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3 n x=(x 1,,x n ) x 1 =(x 1 x 1 2 x 2 11,,x n1 ), x 2 =(x 12,,x n2 ) x 1 x 2 iff (if and only if) x i1 x i2, i=1,,n 1 x 1 x 2 iff (if and only if) x 1 2 i1 > x i2, i=1,,n 1 2 x 22 = x 2 1 x 2 2 x 2 1 x 2 x 1 x 2 1 x 2 x 2 2 x 2 x 1 x 2 1 x 1 1 x 1 x 2 x x 1

4 n n d(x, x ) = (x - x )(x - x ) x R, x 1 x 2 d ( x,x ) = ( x -x )( x -x ) = x x x R 2, d( x,x ) = ( x -x, x -x )( x -x, x -x ) = ( x - x ) + ( x - x ) x 2 x 1

5 (ε- ε-ball) x 0 R n, ε>0 B ε (x 0 )={x d(x, x 0 ) < ε } B ε (x 0 )={x d(x, x 0 ) ε} n=2 x 0 ε x x 0 ε B ε (x 0 ) B * ε(x 0 )

6 A1.5: n S R n x S, ε>0 B B ε (x) S x 2 x 2 x 2 x 1 x 1 x 1 6

7 A1.5: n S R n x S, ε>0 B ε (x) S A1.6 S S c S 7

8 (M.F.3,P943) S S S,, {x 1,x 2,,x n } 1 x n S n=1,2 2 limx n = x n x S x 2 x 2 x 2 x 1 x 1 x 1 8

9 (Bounded Sets) n S ε-, ε>0, x R n, S B ε (x) (compact set) x 2 x S ε x 1

10 x, x x t =tx+(1-t)x t [0,1 x k(x, x t )=(x 2 -x t 2)/(x 1 -x t 1) x 2 x t = [(1-t)(x 2 -x 2 )]/[(1-t)(x 1 -x 1 )] x t 2=tx 2 +(1-t)x 2 x 2 tx 2 ( t)x 2 =(x 2 -x 2 )/(x 1 -x 1 ) x 2 x x 1 =k(x, x ) =k(x, x t ) x 1 x t 1=tx 1 +(1-t)x 1

11 A. S x, x S S x, x S, x t =tx+(1-t)x t [0,1] x x x t S x t x S x

12 B. A1.16 D R f(x) D R A1.10 S R n f: S D x*, x# S, f(x*) f(x) f(x#) x S 12

13 B. L(y)={x f(x)=y, x D} S(y)={x { f(x) y, ) x D} I(y)={x f(x) y, x D} n=1 1 D=[-4,4] L(2) {-3, 3 3.3} 3} y=2 S(2) =[-4,-3] [3.3, 4] I(2) =[-3, 333] 3.3] y=f(x)

14 B. A1.22: f(x) D R D x 1,x 2, D, f(x t ) t f(x 1 )+(1-t)f(x 2 )}, t [0,1]. f(x t ) t f(x 1 )+(1-t)f(x 2 ) x 1 x t x 2 14

15 B. A2.4*: f(x) f(x) D R For all x D, f (x) f (x ) + f (x )(x-x ) z D z H (x) z 0 z D f(x) f(x) f (x) = (,..., ) x x (gradient) f H (x) = f m1 f mn 1 n 11 f1n n 2 f ij f (x) = x x i j

16 B. A1.24: f(x) D R D x 1,x 2, D, f(x t ) min {f(x 1 ), f(x 2 )}, t [0,1]. f(x t ) f(x 1 )=min{f(x 2 ),f(x 2 ) x 1 x t x 2 A

17 B. x 1 x t x 2 1 t x 2 x 1 x t 2 x 1 x t x 2 17

18 1.14 f(x) D R ) S(y) S(y)={x f(x) y, x D} y 1 S(y 1 ) 1 y 2 S(y 2 ) y 3 y 4 S(y 3 ) S(y 4 ) S(y 4 )

