Engineering Optimization: Theory and Practice, Fourth Edition

4.7 Karmarkar’s Interior Method 223 Figure 4.3 Improvement of objective function from different points of a polytope. Karmarkar’ ...

224 Linear Programming II: Additional Topics and Extensions whereX= {x 1 , x 2 ,... , xn}T, c={c 1 , c 2 ,... , cn}T, and [ a]is ...

4.7 Karmarkar’s Interior Method 225 We now define a new vectorzas z=      z zn− 2 zn− 1 zn      andsolve the follo ...

226 Linear Programming II: Additional Topics and Extensions subject to 3 x 1 +x 2 − 2 x 3 = 3 5 x 1 − 2 x 2 = 2 xi≥ 0 , i= 1 , 2 ...

4.7 Karmarkar’s Interior Method 227 2.Test for optimality. Sincef=0 at the optimum point, we stop the procedure if the following ...

228 Linear Programming II: Additional Topics and Extensions Step 1We choose the initial feasible point as X(^1 )=      1 ...

4.8 Quadratic Programming 229 Noting that ∑n r= 1 x(r^1 )yr(^2 )=^13 ( 10834 )+^13 ( 10837 )+^13 ( 10837 )=^13 Eq. (4.71) can be ...

230 Linear Programming II: Additional Topics and Extensions reduces to a LP problem. The solution of the quadratic programming p ...

4.8 Quadratic Programming 231 Multiplying Eq. (4.79) bysiand Eq. (4.80) bytj, we obtain λisi^2 =λiYi= 0 , i= 1 , 2 ,... , m (4.8 ...

232 Linear Programming II: Additional Topics and Extensions phase I. This procedure involves the introduction ofnnonnegative art ...

4.8 Quadratic Programming 233 By comparing this problem with the one stated in Eqs. (4.72) to (4.74), we find that c 1 = − 4 , c ...

234 Linear Programming II: Additional Topics and Extensions According to the regular procedure of simplex method,λ 1 enters the ...

4.9 MATLAB Solutions 235 Basic Variables bi/ais variables x 1 x 2 λ 1 λ 2 θ 1 θ 2 Y 1 Y 2 z 1 z 2 w bi forais> 0 x 1 1 0 0 26 ...

236 Linear Programming II: Additional Topics and Extensions Express the constraints in the formA x≤band identify the matrixAand ...

References and Bibliography 237 Example 4.16 Find the solution of the following quadratic programming problem using MATLAB: Mini ...

238 Linear Programming II: Additional Topics and Extensions 4.3 G. B. Dantzig, L. R. Ford, and D. R. Fulkerson, A primal–dual al ...

Problems 239 Review Questions 4.1 Is the decomposition method efficient for all LP problems? 4.2 What is the scope of postoptima ...

240 Linear Programming II: Additional Topics and Extensions 4.2 Maximizef=^15 x 1 +^6 x 2 +^9 x 3 +^2 x 4 subject to 10 x 1 + 5 ...

Problems 241 x 1 −x 2 ≤ 11 x 1 ≥ 0 , x 2 unrestricted in sign (a)Write the dual of this problem. (b)Find the optimum solution of ...

242 Linear Programming II: Additional Topics and Extensions subject to x 1 +x 2 + 2 x 3 −x 5 −x 6 = 1 − 2 x 1 +x 3 +x 4 +x 5 −x ...