Transcription of Constrained Optimization - Columbia University
{{id}} {{{paragraph}}}
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,..,xn)sub jectto:h(x1,..,xn) = (x1,..,xn).Sincewemightnotb eabletoachievetheun-constrainedmaximaoft hefunctionduetoourconstraint,weseekto ndthevalueofxwhichgets12 ConstrainedOptimizationusontothehighestl evelcurveoff(x)whileremainingonthefuncti onh(x).
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). Call the point which maximizes the optimization problem x , (also referred to as the maximizer ).
Domain:
Source:
Link to this page:
Please notify us if you found a problem with this document:
{{id}} {{{paragraph}}}
Optimization, Objective, Multi, Particle Swarm Optimization, Multi-objective optimization, Multi-Objective Optimization Using Evolutionary, Optimization Methods in Finance, Objective optimization, Optimization with L1-Norm Regularization, Node2vec, Optimization in R, Tutorial of AMPL for Linear Programming