ILOG CPLEX 11.0 User's Manual

Introducing the Barrier OptimizerThe ILOG CPLEX Barrier Optimizer is well suited to large, sparse problems. An alternativeto the simplex optimizers, which are also suitable to problems in which the matrixrepresentation is dense, the barrier optimizer exploits a primal-dual logarithmic barrieralgorithm to generate a sequence of strictly positive primal and dual solutions to a problem.As with the simplex optimizers, it is not really necessary to understand the internal workingsof barrier in order to obtain its performance benefits. However, for the interested reader, hereis an outline of how it works.ILOG CPLEX finds the primal solutions, conventionally denoted (x, s), from the primalformulation:Minimize c T xsubject to Ax =bwith these bounds x +s=u and x ≥ lwhere A is the constraint matrix, including slack and surplus variables; u is the upper and lthe lower bounds on the variables.Simultaneously, ILOG CPLEX automatically finds the dual solutions, conventionallydenoted (y,z,w) from the corresponding dual formulation:Maximize b T y-u T w+l T zsubject to A T y-w+z=cwith these bounds w ≥ 0 and z ≥ 0All possible solutions maintain strictly positive primal solutions (x - l, s) and strictly positivereduced costs (z, w) so that the value 0 (zero) forms a barrier for primal and dual variableswithin the algorithm.ILOG CPLEX measures progress by considering the primal feasibility, dual feasibility, andduality gap at each iteration. To measure feasibility, ILOG CPLEX considers the accuracywith which the primal constraints (Ax =b,x+s=u) and dual constraints (A T y+z-w=c)are satisfied. The optimizer stops when it finds feasible primal and dual solutions that arecomplementary. A complementary solution is one where the sums of the products (x j -l j )z jand (u j -x j )z j are within some tolerance of 0(zero). Since each (x j -l j ), (u j -x j ), and z j isstrictly positive, the sum can be near zero only if each of the individual products is near zero.The sum of these products is known as the complementarity of the problem.On each iteration of the barrier optimizer, ILOG CPLEX computes a matrix based on AA Tand then computes a Cholesky factor of it. This factored matrix has the same number ofnonzeros on each iteration. The number of nonzeros in this matrix is influenced by thebarrier ordering parameter.198 ILOG CPLEX 11.0 — USER’ S MANUAL