱ ♠❀

cj é ❛ ❝♦♥st❛♥t❡ ♦✉ ❝♦❡✜❝✐❡♥t❡ ❞❛ ❥ ✲és✐♠❛ ✈❛r✐á✈❡❧ ❞❛ ❢✉♥çã♦ ♦❜❥❡t✐✈♦✱ ❥❂✶✱ ✷✱ ✳✳✳✱

♥✳

✸✳✸ ▼♦❞❡❧♦ ❞❡ Pr♦❣r❛♠❛çã♦ ▲✐♥❡❛r ♥❛s ❢♦r♠❛s ♣❛✲

❞rã♦ ❡ ❝❛♥ô♥✐❝❛

✸✵ P❘❖●❘❆▼❆➬➹❖ ▲■◆❊❆❘

✸✳✸✳✶ ▼♦❞❡❧♦ ❞❡ Pr♦❣r❛♠❛çã♦ ▲✐♥❡❛r ♥❛ ❢♦r♠❛ ♣❛❞rã♦

❉❡✜♥✐çã♦ ✸✳✶✵✳ ❯♠ ♠♦❞❡❧♦ ❞❡ Pr♦❣r❛♠❛çã♦ ▲✐♥❡❛r ❡stá ♥❛ ❢♦r♠❛ ♣❛❞rã♦ q✉❛♥❞♦ ❛t❡♥❞❡ ❛♦s s❡❣✉✐♥t❡s r❡q✉✐s✐t♦s✿

✶✳ ❖s t❡r♠♦s ✐♥❞❡♣❡♥❞❡♥t❡s ❞❛s r❡str✐çõ❡s ❞❡✈❡♠ s❡r ♥ã♦ ♥❡❣❛t✐✈♦s✳

✷✳ ❚♦❞❛s ❛s r❡str✐çõ❡s ❞❡✈❡♠ ❡st❛r r❡♣r❡s❡♥t❛❞❛s ♣♦r ❡q✉❛çõ❡s ❧✐♥❡❛r❡s ❡ ❛♣r❡s❡♥t❛✲ ❞❛s ♥❛ ❢♦r♠❛ ❞❡ ✐❣✉❛❧❞❛❞❡✳

✸✳ ❆s ✈❛r✐á✈❡✐s ❞❡ ❞❡❝✐sã♦ ❞❡✈❡♠ s❡r ♥ã♦ ♥❡❣❛t✐✈❛s✳

❆ ❢♦r♠❛ ♣❛❞rã♦ s❡rá r❡♣r❡s❡♥t❛❞❛ ♠❛t❡♠❛t✐❝❛♠❡♥t❡ ❝♦♠♦✿

♠✐♥✐♠✐③❛r ♦✉ ♠❛①✐♠✐③❛r z =c1x1+c2x2+...+cnxn✱ s❛t✐s❢❛③❡♥❞♦ ❛s r❡str✐çõ❡s

a11x1 +a12x2+...+a1nxn ❂ b1

a21x1 +a22x2+...+a2nxn ❂ b2

✳ ✳ ✳

✳ ✳ ✳

✳ ✳ ✳

am1x1+am2x2+...+amnxn ❂ bm

❝♦♠ xj ≥0✱ ❥❂✶✱ ✷✱ ✳✳✳✱ ♥✳

bi ≥0i= 1, ..., m.

✸✳✸✳✷ ▼♦❞❡❧♦ ❞❡ Pr♦❣r❛♠❛çã♦ ▲✐♥❡❛r ♥❛ ❢♦r♠❛ ❝❛♥ô♥✐❝❛

❊♠ ✉♠ ♠♦❞❡❧♦ ❞❡ ♣r♦❣r❛♠❛çã♦ ❧✐♥❡❛r ♥❛ ❢♦r♠❛ ❝❛♥ô♥✐❝❛✱ ❛s r❡str✐çõ❡s s❡rã♦ ❛♣r❡✲ s❡♥t❛❞❛s ♥❛ ❢♦r♠❛ ❞❡ ✐♥❡q✉❛çõ❡s✱ ♦♥❞❡ ❛ ❢✉♥çã♦ ♦❜❥❡t✐✈♦ ③ ♣♦❞❡rá s❡r ❞❡ ♠❛①✐♠✐③❛çã♦ ♦✉ ❞❡ ♠✐♥✐♠✐③❛çã♦✳

