Optimizing Optimization: The Next Generation of Optimization Applications and Theory (Quantitative Finance)

88 Optimizing Optimization

Appendix A: BITA Robust optimization

BITA Robust applies a Second-order Cone Programming (SOCP) problem to
the efficient application of quadratic constraints to an objective function.
The standard form in which a second-order cone optimization is expressed
is that of minimizing a linear objective subject to a set of linear equality con-
straints for which the optimization vector variable is restricted to a direct prod-
uct of quadratic cones K. This is the primal SOCP problem.

minimize cxsubject to Ax b

xK tt tt

T

njn

j

n

ii

i







∈

⎧

⎨

⎪⎪

:, 1 ∑

22

1

1

⎪⎪ 0

⎩

⎪⎪

⎪

⎫

⎬

⎪⎪

⎪

⎭

⎪⎪

⎪

∏

i

n

 (^1)
where K is the direct product of n quadratic cones of dimension n i  1, x is
represented as a vector of size nn where nn i ni
n
∑ 1 1. A is an nn by m
matrix where m is the number of conic constraints, and b is of length m and c
nn. We define the dual cone to K as:
Kssx xKd
{:t 0 ,∀∈}
Associated with the primal problem is the dual SOCP problem:
maximize bysubject to s c Ay
sK
TT
d

∈
where s is a vector with the same size as x and y has the length m. It can be
shown that both problems can be solved simultaneously using a primal-dual
interior point method and we have implemented the homogenous method
described in Andersen, Roos, EA and Terlaky (2003) for the programming,
which relies upon the scaling found in Nesterov and Todd (1997). The dual
problem may be written in the more convenient form:
maximize bysubject to Ay c c Ay i
cAy
TTn T
n
T
n
i
ii
i




1 1
1 0
()
()
∀∀
∀ii
A special case where Ajn,(i 1 ) j  1 ... m  0 is the problem:
maximize bysubject to Ay c c i
ci
TTn
n
n
i
ii
i




1 1
1 0
∀∀
∀