凸函数2

W. S. Tang 2Â༠ê†�^ ¼ê 2Âà¼ê  Ð†�^¼ ê 7KêÆÄ: U9ŒÆXÚó§ïĤ /�) 2012.6.25 W. S. Tang W. S. Tang 2Â༠ê†�^ ¼ê 2Âà¼ê  Ð†�^¼ ê 2Âà¼ê . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ½Â1.1: ([à¼ê)½Â3à8X ⊂ Rnþff¢Ł¼êf¡´ [àff,XJé?¿x1 ∈ X , x2 ∈ XÚ�q1,q2 ∈ [0, 1],k f (q1x1 + q2x2) ≤ max(f (x1), f (x2)). XJ−f´[àff¼ê,Kf¡´[]¼ê.�d/k: ½Â1.2: ([]¼ê)½Â3à8X ⊂ Rnþff¢Ł¼êf¡´ []ff,XJé?¿x1 ∈ X , x2 ∈ XÚ�q1,q2 ∈ [0, 1],k min(f (x1), f (x2)) ≤ f (q1x1 + q2x2). w,,z‡¢Łà(])¼êÑ´[à(])ff,�‡ƒØ,. W. S. Tang W. S. Tang 2Â༠ê†�^ ¼ê 2Âà¼ê  Ð†�^¼ ê 'u[à¼êke�(Ø. ½n 1.1 �f´½Â3à8X ⊂ Rnþff¢Ł¼ê,Kéz‡α ∈ R, fffY ²8S(f , α) = {x ∈ X : f (x) ≤ α}Ñ´à8ff¿©7‡^‡ f´[à¼ê. y²:�éz‡α ∈ R, S(f , α)Ñ´à8.-x1 ∈ X , x2 ∈ X , α¯ = max(f (x1), f (x2)),K x1 ∈ S(f , α¯), x2 ∈ S(f , α¯), ϏS(f , α)´àff,¤±é?¿�qk (q1x1 + q2x2) ∈ S(f , α¯). Ïd, f (q1x1 + q2x2) ≤ α¯ = max(f (x1), f (x2)). W. S. Tang W. S. Tang 2Â༠ê†�^ ¼ê 2Âà¼ê  Ð†�^¼ ê ‡ƒ,-S(f , α)´fff?¿Y²8.ex1 ∈ S(f , α), x2 ∈ S(f , α),K f (x1) ≤ α, f (x2) ≤ α. Ϗf´[àff,k f (q1x1 + q2x2) ≤ q1f (x1) + q2f (x2) ≤ α, Ï q1x1 + q2x2 ∈ S(f , α). W. S. Tang W. S. Tang 2Â༠ê†�^ ¼ê 2Âà¼ê  Ð†�^¼ ê 3�gŒ‡¼êffœ/, Arrow!Enthoven‰Ñ ^¼êff \>Hesse ff1�ªL«ff3RnffšK–þ[à5ff¿© 7‡^‡.ù ^‡�5dFerland*¿�˜„à8þff¼ê. ˜‡�gŒ‡¼êf3˜:x ∈ Rn?ff\>Hesse ½Â H (f , x) =  0 ∂f∂x1 · · · ∂f ∂xn ∂f ∂x1 ∂2f ∂x1∂x1 · · · ∂2f∂x1∂xn ... ... . . . ... ∂f ∂xn ∂2f ∂xn∂x1 · · · ∂2f∂xn∂xn  . W. S. Tang W. S. Tang 2Â༠ê†�^ ¼ê 2Âà¼ê  Ð†�^¼ ê ¼êfff\>Hesse ffkffi^SÌfªDk(f , x)½Â Dk(f , x) = det  0 ∂f∂x1 · · · ∂f ∂xk ∂f ∂x1 ∂2f ∂x1∂x1 · · · ∂2f∂x1∂xk ... ... . . . ... ∂f ∂xk ∂2f ∂xk∂x1 · · · ∂2f∂xk∂xk  , k = 1, 2, · · · ,n. W. S. Tang W. S. Tang 2Â༠ê†�^ ¼ê 2Âà¼ê  Ð†�^¼ ê XJ8ÜX ⊂ Rnäkš˜ffSÜ,¡§¢%ff.