Problems 5418.12 Minimizef (X)=x− 12 +^14 x^22 x 3
subject to
3
4 x
2
1 x− 2
2 +3
8 x^2 x− 2
3 ≤^1
xi> 0 , i= 1 , 2 , 38.13 Minimizef (X)=x 1 −^3 x 2 +x 13 /^2 x− 31
subject tox^21 x− 21 +^12 x− 12 x 33 ≤ 1
x 1 > 0 , x 2 > 0 , x 3 > 08.14 Minimizef=x− 11 x 2 −^2 x 3 −^2
subject tox^31 +x 22 +x 3 ≤ 1
xi> 0 , i= 1 , 2 , 38.15 Prove that the functiony=c 1 ea^1 x^1 +c 2 ea^2 x^2 + · · · +cneanxn, c (^) i≥ 0 ,i= 1 , 2 ,... , n, is
a convex function with respect tox 1 , x 2 ,... , xn.
8.16 Prove thatf=lnxis a concave function for positive values ofx.
8.17 The problem of minimum weight design of a helical torsional spring subject to a stress
constraint can be expressed as [8.27]
Minimizef (d, D)=
π^2 ρEφ
14 , 680 M
d^6 +
π^2 ρQ
4
Dd^2
subject to
14. 5 M
d^2.^885 D^0.^115 σmax
≤ 1
wheredis the wire diameter,Dthe mean coil diameter,ρthe density,Eis Young’s
modulus,φthe angular deflection in degrees,Mthe torsional moment, andQthe number
of inactive turns. Solve this problem using geometric programming approach for the
following data:E= 20 × 1010 Pa,σmax= 15 × 107 Pa,φ= 20 ◦, Q= 2 , M= 0 .3 N-m,
andρ= 7. 7 × 104 N/m^3.
8.18 Solve the machining economics problem given by Eqs. (E 2 )and (E 4 )of Example 8.7 for
the given data.
8.19 Solve the machining economics problem given by Eqs. (E 2 ), (E 4 ), and (E 6 )of Example
8.7 for the given data.
8.20 Determine the degree of difficulty of the problem stated in Example 8.8.