REACTOR ANDGREEN CHEMISTRYvvREACTOR ANDGREEN CHEMISTRYREACTOR ANDGREEN CHEMISTRYREACTOR ANDGREEN CHEMISTRYREACTOR ANDGREEN CHEMISTRYREACTOR ANDGREEN CHEMISTRYREACTOR ANDGREEN CHEMISTRYREACTOR ANDGREEN CHEMISTRYREACTOR ANDGREEN CHEMISTRYREACTOR ANDGREEN CHEMISTRYREACTOR ANDGREEN CHEMISTRYREACTOR ANDGREEN CHEMISTRYREACTOR ANDGREEN CHEMISTRY
Chapter Unconstrained optimization Why are numerical methods ? For analytical method 𝛻𝑓 𝑋 ∗ = 𝜕𝑓 𝑋 =0 𝜕𝑥1 𝜕𝑓 𝑋 =0 𝜕𝑥2 ⋯⋯⋯⋯ 𝜕𝑓 𝑋 =0 𝜕𝑥𝑛 “necessary” condition Optimization problem is converted into a problem of solving a set of nonlinear equations in 𝑛 variables for the real roots It’s really not simple Attack to minimization problem directly by the numerical methods What are numerical methods ? Numerical methods that search for an extremum by using function and sometimes derivative values of 𝑓 𝑋 at a sequence of trial points 𝑋1 , 𝑋2 … until 𝑓 𝑋𝑛+1 − 𝑓 𝑋𝑛 < 𝜀 𝑋𝑛+1 = 𝑋𝑛 ± 𝑆𝑐𝑎𝑙𝑎𝑟 ∗ 𝐷𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛 Gradient Step size 𝑋𝑛+1 = 𝑋𝑛 ± ∆𝑋 Procedure for numerical methods Step • Starting at 𝑋𝑜 in a certain direction 𝑠Ԧ𝑜 Step • Move to 𝑋𝑛 with a step size ∆𝑋 = 𝛼𝑜 𝑠Ԧ𝑜 Step • Change the direction (if any) and move to 𝑋𝑛+1 Step • Stop searching until 𝑓 𝑋𝑛+1 − 𝑓 𝑋𝑛