¼êf3 ¢%à8X ⊂ Rnþ[àff7‡^‡:éz‡x ∈ Xk Dk(f , x) ≤ 0, k = 1, 2, · · · ,n. aq/,ef3¢%à8X ⊂ Rnþ´[]ff,Kéz ‡x ∈ Xk (−1)kDk(f , x) ≥ 0, k = 1, 2, · · · ,n. W. S. Tang W. S. Tang 2Â༠ê†�^ ¼ê 2Âà¼ê  Ð†�^¼ ê ‡ƒ,Ä3¹à8X ⊂ Rnff˜‡m8þff�gëYŒ‡ ff¢Ł¼êf . eéz‡x ∈ Xk Dk(f , x) < 0, k = 1, 2, · · · ,n, Kf3Xþ´[àff. eéz‡x ∈ Xk (−1)kDk(f , x) > 0, k = 1, 2, · · · ,n, Kf3Xþ´[]ff. W. S. Tang W. S. Tang 2Â༠ê†�^ ¼ê 2Âà¼ê  Ð†�^¼ ê ½Â1.3: (r[à¼ê)½Â3à8X ⊂ Rnþff¢Ł¼êf¡ r[àff,´é?¿x1 ∈ X , x2 ∈ X , x1 6= x2Ú?¿ �q1 > 0, q2 > 0, q1 + q2 = 1,k f (q1x1 + q2x2) < max(f (x1), f (x2)). ½Â1.4: (î‚[à¼ê)½Â3à8X ⊂ Rnþff¢Ł¼ êf¡î‚[àff,´é?¿x1 ∈ X , x2 ∈ X , f (x1) 6= f (x2)Ú?¿�q1 > 0, q2 > 0, q1 + q2 = 1,k f (q1x1 + q2x2) < max(f (x1), f (x2)). aq/Œ‰Ñr[]¼êÚî‚[]¼êff½Â. W. S. Tang W. S. Tang 2Â༠ê†�^ ¼ê 2Âà¼ê  Ð†�^¼ ê éuù«r[àÚî‚[àff¼ê,dþã½Âá=�Ñe ã(J. ½n 1.2 ½Â3à8X ⊂ Rnþffr[à¼êf ,–õ3˜:ˆ�§3X þ ff4�Ł. ½n 1.3 �f´½Â3à8X ⊂ Rnþffî‚[à¼ê, x∗ ∈ X´fff˜‡ ÛÜ4�Ł:,Kx∗´f3X þff�N4�Ł:. W. S. Tang W. S. Tang 2Â༠ê†�^ ¼ê 2Âà¼ê  Ð†�^¼ ê �f : Rn → R1,î‚[]¼ê,¡Er = {x�Rn | f (x) = r} f ff�Ł¡. ½n 1.4 �S ⊂ Rnà8,ëY¼êf : S → R1´î‚[]¼êff, �…=�éf ff?¿�Ł¡Er ,�x1, x2 ∈ Er …x1 6= x2ž,k f (x1) < f (z),∀z ∈ (x1, x2), Ù¥(x1, x2)L«ë�x1†x2ü:ffm‚ã,= z ∈ (x1, x2)⇔ z = λx1 + (1− λ)x2, 0 < λ < 1. W. S. Tang W. S. Tang 2Â༠ê†�^ ¼ê 2Âà¼ê  Ð†�^¼ ê y²: Iy²�^‡÷vž, f7î‚[]ff. b�(ØØý,=3˜éx, y ∈ S, f (x) < f (y),3m‚ ã(x, y)þk˜:z¦� f (z) 6 f (x) < f (y), dfffëY5Œ:3w ∈ [z, y),¦�f (w) = f (x),=w†x3 Ә�Ł¡þ,…w 6= x, z3‚ã[x,w)þ,b�^‡k f (z) > f (x), gñ. ¤±, fî‚[]ff. W. S. Tang W. S. Tang 2Â༠ê†�^ ¼ê 2Âà¼ê  Ð†�^¼ ê  Ð†�^¼ê . