ized ▁notation ▁may ▁be ▁used ▁as ▁to ▁the ▁function ▁and ▁its ▁input ( s ). ▁More ▁generally , ▁optimization ▁includes ▁finding ▁the ▁best ▁available ▁element ▁of ▁some ▁function ▁given ▁a ▁defined ▁domain ▁and ▁may ▁use ▁a ▁variety ▁of ▁different ▁ ▁computational ▁optimization ▁techniques . ▁ ▁Econom ics ▁is ▁closely ▁enough ▁linked ▁to ▁optimization ▁by ▁agents ▁in ▁an ▁economy ▁that ▁an ▁influ ential ▁definition ▁related ly ▁describes ▁econom ics ▁qu a ▁science ▁as ▁the ▁" stud y ▁of ▁human ▁behavior ▁as ▁a ▁relationship ▁between ▁ends ▁and ▁scar ce ▁means " ▁with ▁alternative ▁uses . ▁Opt im ization ▁problems ▁run ▁through ▁modern ▁econom ics , ▁many ▁with ▁explicit ▁economic ▁or ▁technical ▁constraints . ▁In ▁micro e conom ics , ▁the ▁utility ▁maxim ization ▁problem ▁and ▁its ▁dual ▁problem , ▁the ▁exp end iture ▁minim ization ▁problem ▁for ▁a ▁given ▁level ▁of ▁utility , ▁are ▁economic ▁optimization ▁problems . ▁Theory ▁pos its ▁that ▁consum ers ▁ ▁maxim ize ▁their ▁utility , ▁subject ▁to ▁their ▁budget ▁constraints ▁and ▁that ▁fir ms ▁maxim ize ▁their ▁prof its , ▁subject ▁to ▁their ▁production ▁functions , ▁input ▁costs , ▁and ▁market ▁demand . ▁ ▁Econom ic ▁equilibrium ▁is ▁studied ▁in ▁optimization ▁theory ▁as ▁a ▁key ▁ing red ient ▁of ▁economic ▁the or ems ▁that ▁in ▁principle ▁could ▁be ▁tested ▁against ▁empir ical ▁data . ▁ ▁Ne wer ▁develop ments ▁have ▁occurred ▁in ▁dynamic ▁programming ▁and ▁model ing ▁optimization ▁with ▁risk ▁and ▁uncertainty , ▁including ▁applications ▁to ▁port folio ▁theory , ▁the ▁econom ics ▁of ▁information , ▁and ▁search ▁theory .