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Constrained Optimization - Columbia University

ConstrainedOptimizationJoshuaWilde,revis edbyIsab elTecu,TakeshiSuzukiandMar aJos Bo ccardiAugust13,20131 GeneralProblemConsiderthefollowinggenera lconstrainedoptimizationproblem:maxxi Rf(x1,..,xn)sub jectto:g1(x1,..,xn) b1,..,gk(x1,..,xn) bk,h1(x1,..,xn) =c1,..,hm(x1,..,xn) = (x)iscalledtheob jectivefunction,g(x)iscalledaninequality constraint,andh(x) ,gandhareC1functions, ersfromtheregularunconstrainedoptimizati onprobleminthatinsteadof ndingthemaximumoff(x),weare ndingthemaximumoff(x)onlyoverthep :Maximizef(x) =x2sub jectto0 x :Weknowthatf(x)isstrictlymonotonicallyin creasingoverthedomain,thereforethemaximu m(ifitexists)mustlieatthelargestnumb ,thep ointx= 1isthemaximalnumb erinthedomain, ecausewecouldvisualizethegraphoff(x) ,weseeametho dto ndconstrainedmaximaoffunctionsevenwhenwe can' :maxx Rf(x1.)

2 Constrained Optimization us onto the highest level curve of f(x) while remaining on the function h(x). Notice also that the function h(x) will be just tangent to the level curve of f(x).

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