Quantitative Local Analysis of Nonlinear Systems - University of ...

then set y to ȳ and optimize over z by solving
⎡ ⎤
¯z = argmin z c T ȳ
⎢ ⎥
⎣ ⎦ subject to F (ȳ, z) ≽ 0,
z
and alternate between these two problems as long as there is satisfactory improvement in
the solution. This two-way iterative search, which we call coordinate-wise affine search, is of
course a local search scheme and generated candidate solutions highly depend on the initial
point from which the search starts (it may not reach the optimal solution or may require a
large number of iterations to tightly approximate the optimal solution) [43]. Nevertheless,
it is practically attractive since it only requires an affine SDP solver and it has been widely
used by controls community (for example, D − K iteration in µ-synthesis [6] is based on
alternating between the controller K and the D-scales). Moreover, our experience suggests
that, coupled with efficient methods for generating high quality initial points (see chapter
3), coordinate-wise affine search can be efficiently used to compute suboptimal solutions for
problems with BMI constraints. Consequently, we implement coordinate-wise affine search
schemes throughout this thesis and provide implementation details in the corresponding
chapters.
2.1.3 Linear Programming
A linear program (LP) is an optimization problem with linear objective and affine constraints.
Formally, for c ∈ R n , A ∈ R m×n , and b ∈ R m , a linear program can be written
as
min c T x subject to
x∈R n
Ax ≽ b.
Several optimization problems in the following chapters will have both SDP constraints
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