Derivatives of (2R)-N-Glyoxylbornane-10,2-sultam and Their Use as auxiliaries in the diastereoselective spirocyclization of 2-substituted tryptamines

A crude hydrate 6 and a crystalline hemiacetal 7 of glyoxylamide 4 were prepared from crotonamide 5 (Scheme 2). Particularly hemiacetal 7 , but also 6 and the ‘dimer’ 8 (obtained from 7 ) may serve as homochiral auxiliaries. The structure of 8 was determined by X-ray analysis. By arenesulfonyl halides, tryptimines 12–14 of 4 were diastereoselectively transformed into spirotricycles 15–17 and 19 .     

On Two Measures of Problem Instance Complexity and their Correlation with the Performance of SeDuMi on Second-Order Cone Problems

We evaluate the practical relevance of two measures of conic convex problem complexity as applied to second-order cone problems solved using the homogeneous self-dual (HSD) embedding model in the software SeDuMi. The first measure we evaluate is Renegar's data-based condition measure C(d), and the second measure is a combined measure of the optimal solution size and the initial infeasibility/optimality residuals denoted by S (where the solution size is measured in a norm that is naturally associated with the HSD model). We constructed a set of 144 second-order cone test problems with widely distributed values of C(d) and S and solved these problems using SeDuMi. For each problem instance in the test set, we also computed estimates of C(d) (using Peña’s method) and computed S directly. Our computational experience indicates that SeDuMi iteration counts and log (C(d)) are fairly highly correlated (sample correlation R = 0.675), whereas SeDuMi iteration counts are not quite as highly correlated with S (R = 0.600). Furthermore, the experimental evidence indicates that the average rate of convergence of SeDuMi iterations is affected by the condition number C(d) of the problem instance, a phenomenon that makes some intuitive sense yet is not directly implied by existing theory.