机器学习7 拉格朗日KKT条件

minwf(w)primalconstraints:s.t.{gi(w)<=0,i=1,...,khi(w)=0,i=1,...,l[eq.0]L(w,α,β)=f(w)+∑i=1kαigi(w)+∑i=1kβihi(w)[eq.1]θP(w)=maxα,β:αi>=0L(w,α,β)[eq.2]⇒θP(w)={f(w)ifwsatisfieseq.0∞otherwiseminwθp(w)=minwmaxα,β:αi>=0L(w,α,β)θD(w)=minwL(w,α,β)maxα,β:αi>=0θD(w)=maxα,β:αi>=0minwL(w,α,β)d∗=maxα,β:αi>=0θD(w)=maxα,β:αi>=0minwL(w,α,β)<=minwmaxα,β:αi>=0L(w,α,β)=p∗⇒⎧⎩⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪∂∂wiL(w∗,α∗,β∗)=0,i=1,...,m∂∂βiL(w∗,α∗,β∗)=0,i=1,...,lαi∗gi(w∗)=0,i=1,...,kgi(w∗)<=0,i=1,...,kαi∗>=0,i=1,...,k