On the Behavior of the Homogeneous Self-Dual Model for Conic Convex Optimization

There is a natural norm associated with a starting point of the homogeneous self-dual (HSD) embedding model for conic convex optimization. In this norm two measures of the HSD model’s behavior are precisely controlled independent of the problem instance: (i) the sizes of ε-optimal solutions, and (ii) the maximum distance of ε-optimal solutions to the boundary of the cone of the HSD variables. This norm is also useful in developing a stopping-rule theory for HSD-based interior-point methods such as SeDuMi. Under mild assumptions, we show that a standard stopping rule implicitly involves the sum of the sizes of the ε-optimal primal and dual solutions, as well as the size of the initial primal and dual infeasibility residuals. This theory suggests possible criteria for developing starting points for the homogeneous self-dual model that might improve the resulting solution time in practice

Optimizing Product Line Designs: Efficient Methods and Comparisons

We compare a broad range of optimal product line design methods. The comparisons take advantage of recent advances that make it possible to identify the optimal solution to problems that are too large for complete enumeration. Several of the methods perform surprisingly well, including Simulated Annealing, Product-Swapping and Genetic Algorithms. The Product-Swapping heuristic is remarkable for its simplicity. The performance of this heuristic suggests that the optimal product line design problem may be far easier to solve in practice than indicated by complexity theory.

Behavioral Measures and their Correlation with IPM Iteration Counts on Semi-Definite Programming Problems

We study four measures of problem instance behavior that might account for the observed differences in interior-point method (IPM) iterations when these methods are used to solve semidefinite programming (SDP) problem instances: (i) an aggregate geometry measure related to the primal and dual feasible regions (aspect ratios) and norms of the optimal solutions, (ii) the (Renegar-) condition measure C(d) of the data instance, (iii) a measure of the near-absence of strict complementarity of the optimal solution, and (iv) the level of degeneracy of the optimal solution. We compute these measures for the SDPLIB suite problem instances and measure the correlation between these measures and IPM iteration counts (solved using the software SDPT3) when the measures have finite values. Our conclusions are roughly as follows: the aggregate geometry measure is highly correlated with IPM iterations (CORR = 0.896), and is a very good predictor of IPM iterations, particularly for problem instances with solutions of small norm and aspect ratio. The condition measure C(d) is also correlated with IPM iterations, but less so than the aggregate geometry measure (CORR = 0.630). The near-absence of strict complementarity is weakly correlated with IPM iterations (CORR = 0.423). The level of degeneracy of the optimal solution is essentially uncorrelated with IPM iterations.