19 1.14 f(x) D R S(y) S(y) f(x) x 1, x 2 D f(x 1 ) f(x 2 ) = y 2 =min {f(x 1 ), f(x 2 )} x 1,x 2 S(y 2 ) S(y 2 ) x t S(y 2 ) t [0 [0,1] f(x t ) f(x 2 ) =min {f(x 1 ), f(x 2 )} f(x t ) min {f(x 1 ), f(x 2 )} f(x)

20 1.14 f(x) D R S(y) f(x) S(y) x 1, x 2 S(y) y R, f(x 1 ) y f(x 2 ) y f(x) f(x t ) min {f(x 1 ), f(x 2 )} f(x t ) y x t S(y) )

21 B. y=f(x x 2 = x 0.5 x 0.5 1, 2) 1 2 S (y 1 )={(x 1, x 2 ) x x y 1 } L(y )={(x 1, x 2 ) x 1 x 2 =y 1 } x 2 =(y 1 ) 2 /x 1 x 2 / x 1 =-(y 1 ) 2 /(x 1 ) 2 <0 2 x 2( 2 / x 1 = (y 1 ) 2 /(x 1 ) 3 >0 f(x 1, x 2 ) x 1 x 2 S(y 1 ) L(y 1 ) 21

22 B. M.C.3-4: C34:(MGW) f(x) iff f(x) x -x f(x ) f(x) 1 z H(x) z 0 f(x) z=0 2

23 B. A1.25: f(x) D R D x 1,x 2, D, f(x t ) > min {f(x 1 ), f(x 2 )}, t (0,1) S(y 1 ) S(y 1 ) )={(x 1, x 2 ) x 1 +x 2 =y 1 } L(y 1 ) 23

24 B. A1.25: f(x) D R D x 1,x 2, D, f(x t ) max {f(x 1 ), f(x 2 )}, t (0,1) A1.25: f(x) D R D D x 1,x 2, D, f(x t ) < max {f(x 1 ), f(x 2 )}, t (0,1) 24

25 C *. - Non-binding 25

26 C.1 P1 Max x D y= f(x,a) f(x*) x i =0 i=1,2,..,n H=D 2 f(x*) dx T H(x*) dx 0, dx=(dx 1, dx 2,, dx n ) (-1) i H i >0 i=1,2,..,n P2 Min x D f(x,a) f(x*) x i =0 i=1,2,..,n D 2 f(x*) ) H i >0 i=1,2,..,n 26

27 C.1 n=1 P1 Max x D y= f(x,a) f(x*) f(x*) x =0 dy = f dx =0 dx B (x*,δ) 2 f(x*) 0 d 2 y = f dx 2 0 dx B (x*,δ) 27

28 C.1 n=2 P1 Max x D y= f(x,a) f(x*) f(x*) x i =0 i=1,22 dy = f 1 dx 1 + f 2 dx 2 =0 dx= (dx 1, dx 2 ) H=D 2 f(x*) d 2 y = (f 11 dx 1 + f 12 dx 2 )dx 1 +(f 21 dx 1 + f 22 dx 2 )dx 1 2 f11 f12 dx1 d y = [ dx1, dx2] 0 dx f21 f22 2 f(x,a) x* 28

29 C.1 f(x) x* f(x) x* f(x) 29

30 C.2 P2 Max x f(x,a) s.t: x B ={x: { g j (x,a)=0 (,) i=1,2,..,m},,, } g j (x,a) = 0 j=1,2,..,m M.K.4 (MWG-P962) P2 B 30

31 C.2 P2 Max x f(x,a) s.t: x B ={x: { g j (x,a)=0 (,) i=1,2,..,m},,, } g j (x,a) = 0 j=1,2,..,m L= f(x,a) +λ 1 g 1 (x,a)+ +λ m g m (x,a) L x i 0, i=1,2,..,n L λ j 0, j=1,2,,m 31

32 C.2 * D 2 L(x) dx T D 2 L(x) ()dx 0, dx Dg j (x)dx=0 ()d j=1,,m H m+i (-1) H m + i< 0, i=1,2,...,n,, n 2 m 1: H < 0, H > 0 H L L L g g 2 0 λ λ x1 λ x2 x1 x L L L L L g = = x1 λ x1 x1 x1 x2 x1 x1 x1 x1 x L L L 2 2 g L L x2 λ x2 x1 x2 x x 2 2 x2 x1 x2 x 2 32

