最大最小对偶 Still waters run deep.流静水深流静水深,人静心深人静心深 Where there is life,there is hope有生命必有希望有生命必有希望x Xy YF x yminmax(,)F x yxX yY(,),x Xy YF x ymaxmin(,)x Xy YF x yF x yF x yxX yY1)min(,)(,)max(,),x Xx Xy Yy YF x yF x y2)maxmin(,)minmax(,)x Xx Xy Yy YgapF x yF x y3)minmax(,)maxmin(,)min maxinfsup2x Xy YF x yminmax(,)2 xXyY1,2,1,2,x Xy YF x ymaxmin(,)2 xyF x ya(,),ijAa12()43 gap0 ijAa12()41 x Xy YF x yminmax(,)2 x Xy YF x ymaxmin(,)1 gap10 3(,),F x yxX yYx Xy YF x yF x yF x yxX yY1)min(,)(,)max(,),x Xx Xy Yy YF x yF x y2)maxmin(,)minmax(,)x Xx Xy Yy YgapF x yF x y3)minmax(,)maxmin(,)*,xX yYF xyF xyF x yxX yY*(,)(,)(,),x Xx Xy Yy YF x yF x y4)minmax(,)maxmin(,)xy*(,)4f xs tg(x)h xmin ().0,()0 TTL x u vf xu g xv h xu(,)()()()(0)nTTx Ru vf xu g xv h xu(,)min()()(),0 其其中中nnuvuvuvx Rx RL x u vu vL x u v 0,0,0,2)maxmin(,)max(,)minmax(,)nuvx Rf xxSL x u vL x u vf xL x u vxS0,(),min(,)(,)()max(,),xS uv1),0,nnuvuvx Rx RL x u vL x u v0,0,3)minmax(,)maxmin(,)原规划原规划uvu v 0,max(,)凹函数凹函数5xs tx2min .10 20,2min .10 xxs tx nnuvuvx Rx RL x u vL x u v0,0,minmax(,)maxmin(,)xxs txxxx22121212min .40,0 xxs txxxx121212min -2.30(,)(0,0),(0,4),(4,4),(4,0),(1,2),(2,1)22121212min .40,0 xxs txxxx 6像集像集722121211min.40,0 xxs txxx x2()4,02uuu u 89f xs tg(x)h xmin ().0,()0 x,u v10f xs tg(x)h xmin ().0,()0 TTL x u vf xu g xv h xu(,)()()()(0)x u vxL x u vs tL x u vu,max (,).(,)0,0.111min2.0nTTx Rx Qxc xs tAxd1min2.00nTTx Rx Qxc xs tAxbCxd1min2.00nTTx Rx Qxc xs tAxbCxdlxu1min2.0nTTTx Rx HH xc xs txC,1min2.,0nTTx RyTy yc xs tyH xxC1max2Tuu uC Huc12共轭对偶、广义共轭对偶、广义Lagrange对偶对偶参阅参阅非线性规划及其理论非线性规划及其理论第第6章章应玖茜、魏权龄应玖茜、魏权龄13。