凸分析的基本概念

1.......1.1.1.A.(1.1.1).1.1.1nC®x+(1¡®)y2C;8x;y2C;8®2[0;1]...1.1.1.1.1.1(a)fCiji2Ig\i2ICi.21.1.1..(b)C1C2C1+C2.(c)C¸¸C.¸1,¸2(¸1+¸2)C=¸1C+¸2C:(d)(closure)(interior).(e)..(a)\i2ICixy.CixyCi,.(b)C1+C2x1+x2y1+y2x1;y12C1x2;y22C2.®2[0;1]®(x1+x2)+(1¡®)(y1+y2)=¡®x1+(1¡®)y1¢+¡®x2+(1¡®)y2¢:C1C2C1C2C1+C2.C1+C2.(c).(e)(b).(d)CCxy.Cfxkg½Cfykg½C13xyxk!xyk!y.®2[0;1]®x+(1¡®)y©®xk+(1¡®)ykªCC.®x+(1¡®)yCC.CxyxyrC.®2[0;1]®x+(1¡®)yr.Cx+zy+z®(x+z)+(1¡®)(y+z)kzkr.C®(x+z)+(1¡®)(y+z)C.C®x+(1¡®)yC.¤.(hyperplane)fxja0x=bgab.(halfspace)fxja0x6bgab..(polyhedral)fxja0jx6bj;j=1;¢¢¢;rg;a1;¢¢¢;arb1;¢¢¢;brn.[1.1.1(a)].C(cone)x2C¸0¸x2C.(1.1.2).(polyhedralcone)C=fxja0jx60;j=1;¢¢¢;rg;a1;¢¢¢;arn..1.1.1(1.1.3).1.1.2Cnf:C7!(convex41.1.2.(a)(b)(c).(a).(b).1.1.3f:C7!.®f(x)+(1¡®)f(y)f¡®x+(1¡®)y¢®[0;1].function)f¡®x+(1¡®)y¢6®f(x)+(1¡®)f(y);8x;y2C;8®2[0;1](1.1)Cf:C7!...f:C7!(strictlyconvex)(1.1)(1.1)x6=yx;y2C®2(0;1).f:C7!(concave)(¡f)C.(a±nefunction)f(x)=a0x+ba2nb2.k¢k.x;y2n®2[0;1]15k®x+(1¡®)yk6k®xk+k(1¡®)yk=®kxk+(1¡®)kyk;k¢k.f:C7!°fx2Cjf(x)6°gfx2Cjf(x)°gf(levelsets).f.x;y2Cf(x)6°f(y)6°C®2[0;1]®x+(1¡®)y2C.ff¡®x+(1¡®)y¢6®f(x)+(1¡®)f(y)6°;fx2Cjf(x)6°g.ffx2Cjf(x)°g.f(x)=pjxjf.n()..f(x)=supi2Ifi(x);Ifif1(1.6).f[f:(0;1)7!f(x)=1=x].fCnf.(extendedreal-valued)n¡11.(epigraph).6X½nf:X7![¡1;1]n+1epi(f)=©(x;w)jx2X;w2;f(x)6wª:f(e®ectivedomain)dom(f)=©x2Xjf(x)1ª(1.1.4).dom(f)=©xjw2(x;w)2epi(f)ª;dom(f)epi(f)n(x).f.fnx=2Xf(x)=1.1.1.4.f1[epi(f)]¡1[epi(f)].x2Xf(x)1x2Xf(x)¡1f(proper)f(improper).f.f¡11®f(x)+(1¡®)f(y)¡1+1(f).171.1.3Cnf:C7![¡1;1]epi(f)n+1.1.1.3fdom(f)©x2Cjf(x)6°ª©x2Cjf(x)°ª°.ff(x)1f(x)¡1f¡®x+(1¡®)y¢6®f(x)+(1¡®)f(y);8x;y2C;8®2[0;1];(1.2)1.1.31.1.2.().X½n(indicatorfunction)±:n7!(¡1;1]±(xjX)=(0,x2X,1,..f:C7!(¡1;1](strictlyconvex)(1.2)x6=yx;y2dom(f)®2(0;1).f:C7:C7![¡1;1]C.CC..1.1.4CXnnCXC½X.f:X7![¡1;1]C(convexoverC)fC~f:C7![¡1;1]x2C~f~f(x)=f(x)..1.,8.Rockafellar[Roc70]..1.1.2f:X7![¡1;1]f..fx2Xf(x)6liminfk!1f(xk)xk!xfxkg½X.f(lowersemicontinuous)Xx.f(uppersemicountinous)¡f.[A.2.4(c)]..1.1.5.1.1.5.fxjf(x)6°gepi(f)\f(x;°)jx2ng.epi(f).1.1.2f:n7V°=©xjf(x)6°ª°.(ii)f.(iii)epi(f).19f(x)=1x.f(x)1x2n.epi(f)f.(i)(ii).V°°.f(x)liminfk!1f(xk)xxfxkg°f(x)°liminfk!1f(xk).fxkgKf(xk)6°k2K.fxkgK½V°.V°xV°f(x)6°.(ii)(iii).fn(x;w)©(xk;wk)ª½epi(f).f(xk)6wkk!1fxf(x)6liminfk!1f(xk)6w(x;w)2epi(f)epi(f).(iii)(i).epi(f)fxkgx°V°.(xk;°)2epi(f)k(xk;°)!(x;°)epi(f)(x;°)2epi(f).xV°..¤..f(x)=(0;x2(0;1);1;x=2(0;1):f:7!(¡1;1](0;1).f:X7![¡1;1]dom(f)x2dom(f)f..1.1.2(ii)(iii).101.1.3f:X7![¡1;1].dom(f)fx2dom(f)f.XX().fXfX(x)=(f(x);x2X1;f:n7!fXX.f(x)=(¡1;x2dom(f);1;x=2dom(f):f:n7![¡1;1]xf(x).xf(x)=¡1(ff1).fxk=k¡1kx+1kx;8k=1;2;¢¢¢f(xk)=¡1xk!x.ff(x)=¡1..1.1.3.(a²inefunc-tions)(,norms).(poly-hedralfunction)(polyhedralset)..(a).(b).