33 C.2 * D 2 L(x) dx T D 2 L(x) dx 0, dx Dg j (x)dx=0 j=1,,mm H H m + i< 0, i=1,2,...,n 33

34 C.3 P3 Max x f(x,a) s.t: x B ={x: g j (x,a) 0 i=1,2,..,m} g j (x,a) 0 j=1,2,..,m P4 Min x f(x,a) st: s.t: x B ={x: g i (x,a) 0 i=1,2,..,m} 2 g j (x,a) 0 j=1,2,..,m M.K.4 (MWG-P962) P3-4 B 34

35 C.3 g(x,a)=k-h(x) 0 Binding constraint: k = k 1 h(x*)=k( ) 1 Non-binding constraint: k=k 2 h(x*)<k( ) 2 h(x,a) h(x,a) f(x,a) k 2 f(x,a) k 1 x* x max x*=x max 35

36 C.3 g(x,a)=k-h(x) 0 1 f (x,a)>0 g (x*,a) a)=0 binding constraint g j (x,a) 0 g j (x,a)=0 k 2 f(x,a) h(x,a) k 1 x* x* 36

37 C.3* Kuhn-Tucker u(x) g i (x,a) L= u(x) +λ 1 g 1 (x,a)+ +λ k g k (x,a) Kuhn-Tucker L x 0 x * i, i ( L x i i) )=0, x i i 0,, i=1,2,..,n,,, L λ j 0, λ * j ( L λ j )=0, λ j 0, j=1,2,,k x * 0 L x i 0, i=1,2,..,n L λ j 0 j=1,2,,k D 2 L(x) z T D 2 L(x) z 0, z Dg(x)z=0 37

38 C.3* Kuhn-Tucker u(x) g i (x,a) L= u(x) +λ 1 g 1 (x,a)+ +λ k g k (x,a) Kuhn-Tucker L x 0 x * i, i ( L x i i) )=0, x i i 0,, i=1,2,..,n,,, L λ j 0, λ * j ( L λ j )=0, λ j 0, j=1,2,,k x * 0 L x i 0, i=1,2,..,n L λ j 0 j=1,2,,k D 2 L(x) z T D 2 L(x) z 0, z Dg(x)z=0 38

39 D.1* x * 0 L(x, λ,a) x i 0, i=1,2,..,n (1) L(x, λ,a) λ j 0 j=1,2,,m (2) j (1-2) a k 2 λ 1/ ak L / λ1 a k 2 DL(x, λ,a)*... =... 2 x / / n a k L xn a k x / a = i k 2 DL (x,,a) i λ 2 DL (x, λ,a) Ax=b x i = A i / A 39

40 D.2 (maximum-value function) M(a) max x f(x,a) s.t. g(x,a)=0, x 0. M(a)= f(x*(a),a) = L(x*(a), λ*(a),, a) = f(x*(a),a)+λ*(a)g(x*(a),a) M(a)/ a j [L(x*(a), λ*(a), a)]/ a j =0 [ ] = Σ i=1,,n L(x*(a) (a), λ*(a) (a), a)/ x i x i *(a)/ a j + L(x*(a), λ*(a), a)/ λ λ *(a)/ a j + L(x,λ, a)/ a j (x*,λ*) = L(x,λ, (,λ, a)/ a j (x*,λ*)

41 D.2 A2.21: f g a a x(a) 0 x(a) 0 x(a) a M(a) a λ(a) x(a) M( a) f (x,a) g(x,a) L(a) = λ = a a a a j j j j x(a), λ(a)

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州盟 ) 县 ( ) 组织机构代码企业详细名称武汉东湖开发武汉新芯集成电路制造有限公司黄石市下陆大冶有色自强服务有限责任公司黄石市下陆附件 2016 年国家重点监控企业名单一废水国家重点监控企业名单 ( 共 2660 家 ) 湖北省 (132 家 ) 市 ( 地州盟 ) 县 ( ) 组织机构代码企业详细名称汉阳 616400141 百威 ( 武汉 ) 国际啤酒有限公司青山 177681913 武汉钢铁 ( 集团 ) 公