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??7. ????(????????)active constraint
?x??
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x????????
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???????????
Karush-Kuhn-Tucker ??KKT??
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Lagrange??
Lagrange??
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Karush-Kuhn-Tucker??KKT??
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17
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Karush-Kuhn-Tucker??KKT??
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?? (Fritz John??)
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Fritz John ???KKT????? Fritz John
??????w00??????Fritz John ??????????????????,????
???????????????????,?????????,??????????????w0?0??
?,?????? w0?0 ,???????????????????????????????????
,???????????????????,??????KKT???
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1) ?????????????KKT??????
2) ????????????KKT??
?????????????????????????(Farkas??
???????????????,????????????????,???????Langrang
e??
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Lagrange??
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Lagrange??
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Lagrange??
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Lagrange????
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Lagrange????????
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f?g???,h??
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1)??????,????????
2)???????????????,?????????????????????????(?????
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3)???????,?????????????,?????????
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38
Wolfe??
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Lagrange??
Wolfe??
Wolfe???????????,???????,?
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2)???????????????,?????????????????????????(?????
0).
39
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Lagrange??
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1?????
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Unconstrained Minimization Technique - SUMT)
2?????

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3???
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4????(???????)
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(No Transcript)
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(1)????
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(2)????(???)
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(3)?????
yes
No
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(4)??????
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?
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??P207?2??2??????!
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(5)????????

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(6)?????
??P202,??8.1,??8.2.
??P203, ??8.4.
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(3)????
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(4)????(???)
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yes
No
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?????P217-218.
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(5)?????
(6)???????
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(7)????????????
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1. x(0)?S 0(??P220???????) 2.??????? 3.????B(x)
?S 0??????gi(x)??(?????????????) 4.???x(k)?R
(????x(k)x) 5.??????
1.??x(0)?Rn 2.?????? 3.?????????S???????
4.???x(k) Ï R 5.??????
??P246. 1,2,4,7,8,9,10.
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6. ???
(1)????
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??????Langrange??????????????
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(2)????????
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?????x (0.25, 0.75),???????????? ????????

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?????,xk? xk ??x??????,??????6???????????15?
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66.4. ????,??????????????,????????????.
69
Matlab?????????
?? x?b?beq?lb?ub???,A?Aeq???,C(x)?Ceq(x)????????,
f(x)?????,f(x)?C(x)?Ceq(x)?????????
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?? fmincon ?? x fmincon(fun,x0,A,b) x
fmincon(fun,x0,A,b,Aeq,beq) x
fmincon(fun,x0,A,b,Aeq,beq,lb,ub) x
fmincon(fun,x0,A,b,Aeq,beq,lb,ub,nonlcon) x
fmincon(fun,x0,A,b,Aeq,beq,lb,ub,nonlcon,options)
x,fval fmincon() x,fval,exitflag
fmincon() x,fval,exitflag,output
fmincon() x,fval,exitflag,output,lambda
fmincon() x,fval,exitflag,output,lambda,grad
fmincon() x,fval,exitflag,output,lambda,grad,hes
sian fmincon()
71
?1
? (1)??????
72
(2)???M-?? fun1.m function ffun1(x)
f-x(1)-2x(2)(1/2)x(1)2(1/2)x(2)2
(3)??????youh1.m x011 A2 3 1
4 b65 Aeqbeq LB00
UB x,fvalfmincon('fun1',x0,A,b,Aeq,beq,L
B,UB)
(4)????????youh1,?????? x 0.7647
1.0588 fval -2.0294
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???????????
(1)?MATLAB??????????????? function c,
ceqnlcon (x) c(x(1)-1)2-x(2) ceq
?????
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(1)?MATLAB??????????????? function c,
ceqnlcon (x) c(x(1)-1)2-x(2) ceq
?????
(2)??????????????M?? fun2'x(1)2x(2)2-x(1)x(2
)-2x(1)-5x(2)' ???? x00 1 A-2 3
??????? b6 Aeq ??????? beq lb
x?????? ub x,fval,exitflag,output
,lambda,grad,hessian fmincon(fun2,x0,A,b,Aeq,beq
,lb,ub,_at_nlcon)
75
???? x 3 4 fval -13 exitflag
??? 1 output iterations 2
funcCount 9 stepsize 1
algorithm 'medium-scale SQP, Quasi-Newton,
line-search' firstorderopt
cgiterations lambda lower 2x1
double x??????,??lambda.lower????
upper 2x1 double x??????,?0???????
eqlin 0x1 double ??????????,??0???????
eqnonlin 0x1 double ????????????
ineqlin 2.5081e-008 ????????????
ineqnonlin 6.1938e-008 ????????????? grad
???????????? 1.0e-006 -0.1776
0 hessian ??????????Hessian? 1.0000
-0.0000 -0.0000 1.0000
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??????(quadratic programming)?Matlab?
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?? quadprog
?? x quadprog(H,f,A,b) ??H,f,A,b????????,x?
????????? x quadprog(H,f,A,b,Aeq,beq)
Aeq,beq???????
x quadprog(H,f,A,b,Aeq,beq,lb,ub)
lb,ub????x????? x quadprog(H,f,A,b,Aeq,beq,lb,ub
,x0) x0?????? x quadprog(H,f,A,b,Aeq,beq,lb,u
b,x0,options) options???????? x,fval
quadprog() fval???????? x,fval,exitflag
quadprog() exitflag???????????? x,fval,exitf
lag,output quadprog() output????????????
x,fval,exitflag,output,lambda quadprog()
lambda????????????
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?MATLAB????? H 1 -1 -1 2 f -2 -6 A
1 1 -1 2 2 1 b 2 2 3 lb
zeros(2,1) x,fval,exitflag,output,lambda
quadprog(H,f,A,b, , ,lb)
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