4. Equality constrained optimization problem and the method of Lagrange

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1、EqualityconstrainedoptimizationproblemandthemethodofLagrangeInstructor:MinWang1.Equalityconstrainedoptimizationproblem(+)maxF(x)w:r:txs:t:x2XwhereX=fx2Rn;g(x)=b;g(x)=b;:::;g(x)=b;g1122mmNotes:–n>m:–Considerthecaseofn=2andm=1.–Strategy:translatetheequalityconstrainedprob

2、lemwithn=2andm=1intoequivalentunconstrainedproblemwithn=1,i.e.,substitutetheconstraintintotheobjectivefunction.–Ex.maxx2x+xx1212w:r:tx1;x2s:t:4x1+3x2=22.LagrangefunctionConsiderthemaximizationproblem(1)maxF(x1;x2)w:r:tx1;x2s:t:g(x1;x2)=b–Assume(x;x)isalocalsolutionto(

3、1).12–Assumeconstraintquali…cationholdsatx=(x;x)if12@g@g(x)6=0and/or(x)6=0:@x1@x2Withoutlossofgenerality,weassume@g(x)6=0.@x2–Applyimplictfunctiontheoremtog(x;x)=baboutx=(x;x)(withxendogenous,x121221exogenous).Thereexists">0andh:R!Rdi¤erentiablesuchthatx2=h(x1),

4、g(x1;x2)=bforallx2B(x)and11"@g(x)h0(x)=@x11@g(x)@x2Translatetheequalityconstrainedproblemintounconstrainedproblem(2)maxH(x1)w:r:tx1whereH(x1)=F(x1;h(x1)):–(x;x)isalocalsolutionto(1)=)xisalocalsolutionto(2)=)Necessaryconditions:121H0(x)=0andH00(x)0.111–Writedo

5、wnthe…rstordernecessarycondition.Since0d@F@F0H(x1)=F(x1;h(x1))=(x1;h(x1))+(x1;h(x1))h(x1);dx1@x1@x2wehave@g(x)0@F@F@x1H(x1)=(x)(x)@g=0:@x1@x2(x)@x2–Wefurthercanwritethe…rstordernecessaryconditionas@F(x)@F@x2@g(x)(x)=0:@x1@g(x)@x1@x2Inaddition,itisobviousthat@F

6、(x)@F@x2@g(x)(x)=0:@x2@g(x)@x2@x2Comibingthesetwoconditions,wehave@F@g(x)=(x)@x@x@F(x)@x2@F@gwhere=@g:Notethat(x)and(x)arecolinear.(x)@x@x@x2Notes:–@F(x)pointsinthedirectionofgreatestrateofincreaseinF(
).@x–@F(x)isperpendiculartolevelcurveofF(
)throughx

7、;@g(x)isperpendiculartoconstrained@x@xcurveatx;thesetwocurvesarecolinearatx=)levelcurveofF(
)andconstrainedcurvemustbetangentatx.Sotheoptimalsolutionxneedtosatisfy@F(x)@g(x)=0andbg(x)=0.@x@xLagrangefunctionL(x1;x2;)=F(x1;x2)+[bg(x1;x2)]Conditionsfor(x;

8、x;)tobeastationarypointofL(
):12@L@F@g(x;)=(x)(x)=0@x1@x1@x1@L@F@g(x;)=(x)(x)=0@x2@x