v2010.10.26 - Convex Optimization

276 CHAPTER 4. SEMIDEFINITE PROGRAMMING⎡A svec X = ⎣〈A 1 , X〉.〈A m , X〉(651)∑svec −1 (A T y) = m y i A iThe vector inner-product for matrices is defined in the Euclidean/Frobeniussense in the isomorphic vector space R n(n+1)/2 ; id est,i=1〈C , X〉 tr(C T X) = svec(C) T svec X (38)where svec X defined by (56) denotes symmetric vectorization.In a national planning problem of some size, one may easily run into severalhundred variables and perhaps a hundred or more degrees of freedom. ...Itshould always be remembered that any mathematical method and particularlymethods in linear programming must be judged with reference to the type ofcomputing machinery available. Our outlook may perhaps be changed whenwe get used to the super modern, high capacity electronic computor that willbe available here from the middle of next year. −Ragnar Frisch [147]Linear programming, invented by Dantzig in 1947 [94], is now integralto modern technology. The same cannot yet be said of semidefiniteprogramming whose roots trace back to systems of positive semidefinitelinear inequalities studied by Bellman & Fan in 1963. [31] [102] Interior-pointmethods for numerical solution can be traced back to the logarithmic barrierof Frisch in 1954 and Fiacco & McCormick in 1968 [141]. Karmarkar’spolynomial-time interior-point method sparked a log-barrier renaissancein 1984, [275,11] [381] [363] [277, p.3] but numerical performanceof contemporary general-purpose semidefinite program solvers remainslimited: Computational intensity for dense systems varies as O(m 2 n)(m constraints ≪ n variables) based on interior-point methods that producesolutions no more relatively accurate than 1E-8. There are no solverscapable of handling in excess of n=100,000 variables without significant,sometimes crippling, loss of precision or time. 4.3 [35] [276, p.258] [67, p.3]4.3 Heuristics are not ruled out by SIOPT; indeed I would suspect that most successfulmethods 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 manyapplications a few digits of accuracy suffices and overnight runs for non-real-time deliveryis acceptable.−Nicholas I. M. Gould, Editor-in-Chief, SIOPT⎤⎦