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 sample correlation (CORR) between these measures and IPM iteration counts (solved using the software SDPT3) when these measures have finite values. Our conclusions are roughly as follows: the aggregate geometry measure is highly correlated with IPM iterations (CORR = 0.901), and provides a very good explanation 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.

[1]  Michael L. Overton,et al.  Primal-Dual Interior-Point Methods for Semidefinite Programming: Convergence Rates, Stability and Numerical Results , 1998, SIAM J. Optim..

[2]  Kim-Chuan Toh,et al.  Solving Large Scale Semidefinite Programs via an Iterative Solver on the Augmented Systems , 2003, SIAM J. Optim..

[3]  Javier Peña,et al.  Two properties of condition numbers for convex programs via implicitly defined barrier functions , 2002, Math. Program..

[4]  J. Renegar.  Some perturbation theory for linear programming , 1994, Math. Program..

[5]  Jun Ji,et al.  On the Local Convergence of a Predictor-Corrector Method for Semidefinite Programming , 1999, SIAM J. Optim..

[6]  Henry Wolkowicz,et al.  Handbook of Semidefinite Programming , 2000 .

[7]  Robert M. Freund,et al.  On Two Measures of Problem Instance Complexity and their Correlation with the Performance of SeDuMi on Second-Order Cone Problems , 2004, Comput. Optim. Appl..

[8]  Stephen M. Robinson,et al.  A Characterization of Stability in Linear Programming , 1977, Oper. Res..

[9]  Robert M. Freund,et al.  On the Primal-Dual Geometry of Level Sets in Linear and Conic Optimization , 2001, SIAM J. Optim..

[10]  Kim-Chuan Toh,et al.  SDPT3 — a Matlab software package for semidefinite-quadratic-linear programming, version 3.0 , 2001 .

[11]  R. Freund,et al.  On the Primal-Dual Geometry of Level Sets in Linear and Conic Optimization , 2001 .

[12]  Z. Luo,et al.  Superlinear Convergence of a Symmetric Primal-Dual Path Following Algorithm for SDP , 1998 .

[13]  Robert M. Freund,et al.  Complexity of convex optimization using geometry-based measures and a reference point , 2015, Math. Program..

[14]  Renato D. C. Monteiro,et al.  Error Bounds and Limiting Behavior of Weighted Paths Associated with the SDP Map X1/2SX1/2 , 2005, SIAM J. Optim..

[15]  Stephen J. Wright,et al.  Primal-Dual Interior-Point Methods , 1997 .

[16]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[17]  Joo-siong Chai,et al.  Computation of condition numbers for linear programming problems using Peña’s method , 2006, Optim. Methods Softw..

[18]  Masakazu Kojima,et al.  Local convergence of predictor—corrector infeasible-interior-point algorithms for SDPs and SDLCPs , 1998, Math. Program..

[19]  Robert M. Freund,et al.  Some characterizations and properties of the “distance to ill-posedness” and the condition measure of a conic linear system , 1999, Math. Program..

[20]  M. Overton,et al.  Primal - dual interior - point methods for semidefinite programming : Stability, convergence, and nu , 1998 .

[21]  Fernando Ordóñez,et al.  Computational Experience and the Explanatory Value of Condition Measures for Linear Optimization , 2003, SIAM J. Optim..

[22]  Kim-Chuan Toh,et al.  Solving semidefinite-quadratic-linear programs using SDPT3 , 2003, Math. Program..

[23]  James Renegar,et al.  Linear programming, complexity theory and elementary functional analysis , 1995, Math. Program..

[24]  Michael L. Overton,et al.  Complementarity and nondegeneracy in semidefinite programming , 1997, Math. Program..

[25]  Robert M. Freund,et al.  On the Complexity of Computing Estimates of Condition Measures of a Conic Linear System , 2003, Math. Oper. Res..

[26]  F. Potra,et al.  Superlinear Convergence of Interior-Point Algorithms for Semidefinite Programming , 1998 .

[27]  Yuval Rabani,et al.  Linear Programming , 2007, Handbook of Approximation Algorithms and Metaheuristics.