❉❡✜♥✐çã♦ ✸✳✶✶✳ P❛r❛ ✉♠❛ ❢✉♥çã♦ ♦❜❥❡t✐✈♦ ③ ❞❡ ♠❛①✐♠✐③❛çã♦✱ t♦❞❛s ❛s r❡str✐çõ❡s ❞❡✲ ✈❡♠ s❡r r❡♣r❡s❡♥t❛❞❛s ❝♦♠ s✐♥❛❧ ❞♦ t✐♣♦ ≤✱ ❥á ♣❛r❛ ✉♠❛ ❢✉♥çã♦ ♦❜❥❡t✐✈♦ ③ ❞❡ ♠✐♥✐♠✐✲ ③❛çã♦✱ ❛s r❡str✐çõ❡s ❞❡✈❡♠ ❡st❛r ❝♦♠ s✐♥❛❧ ❞♦ t✐♣♦ ≥✳

P❛r❛ ✉♠ ♣r♦❜❧❡♠❛ ❞❡ ♠❛①✐♠✐③❛çã♦✱ ❛ ❢♦r♠❛ ❝❛♥ô♥✐❝❛ ♣♦❞❡ s❡r r❡♣r❡s❡♥t❛❞❛ ♠❛t❡✲ ♠❛t✐❝❛♠❡♥t❡ ❝♦♠♦✿

♠❛①✐♠✐③❛r z =c1x1+c2x2+...+cnxn✱ s❛t✐s❢❛③❡♥❞♦ ❛s r❡str✐çõ❡s

a11x1+a12x2+...+a1nxn ≤ b1

a21x1+a22x2+...+a2nxn ≤ b2

✳ ✳ ✳

✳ ✳ ✳

✳ ✳ ✳

am1x1 +am2x2+...+amnxn ≤ bm

▼♦❞❡❧♦ ❞❡ Pr♦❣r❛♠❛çã♦ ▲✐♥❡❛r ♥❛s ❢♦r♠❛s ♣❛❞rã♦ ❡ ❝❛♥ô♥✐❝❛ ✸✶

❏á ♣❛r❛ ✉♠ ♣r♦❜❧❡♠❛ ❞❡ ♠✐♥✐♠✐③❛çã♦✱ ❛ ❢♦r♠❛ ❝❛♥ô♥✐❝❛ ♣❛ss❛ ❛ s❡r✿ ♠✐♥✐♠✐③❛r z =c₁x₁+c₂x₂+...+c_nx_n✱ s❛t✐s❢❛③❡♥❞♦ ❛s r❡str✐çõ❡s

a₁₁x₁+a₁₂x₂+...+a_1nx_n ≥ b₁ a₂₁x₁+a₂₂x₂+...+a_2nx_n ≥ b₂

✳ ✳ ✳

✳ ✳ ✳

✳ ✳ ✳

a_m1x₁+a_m2x₂+...+a_mnx_n ≥ b_m

❝♦♠ x_j ≥0✱ ❥❂✶✱ ✷✱ ✳✳✳✱ ♥✳

✸✳✸✳✸ ❚r❛♥s❢♦r♠❛çõ❡s ♣❛r❛ ❛ ❢♦r♠❛ ♣❛❞rã♦ ♦✉ ❝❛♥ô♥✐❝❛

❆ ♣❛rt✐r ❞❡ ❛❧❣✉♠❛s ♦♣❡r❛çõ❡s ❡❧❡♠❡♥t❛r❡s✱ ♣♦❞❡♠♦s ❡s❝r❡✈❡r ✈ár✐❛s ❢♦r♠✉❧❛çõ❡s ❡q✉✐✈❛❧❡♥t❡s✱ ♣❛r❛ ✉t✐❧✐③❛çã♦ ❞❛ ♠❛✐s ❝♦♥✈❡♥✐❡♥t❡✳ ❚❛✐s ❝♦♠♦✿

