v2009.01.01 - Convex Optimization

246 CHAPTER 4. SEMIDEFINITE PROGRAMMING
generic convex optimization problem
minimize g(X)
X
subject to X ∈ C
(582)
where constraints are abstract here in the membership of variable X to
feasible set C . Inequality constraint functions of a convex optimization
problem are convex while equality constraint functions are conventionally
affine, but not necessarily so. Affine equality constraint functions (convex),
as opposed to the larger set of all convex equality constraint functions having
convex level sets, make convex optimization tractable.
Similarly, the problem
maximize g(X)
X
subject to X ∈ C
(583)
is convex were g a real concave function. As conversion to convex form is not
always possible, there is much ongoing research to determine which problem
classes have convex expression or relaxation. [31] [51] [126] [239] [300] [124]
4.1 Conic problem
Still, we are surprised to see the relatively small number of
submissions to semidefinite programming (SDP) solvers, as this
is an area of significant current interest to the optimization
community. We speculate that semidefinite programming is
simply experiencing the fate of most new areas: Users have yet to
understand how to pose their problems as semidefinite programs,
and the lack of support for SDP solvers in popular modelling
languages likely discourages submissions.
−SIAM News, 2002. [98, p.9]
Consider a conic problem (p) and its dual (d): [254,3.3.1] [206,2.1]
[207] (confer p.146)
(p)
minimize c T x
x
subject to x ∈ K
Ax = b
maximize b T y
y,s
subject to s ∈ K ∗
A T y + s = c
(d) (275)