v2009.01.01 - Convex Optimization

248 CHAPTER 4. SEMIDEFINITE PROGRAMMING
The vector inner-product for matrices is defined in the Euclidean/Frobenius
sense in the isomorphic vector space R n(n+1)/2 ; id est,
〈C , X〉 ∆ = tr(C T X) = svec(C) T svec X (31)
where svec X defined by (49) denotes symmetric vectorization.
Linear programming, invented by Dantzig in 1947, is now integral to
modern technology. But the same cannot yet be said of semidefinite
programming introduced by Bellman & Fan in 1963. [86] Numerical
performance of contemporary general-purpose semidefinite program solvers
is limited: Computational intensity varies as O(n 6 ) based on interior-point
methods that produce solutions no more relatively accurate than1E-8. There
are no solvers available capable of handling in excess of n=100 variables
without significant, sometimes crippling, loss of precision or time. 4.2
Nevertheless, semidefinite programming has emerged recently to
prominence primarily because it admits a new class of problem previously
unsolvable by convex optimization techniques, [51] secondarily because it
theoretically subsumes other convex techniques such as linear, quadratic, and
second-order cone programming. Determination of the Riemann mapping
function from complex analysis [248] [28,8,13], for example, can be posed
as a semidefinite program.
4.1.1.1 Maximal complementarity
It has been shown that contemporary interior-point methods (developed
circa 1990 [126]) [53,11] [334] [251] [239] [10] [121] for numerical
solution of semidefinite programs can converge to a solution of maximal
complementarity; [150,5] [341] [218] [133] not a vertex-solution but a
solution of highest cardinality or rank among all optimal solutions. 4.3
[342,2.5.3]
4.2 Heuristics are not ruled out by SIOPT; indeed I would suspect that most successful
methods have (appropriately described) heuristics under the hood - my codes certainly do.
...Of course, there are still questions relating to high-accuracy and speed, but for many
applications a few digits of accuracy suffices and overnight runs for non-real-time delivery
is acceptable.
−Nicholas I. M. Gould, Editor-in-Chief, SIOPT
4.3 This characteristic might be regarded as a disadvantage to this method of numerical
solution, but this behavior is not certain and depends on solver implementation.