eldo_user

Optimization
Eldo Optimizer SQP Algorithm
Refer to Nocedal and Wright [5], and Fletcher [3] for recent books giving a broader view of
optimization techniques. For more advanced books consult Bonnans, Gilbert, Lemarechal and
Sagastizabal [1] and Gill, Murray and Wright [4].
The design variables are denoted by:
x =(x (1) , x (2) , ... , x (N) )
a vector of real numbers of dimension N. Therefore the space of variables is denoted by R N . The
scalar product is denoted by:
onto the space R Ν , and its associated norm is ||u||.
A simplified structure of the kth iteration can be depicted as:
1. Step Computation: determine a direction of search s k (by solving a tangent quadratic
subproblem (QP) k
2. Step Assessment: find α k > 0 to minimize a chosen merit function x → Φ(x) along the
line α→ x k + αs k
3. Update: x k+1 = x k + α k s k
This approach is referred to as a descent method since the search direction s k satisfies a descent
property:
where the vector:
represents the gradient of the chosen merit function. The role of the line search algorithm is to
dampen the displacement s k .
During the optimization process, simulations are required at two distinct stages:
• The computation of derivatives of the extracted measures involved in the problem
statement.
• The step assessment procedure, or line search algorithm.
Eldo® User's Manual, 15.3 623