Engineering Optimization: Theory and Practice, Fourth Edition

8.6 Primal–Dual Relationship and Sufficiency Conditions in the Unconstrained Case 503 where w=      w 0 w 1 .. . wn   ...

504 Geometric Programming By substituting Eq. (8.45) into Eq. (8.36), we obtain L(,w)= − ∑N j= 1 jln j cj +( 1 −w 0 )   ∑N ...

8.6 Primal–Dual Relationship and Sufficiency Conditions in the Unconstrained Case 505 Primal and Dual Problems. We saw that geom ...

506 Geometric Programming The pumping cost is given by (300Q^2 /D^5 ) Find the optimal size of the pipe and the. amount of fluid ...

8.6 Primal–Dual Relationship and Sufficiency Conditions in the Unconstrained Case 507 Since lnvis expressed as a function of 4 ...

508 Geometric Programming The optimum values of the design variables can be found from U 1 ∗=∗ 1 f∗= ( 0. 385 )( 242 )= 92. 2 U ...

8.8 Solution of a Constrained Geometric Programming Problem 509 andakij (k= 1 , 2 ,... , m; i= 1 , 2 ,... , n;j= 1 , 2 ,... , Nk ...

510 Geometric Programming If the functionf(X) is known to possess a minimum, the stationary valuef∗given by Eq. (8.59) will be t ...

8.9 Primal and Dual Programs in the Case of Less-Than Inequalities 511 gk( X)≤ 1 , the signum functionsσk are all equal to+ 1 , ...

512 Geometric Programming Table 8.2 Corresponding Primal and Dual Programs Primal program Dual program FindX=      x 1 ...

8.9 Primal and Dual Programs in the Case of Less-Than Inequalities 513 Table 8.2 (continued) Primal program Dual program the exp ...

514 Geometric Programming a 021 λ 01 +a 022 λ 02 +a 023 λ 03 +a 121 λ 11 = 0 a 031 λ 01 +a 032 λ 02 +a 033 λ 03 +a 131 λ 11 = 0 ...

8.9 Primal and Dual Programs in the Case of Less-Than Inequalities 515 λ∗ 03 = c 03 (x 1 ∗)^13 a^0 (x 2 ∗)^23 a^0 (x∗ 3 )a^330 x ...

516 Geometric Programming a 142 ,= 1 a 211 = , 1 a 221 = , 1 a 231 = , and 0 a 241 = 0,Eqs. (E 1 ) ecomeb Maximizev(λ)= [ c 01 λ ...

8.9 Primal and Dual Programs in the Case of Less-Than Inequalities 517 λ 12 =λ 01 −λ 03 −λ 11 (E 7 ) λ 12 = 2 λ 11 −λ 02 (E 8 ) ...

518 Geometric Programming To find the maximum ofv, we set the derivative ofvwith respect toλ 01 equal to zero. To simplify the c ...

8.10 Geometric Programming with Mixed Inequality Constraints 519 The constraints are given by (see Table 8.2) ∑N^0 j= 1 λ 0 j= 1 ...

520 Geometric Programming By using Eqs. (E 3 ) the dual objective function of Eq. (E, 1 ) an be expressed asc v(λ 01 )= ( 1 λ 01 ...

8.11 Complementary Geometric Programming 521 whereAk( ,X) Bk( ,X) Ck( , andX) Dk( are posynomials inX) Xand possibly some of the ...

522 Geometric Programming Solution Procedure. 1.Approximate each of the posynomialsQ(X)†by a posynomial term. Then all the const ...