v2010.10.26 - Convex Optimization

332 CHAPTER 4. SEMIDEFINITE PROGRAMMING〈Z , W 〉rank Zw ck0wf(Z)Figure 98: Regularization curve, parametrized by weight w for real convexobjective f minimization (756) with rank constraint to k by convex iteration.provide some impetus to focus more research on computational intensity ofgeneral-purpose semidefinite program solvers. An approach different frominterior-point methods is required; higher speed and greater accuracy from asimplex-like solver is what is needed.4.4.1.3 regularizationWe numerically tested the foregoing technique for constraining rank on a widerange of problems including localization of randomized positions (Figure 97),stress (7.2.2.7.1), ball packing (5.4.2.3.4), and cardinality problems (4.6).We have had some success introducing the direction matrix inner-product(753) as a regularization term 4.30minimizeZ∈S N f(Z) + w〈Z , W 〉subject to Z ∈ CZ ≽ 0(756)4.30 called multiobjective- or vector optimization. Proof of convergence for this convexiteration is identical to that in4.4.1.2.1 because f is a convex real function, hence boundedbelow, and f(Z ⋆ ) is constant in (757).