Chapter 5 Robust Performance Tailoring with Tuning - SSL - MIT

algorithms begin at an initial guess for the design variables and then try some number
of configurations until an optimal design is achieved or some termination conditions
are met. The approaches differ in the methods used to move from one iteration to
the next. Gradient-based methods use gradient information to guide the search for
points that satisfy the necessary conditions for optimality. However, if the solution
space is not convex there is no guarantee that a global optimum is found. Heuristic
methods do not require gradients and use randomized algorithms to search for good,
but not necessarily optimal, solutions. These methods are most useful when the
solution space is non-convex and/or highly constrained as the search is able to jump
out of local minima and across islands of feasibility. A more detailed discussion of
the gradient-based algorithms and a simple example is given in Appendix A. Both
a gradient-based method, sequential quadratic programming (SQP), and a heuristic
method, simulated annealing (SA), are applied to the PT SCI problem.
2.4.1 Sequential Quadratic Programming
Sequential quadratic programming (SQP) is a quasi-Newton method developed to
solve constrained optimization problems such as that of Equation 2.1. Constraints
are handled by augmenting the objective function to produce the Lagrangian function:
L (�x, λ) =f (�x)+
m�
λigi (�x) (2.39)
The general goal of SQP is to find the stationary point of this Lagrangian using New-
ton’s method. Therefore, the algorithm is also referred to as the Lagrange-Newton
method. For a detailed discussion of SQP the reader is referred to Fletcher [44].
SQP is chosen as the gradient-based optimization algorithm for the SCI develop-
ment model because it can handle constrained problems with nonlinear objectives and
constraints, and is already implemented in the MATLAB optimization toolbox [108].
Also, analytical gradients of the cost function are available, as described above, en-
abling a computationally efficient search.
In the MATLAB SQP implementation there are two levels of iterations. At each
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i=1