Multi-Parameter Surfaces of Analytic Centers and Long-Step Surface-Following Interior Point Methods

(q ú 0 is fixed); here in order to get close to the optimal set one should approach the parameter t to the optimal value of the problem. As it is well-known, both the parameterizations, under appropriate choice of F , imply polynomial-time interior-point methods for (1) . If G is a polytope, then it is reasonable to choose as F the standard logarithmic barrier for G ; polynomiality of the associated path-following methods for Linear Programming was first established in the seminal papers of Renegar (1988), parameterization (3) , and Gonzaga (1989), parameterization (2) . For the nonpolyhedral case polynomial time results for both the parameterizations can be obtained if F is a self-concordant barrier for G (see below), as it is the case with the standard logarithmic barrier for a polytope. Now, in order to trace the path of analytic centers F one should first get close to the path; this is the aim of a special preliminary phase of a path-following method, which, theoretically, is of the same complexity as following the path itself. Moreover, to initialize the preliminary phase one should know in advance a strictly feasible solution x̂ √ int G . To get such a point, it, generally speaking, again requires an additional phase of the method; at this phase we, basically, solve an auxiliary problem of the same type as (1) , but with a known in advance strictly feasible solution. There are numerous strategies of