4. Statement of the mathematical programming problem and types of solutions

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1、StatementofthemathematicalprogrammingandtypesofsolutionsInstructor:MinWang1.ElementsofamaximizationproblemObjectivefunction:F:Rn!Rdi¤erentiable.0Vectorofchoicevariable:x=(x1x2x3...xn).Constraintset(orchoiceset):XRn:2.ProblemmaxF(x)w:r:txsubjecttoxX:Notes:Min

2、imizeF(x)bymaxF(x).Di¤erentformsofXgivevarietyofspecialcases:–X=Rn,unconstrainedoptimization.–EqualityconstrainedoptimizationX=fx2Rn:g(x)=b;g(x)=b;:::;g(x)=bg1122mmwhereg:Rn!Risdi¤erentiableandb2Rforallj=1;2;:::;m:Expectn>m.jj–gj(x)bjorxi0:–Kuhn-Tuckerformulat

3、ionX=fx2Rn:x0fori=1;2;:::;n;g(x)bforj=1;2;:::;mg:ijjTypesofsolutions–xisalocalsolutionto(+)((+)denotesthemaximizationproblem)ifthereexists">0suchthatF(x)F(x)forallx2XB(x):"–xisastrictlylocalsolutionto(+)ifthereexists">0suchthatF(x)>F(x)forallx2XB(x)and

4、x6=x."–xisaglobalsolutionto(+)ifF(x)F(x)forallx2X.–xisastrictlyglobalsolutionto(+)ifF(x)>F(x)forallx2Xandx6=x.3.FirstordernecessaryconditionWritex2XB(x)asx=x+h
x,where
x2Rn,k
xk=1,h2(";").Thenxisalocal"solutiontowithX=Rnifthereexists">0suchthatF(x)F

5、(x+h
x)forall
x2Rn,k
xk=1,andforallh2(";"):ConsiderTaylorexpansionaroundx,@F1@2FF(x+h
x)=F(x)+h(x)
x+h2
x0(x+h
x)
x;@x2@x2forsome2(0;1):1Combiningtheabovetwoconditions,xisalocalsolutionto(+)withX=Rnifthereexists">0suchthat@F1@2Fh(x)
x+h2
x0(x+h
x)
x

6、0(Fundamentalinequality)@x2@x2forall
x2Rn,k
xk=1,andforallh2(";"):Let
x=ithunitcoordinatorvector(onlytheithelementofthevectortakesthevalueof1andallothersarezero).Thentakeh2(0;"),dividethefundamentalinequalitybyh>0andtakethelimitash!0+,wehave@F(x)0;@xiTakeh2(

7、";0),dividethefundamentalinequalitybyh<0andtakethelimitash!0,wehave@F(x)0:@xiHencethelocalsolutionto(+)needtosatisfythe…rstordernecessarycondition@F(x)=0:@x4.SecondordernecessaryconditionForanygiven
xandh6=0,dividethefundamentalinequalityby1h2>0andtakethelim

8、itas2h!0,wehave@2F
x0(x)
x0@x2forall
x2Rnandk
xk=1.2Hencethelocalsolutionto(+)needtosatisfythesecondordernecessarycondition@F(x)is@x2negativesemi-de