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日期：2023-01-16 10:13

Final Quiz Optimization Theory and Algorithms

1. Suppose S ⊂ Rn is convex. f is differential on S. Please show that f is convex if

and only if

f(y)≥f(x)+∇f(x)T(y−x), ∀x, y∈S.

2. The Cauchy–Schwarz inequality states that for any vectors u and v, we have

|uT v|2 ≤ (uT u)(vT v)

with equality only when u and v are parallel. When B is positive definite, show the

following inequality

∥g∥4

(gT Bg)(gT B−1g) ≤ 1.

3. The vector p∗ is a global solution of the trust-region problem minm(p)=f+gTp+1pTBp, s.t.∥p∥≤∆

conditions are satisfied:

(B+λI)p∗ =−g,

λ(∆ − ∥p∗∥ = 0),

(B + λI) is positive semidefinite.

Suppose pLM is the global solution of the following specific trust region subproblem in the Levenberg-Marquard method

min∥Jp+r∥2, s.t.∥p∥≤∆ p∈Rn

Please use the result above to give the necessary and sufficient conditions for pLM .

4. Considering the following optimization problem minf(x)=x21 +x1x2;

 p∈Rn 2

if and only if p∗ is feasible and there is a scalar λ ≥ 0 such that the following

c1(x)=1−x21 −x2 ≥0, 49

s.t.

c2(x)=x1 −x2 −1≥0.

(a) solve the corresponding KKT system to obtain a local solution x∗ (b) verify the equivlence of feasible set and tangent cone at x∗.

5. Use the active method to compute

min x21 +2x2 −2x1 −6x2 −2x1x2

s.t. 21x1 +x2 ≤1, −x1 +2x2 ≤2, x1,x2 ≥0.

Choose three initial starting points : one in the interior of the feasible region ( 14 , 13 ), one at a vertex (2, 0), and one at a non-vertex point on the boundary of the feasible region (1, 12 ).

Dec 29, 2022 1

Final Quiz Optimization Theory and Algorithms

6. Using (i)quadratic penalty method, (ii) classical l1 penalty method and (iii) argumented Lagrangian method to solve the above problem. Report numerical results with different methods and Compare these three methods. (Try various non- constrains optimization methods.)

Dec 29, 2022 2