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ½Â2.1: ½Â3ãÀ˜mRnþff��'X�¡ Ð'X, ´§÷vXe^‡: (1)‡fi5:é?¿x ∈ Rn, x � x; (2)D45:�x � y, y � z,Kkx � z; (3)��5:é?¿x, y ∈ Rn,Kx � y½öy � x. ùp“x � y”ÖŁ“y–�Úx˜�Д. XJx � y…y � x,KÖŁ“x†yÃ�O”,¿PŁx ∼ y. XJx � y�x ∼ yؤá,KÖŁ“y'xД,¿PŁx ≺ y. W. S. Tang W. S. Tang 2Â༠ê†�^ ¼ê 2Âà¼ê  Ð†�^¼ ê ½Â¥ffn‡^‡qk²wff†*)º,AO´��5L²,  Ð'X(½ ˜mRnfffiS'X,=e�œ/: x ≺ y, x ∼ y, y ≺ x 7L …Uk˜«¤á. æ^8ÜØ.^ff{,éz‡x¯ ∈ RnŒ±½Â�da Ex¯ = {x ∈ Rn | x ∼ x¯}, KRnŒ�y©¤p؃�ff�da.  Ð'X´˜«S'X,J±ë†E,ffêÆín. k �ff¦^êÆóä,<‚Ú? y Ð'Xff¼ê. W. S. Tang W. S. Tang 2Â༠ê†�^ ¼ê 2Âà¼ê  Ð†�^¼ ê ½Â2.2: ��´½Â3˜mRnþff Ð'X,¡¢Ł¼ êu : Rn → R´Ly Ð'X�ff�^¼ê,´x � y�…= �u(x) 6 u(y). w,, x ∼ y�…=�u(x) = u(y); x ≺ yK�d uu(x) < u(y). éu,(½ff Ð�,Ly§ff�^¼ê¿Ø´˜ff.�¼ êρ : R → R´î‚4Off,XJu(x)´Ly Ð�ff�^¼ê, Kρ(u(x))½,. é�^¼êu(x)ŒÚ?�Ł¡ Eq = {x ∈ Rn | u(x) = q}, ¡§Ã�O­¡. ��q = u(x¯)ž,Ã�O­¡EqÒ´c¡¤ãff�daEx¯ . W. S. Tang W. S. Tang 2Â༠ê†�^ ¼ê 2Âà¼ê  Ð†�^¼ ê   �^¼êffëY5,·‚Ú?Xe½Â. ½Â2.3: (ëY5)½Â3˜mRnþff Ð'X�¡ëYff, XJé?Ûx ∈ Rn,8Ü U+(x) = {y ∈ Rn | x � y} Ú U−(x) = {y ∈ Rn | y � x} ´Rn¥ff48. ½�d/: {x | x < y}Ú{x | y < x}m8. ½Â2.4: (üN5)��´½Â3˜mRnþff Ð'X.¡  Ð'X�´üNff,XJ�x 6 y…x 6= yž,okx ≺ y. W. S. Tang W. S. Tang 2Â༠ê†�^ ¼ê 2Âà¼ê  Ð†�^¼ ê ëY�^¼ê3½n ½n 2.1 �Rn+þff Ð'X�üNëY,K˜½3˜‡ëYff�^¼ êu : Rn+ → R1,¦�x � y�duu(x) 6 u(y),∀x, y ∈ Rn+. y²:é∀x ∈ Rn+,�E¼ê u(x) = max{r | re � x}, Ù¥e = (1, 1, · · · , 1)T ∈ Rn+, r¢ê. 1)e¡ky²e��ª u(x)e ∼ x, ∀x ∈ Rn+ ¤á. W. S. Tang W. S. Tang 2Â༠ê†�^ ¼ê 2Âà¼ê  Ð†�^¼ ê X`Ø,,‡�u(x)e ≺ x, ∵ {y | y ≺ x}m8, …u(x)e ∈ {y | y ≺ x}, ∴ ∃δ > 0,¦� B(u(x)e, δ) ⊂ {y | y ≺ x}, Ù¥m¥B(u(x)e, δ) = {z ∈ Rn+ :‖ z − u(x)e ‖< δ}. 