Computer Aided Engineering Design

OPTIMIZATION 365

the objective using line search. If the inequality constraints are violated, αk is reduced so that the
solution lies on the surface of the most violated constraint thus retaining feasibility. The most
violated constraint is then added to the set of active constraints.
(d) The variables are updated from the results of the line search.
(e) The signs of the Lagrange multipliers are checked and if they are negative, the constraint
corresponding to the most negative multiplier is dropped from the active set.
(f) Termination criteria are evaluated and if they are not satisfied, the algorithm resumes to step (b).
Schematic of the implementation is given in Figure 12.13.

Design specifications

feasible initial design

define a constraint set

Compute search vector, ΔX and

lagrange multipliers ΛΛΛΛΛ

Compute step length along the search

vector using line search

If constraints are violated, reduce step length till the design

reaches the surface of the most violated constraint. Add that

constraint in the active set

Ye s

Constraint added

no

Update design variables

IfΛΛΛΛΛ are negative, detele the constraint

with the most negative multiplier from

the active set

No Ye s

Check for convergence STOP

Figure 12.13 Implementation details for sequential quadratic programming

12.7 Stochastic Approaches (Genetic Algorithms and Simulated Annealing)

Quite a few methods exist that are probabilistic in nature and rely only on the function values (and
not its gradient or Hessian) to reach an optimal solution, two of them being the genetic algorithms and
simulated annealing. A genetic algorithm mimics three operations of nature, namely, reproduction,
crossover, and mutation to form a new and better population or the generation of variable vectors X
from the previous one. The algorithm commences with a constant even size population that is a
predetermined number of candidate variable vectors. These vectors are initially generated randomly
such that the variable values lie between the lower and upper limits. For random generation of
variable values, a normal distribution scheme may be incorporated.