✶✳ ▼✐♥✐♠✐③❛r ✉♠❛ ❢✉♥çã♦ ❡q✉✐✈❛❧❡ ❛ ♠❛①✐♠✐③❛r s❡✉ ♥❡❣❛t✐✈♦✱ ♦✉ s❡❥❛✱ ♠✐♥✐♠✐③❛r

z = c₁x₁ +c₂x₂ +...+c_nx_n ❡q✉✐✈❛❧❡ ❛ ♠❛①✐♠✐③❛r (−z) = (−c₁)x₁ + (−c₂)x₂ +

...+ (−c_n)x_n✳

✷✳ ❙❡ t✐✈❡r♠♦s ❞❡s✐❣✉❛❧❞❛❞❡s ❡♠ ✈❡③ ❞❡ ❡q✉❛çõ❡s✱ ♣♦❞❡♠♦s ✐♥s❡r✐r ♥♦✈❛s ✈❛r✐á✈❡✐s✱ ❝♦♥❤❡❝✐❞❛s ❝♦♠♦ ✈❛r✐á✈❡✐s ❞❡ ❢♦❧❣❛✱ ♦❜t❡♥❞♦ ✐❣✉❛❧❞❛❞❡s✳

❙❡❣✉❡ ✉♠ ❡①❡♠♣❧♦ ❞❡ ❛♣❧✐❝❛çã♦ ❛❜♦r❞❛❞♦ ❡♠ ▼❛❝✉❧❛♥✭✶✾✽✵✮✳

❊①❡♠♣❧♦ ✸✳✷✳ ❙❡ t✐✈❡r♠♦s✿ ✹x1 ✰ ✷x2 ≤ ✺ ❛ ✈❛r✐á✈❡❧ ❞❡ ❢♦❧❣❛ s1 ✭s1 ≥0✮ ♣♦❞❡

s❡r ✐♥tr♦❞✉③✐❞❛ ♣❛r❛ ❝♦♥s❡❣✉✐r♠♦s ✹x1 ✰ ✷x2 ✰ s1 ❂ ✺✳

P♦❞❡♠♦s ✈❡r ✐♠❡❞✐❛t❛♠❡♥t❡ q✉❡✱ ♣❛r❛ ❡❧✐♠✐♥❛r ❞❡s✐❣✉❛❧❞❛❞❡s ❞♦ t✐♣♦ ✭≥✮✱ ❜❛st❛ ❝♦❧♦❝❛r ✉♠❛ ✈❛r✐á✈❡❧ ❞❡ ❢♦❧❣❛ ❝♦♠ s✐♥❛❧ ♥❡❣❛t✐✈♦✱ ✐st♦ é✱ ✹x₁ ✰ ✷x₂ ≥ ✺ t♦r♥❛✲s❡ ✹x₁ ✰ ✷x₂ ✲ s₁ ❂ ✺ ✭s₁ ≥0✮✳

➱ ♥❡❝❡ssár✐♦ ❝♦❧♦❝❛r ✉♠❛ ✈❛r✐á✈❡❧ ❞❡ ❢♦❧❣❛ ❞✐❢❡r❡♥t❡ ♣❛r❛ ❝❛❞❛ ❧✐♥❤❛✳ ❙❡ ❢♦r ✉s❛❞❛ ❛ ♠❡s♠❛ ✈❛r✐á✈❡❧ ❞❡ ❢♦❧❣❛✱ ♦❜t❡rí❛♠♦s ✉♠❛ ♥♦✈❛ r❡❧❛çã♦ ❡♥tr❡ ❛s ❞❡s✐❣✉❛❧❞❛❞❡s q✉❡ ♥ã♦ ❢♦✐ ✐♠♣♦st❛ ♥♦ ♣r♦❜❧❡♠❛✳

✸✳ ❯♠❛ ✐❣✉❛❧❞❛❞❡ ✹x₁❂ ✺✱ ♣♦r ❡①❡♠♣❧♦✱ ♣♦❞❡ s❡r s✉❜st✐t✉í❞❛ ♣♦r ❞✉❛s ❞❡s✐❣✉❛❧❞❛❞❡s

✹x₁ ≤ ✺ ❡ ✹x₁ ≥ ✺✳

✹✳ ❯♠❛ ✈❛r✐á✈❡❧ xj é ❝♦♥s✐❞❡r❛❞❛ ❧✐✈r❡✱ ♦✉ s❡❥❛✱ s❡♠ r❡str✐çã♦ ❞❡ s✐♥❛❧✱ s❡ ♣♦❞❡ s❡r

s✉❜st✐t✉í❞❛ ♣❡❧❛ ❞✐❢❡r❡♥ç❛ ❞❡ ❞✉❛s ✈❛r✐á✈❡✐s ♥ã♦ ♥❡❣❛t✐✈❛s✿