上海交大研究生《最优化理论基础》试卷A卷.pdf

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ˇ˘˜)§ 5‘zn˜:6`£ A ⁄ £ 2010 2011 ˘c1 1 ˘ˇ⁄ ˘ 6¶ ⁄1 !JK£15'⁄. ((J. 1. –eØf (x1, x2) = x2 1 − 2x1x2 + x2 2 + x1 + x2‘{§(·£ ⁄ A. T…Œ·…Œ¶ C. T…ŒHesse‰¶ 2. “‘zﬂK{min x2 3:x = (0, 0)T ?e1£ 1+x2 A. (1, 1)T ; C. (1,−2)T ; B. f (¯x) ≤ f (ˆx) + ∇f (ˆx)T (¯x − ˆx), ∀¯x, ˆx ∈ R2¶ D. T…Œ4:·4:. 2−2x1−4x2+6 | −2x1+x2+1 ≥ 0;−x1−x2+2 ≥ 0; x1, x2 ≥ 0} ⁄ B. (−2, 1)T ; D. (−2,−1)T . 3. “‘zﬂK{min f (x) = (x1− 1)2 + x2 | − x1− x2 + 2 ≥ 0; x2 ≥ 0}K-T:£ ⁄ A. (1, 0)T ¶ B. (0, 1)T ¶ C. (2, 0)T ¶ D. (0, 2)T . 4. ˜8S = {(x1, x2) | 0 < x1 < 2; 0 < x2 < 1}, K£ ⁄ A. S8, k4:; C. S8, ˆ4:; B. S·8, k4:; D. S·8, ˆ4:. 5. ˜55y{min z = −x1 − 2x2 | − 2x1 + x2 ≤ 2;−x1 + x2 ≤ 3; x1, x2 ≥ 0}, K£ ⁄ A. (1, 2)T ·4; C. (2, 1)T ·4; B. (1, 0)T ·ˆ.; D. (0, 1)T ·ˆ.. !WK£24'⁄. 1. 55yﬂK{min cT x | Ax = b; x ≥ 0}ØﬂK 2. ƒ)ˆ‘zﬂKmin f (x) = x2 {|¢d(k) = 3. A n Ø¡‰§pi(i = 1,··· , n)’uAn. ex = n , {|¢d(k) = . 1 − 2x1x2 + 4x2 . 2 + x1 − 3x2. K3x(k) = (1, 1)T ?§e i=1 αipi§ £^x, pi, AL«⁄. Kαi = 4. …Œf (x1, x2) = (x1− 1 2 4:. 5. ˜ﬂK{min f (x) | l ≤ x ≤ u; x ∈ Rn}, ¥li < ui(i = 1,··· , n). f (x)·ºY 2−3x2 ›‰:k x2)2+x3 §¥ 1

…Œ§x∗·‘). ex∗i = li, K ∂f (x∗) ∂xi 0. ex∗i = ui, K ∂f (x∗) ∂xi 0. ∂f (x∗) 0. eli < x∗i < ui, K 6. f : Rn → RgºY. Psk = x(k+1) − x(k), yk = ∇f (x(k+1)) − ∇f (x(k)). Bk+1∇2f (x(k+1))Cq§Kv[§ ∂xi .   β β β min x1 + βx2 s.t. − x1 + x2 ≤ 1, −x1 + 3x2 ≤ 4, x1 ≥ 0, x2 ≥ 0, K 7. ˜55yﬂK §k‘)¶ §kˆ¡‘)¶ §ˆ‘). 8. ^0.618{(7'{⁄ƒ)minx∈R 2x2 − x − 1. —'«m[a1, b1] = [−1, 1]§K«m [a2, b2]= (3nŒ⁄. n!OK£37'⁄. 1£13'⁄. ‰|–3«‹. z«¨d'O1200, 1000, 700 /Z. ¥zZªn«‹„E⁄'A'O3§4§2§„E⁄'B'O2§1, 1. qz

2£12'⁄. ^F{ƒ)min f (x) = 2x2 1 + 2x1x2 + x2 2 + 3x1 − 4x2§—':x(1) = (0, 0)T . 4

Rn → RºY§A ∈ Rm×n, B ∈ Rs×n, bc'Oms.  min f (x) s.t. Ax ≥ b, Bx = c, 3£12'⁄. ƒ)eª55yﬂKZoutendijk1{§ ¥f (x) : 5

o!'K£24'⁄. 1£12'⁄. (a) y†‰nØu5yﬂK§ex∗ ·ﬂK4:§Kx∗‰·4:. e8I…Œ…Œ§Kx∗ 4:. (b) A m × n1. ˜‘zﬂK min f (x) = 1 2 xT x s.t. Ax = b. ØTﬂK)§‘†T)·4:. 6

2£12'⁄. ¤v…ŒS:{‰´¤v…ŒG(x, r) = f (x) + rB(x)§¥B(x) > 0. 0 < rk+1 < rk, xk xk+1 'Ovˇfr rk rk+1 ˆﬂKminx G(x, r) 4:§y†e “⁄Æ (1) B(xk) ≤ B(xk+1); (2) f (xk+1) ≤ f (xk); (3) G(xk+1, rk+1) ≤ G(xk, rk). 7

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上海交大研究生《最优化理论基础》试卷A卷.pdf

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