3m¥B(u(x)e, δ)¥�˜AÏ�ƒ u(x)e + 1 2 δe, ∴ u(x)e + 12δe ∈ {y | y ≺ x},= (u(x) + 1 2 δ)e ∈ {y | y ≺ x}, Ïd(u(x) + 12δ)e � x,w,ku(x) + 12δ > u(x),ù†u(x)ff½ Âgñ. ∴ u(x)e ∼ x. W. S. Tang W. S. Tang 2Â༠ê†�^ ¼ê 2Âà¼ê  Ð†�^¼ ê 2)e¡y²u(x) 6 u(y)⇐⇒ x � y. i)�u(x) 6 u(y)ž,kx � y.X`Ø,,‡�y ≺ x,Kdu½Â Œ u(y)e ∼ y,u(x)e ∼ x, Ïdk y ≺ x ∼ u(x)e � u(y)e ∼ y, gñ. ∴ x � y. ii)�x � yž,ku(x) 6 u(y).‡�u(x) > u(y),düN5Œ u(y)e ≺ u(x)e, 2du(y)e ∼ y,u(x)e ∼ x�y ≺ x,gñ. ∴ u(x) 6 u(y). W. S. Tang W. S. Tang 2Â༠ê†�^ ¼ê 2Âà¼ê  Ð†�^¼ ê 3)e¡y²uffëY5. ¼êffëY5⇔¼êm8ff�”m8⇔¼ê48ff�” 48. éR1¥ff48[a,b],k u−1([a,b]) = {x | u(x) ∈ [a,b]} = ⋂ a≤r≤b {x | re � x}, d ÐffëY5,Œéff½ffr , {x | re � x}48, ∴ ⋂ a≤r≤b {x | re � x} E48. ∴¼êu´ëYff. W. S. Tang W. S. Tang 2Â༠ê†�^ ¼ê 2Âà¼ê  Ð†�^¼ ê ù‡½nffnØdŁ,Ò3u²( |^�^¼êLy Ð 'XffŒU5,�vk‰ÑÏé�^¼êffk�{.Œ±Ž„, ù´˜‡é(Jff¯œ.�^¼ê̇´^5½5ffxž¤1 Ú©Û²L5Æ,�^¼ê©ÛóäïIJLƋm Ï �. ob½¤Øff Ð'X´ëYff,l ƒAff�^¼ê´ ëYff. W. S. Tang W. S. Tang 2Â༠ê†�^ ¼ê 2Âà¼ê  Ð†�^¼ ê ½Â2.5: ( Ð'Xffà5) 1)¡ Ð'X�´àff,´é?¿x ∈ Rn,8 Ü{y ∈ Rn | x � y}´à8; 2)¡ Ð'X�´î‚àff,´é? ¿x1, x2 ∈ Rn, x1 6= x2, x1 � x2,Kk x1 ≺ λx1 + (1− λ)x2, Ù¥0 < λ < 1. W. S. Tang W. S. Tang 2Â༠ê†�^ ¼ê 2Âà¼ê  Ð†�^¼ ê ƒA/,'u�A¼êkXe(Ø. ·K2.1 �u(x)´Ly Ð'X�ff�^¼ê,Kk 1) Ð'X�´üNff�…=�u(x)´î‚4O¼ê,= �x < yž,ku(x) < u(y); 2) Ð'X�´àff�…=�u(x)´[]ff¼ê; 3) Ð'X�´î‚àff�…=�u(x)´î‚[]ff¼ê. ù‡·Kffy²,3‰Öö��¤. W. S. Tang 广义凸函数与效用函数 广义凸函数 偏好与效用函数