Section 8
本周的第一个diagram。
Test 8
Review for last week
f Hessian matrix,
Quadratic form,
Leading principle minors) id1[Concave programming]-->id2(Quasi concavity:
f borderd Hessian,
constrained quadratic form,
LPM) id3-->id7[Concave: negative definite] id3-->id8[Convex: positive definite] id2-->id5[Quasi concave:
negative definite] id2-->id6[Quasi convex:
positive definite] id5--> id9[Unique and global Solution] id6--> id9[Unique and global Solution] style id1 fill:#F8E80A,stroke:#333,stroke-width:4px; style id2 fill:#609256,stroke:#333,stroke-width:4px
Outline for this week
Concavity, Quadratic form , Hessian
Necessary FOC, SOC Sufficient] SOC-->id4[40: constrained:
Hessian of Lagrangean,
constrained Quadratic form,
Lagragean Bordered Hessian,
] id3-->id5[Max] id3-->id6[Min] id4-->Max id4-->Min id1-->id7(39: unconstrained:
FOC, Total differenial of I,
Cramer rule ) id1-->id8(42: constrained:
KKT FOC, Total differenial of I,
not testable yet)
38. SOC - unconstrainted optimization
For all of these exercises, solve for the optimum point using the FOC’s and prove that the optimum is a maximum or minimum using SOC.
Hint: Just as sufficient SOC for maximum is the same as the test for strict concavity, the sufficient SOC for minimum is the same as the test for strict convexity, i.e.,
Concavity
Hessian
for example:
Since 1 H = −6 < 0 and 2 H = 47 0 , x is a maximum.
Since 1 H = 4 0 and 2 H =12 0 , x is a minimum.
Max or min
39. Solution static analysis - unconstrainted optimization
就是要知道，最优解的\(\bar x\) 们，是怎么根据参数parameters \( \theta \) 们来变化的。
比如说，如果我妈给我多一块钱的预算，那我买苹果香蕉会不会变。
-
FOC
-
Totla differentiate of I implicite function of foc.
- SOC.
40. SOC with constrainted optimization
p109, the crucial point is the relative curvature of the two contours through the optimum \( \bar x \) the contour of F should be more convex than that of G.
p110, the second - order sufficient condition for \(\bar x \) to be a local optimum is that $\frac{ d \bar x_1}{ d \alpha_1 }$ along the F contour should be greater than that along the G contour.
-
KKT FOC
-
SOC for F
-
SOC for G
最后，发现还是 Lagrangean 来救场。 Quadratic form of Lagrangean。
加上，预算不变，就是constraints 不变。
p111, The second-order sufficient condition for local maximum then impose restrictions on the signs of their determinants. The signs are required to alternate, the first one (that formed by the last 2m rows and the corresponding columns ) having the sigh of $(-1)^m$.
41. Lagrangean Bordered Hessian.
和上周的C matrix 类似。 从右下角往左上角，计算C1, C2, C3. 一般，考试也就是C2了。
42. Solution statics – constraited optimization
- KKT FOC
here, $\bar x$ , $\theta$, and $\bar \lambda$ are free variables, which are needed to calculated.
Furthermore,
${d{\bar x}} $, ${d{\bar \lambda}}$ are vectors. ${d{\theta}}$ is a scalar.
Totla differentiate of I implicite function of foc.
- CS : K implicite function for CS.
Taylor expansion.
- System of equations
SOC.
Crammer’s rule to solve