What Is Optimization Toolbox?

Large-Scale Linear ProgrammingThe algorithm is a variant of the predictor-corrector algorithm proposed byMehrotra. Consider an iterate,whereFirst compute the so-called prediction directionwhich is the Newton direction; then the so-called corrector directionwhere is called the centering parameter and must be chosen carefully. isa zero-one vector with the ones corresponding to the quadratic equationsin F(v) ,i.e., the perturbations are only applied to the complementarityconditions, which are all quadratic, but not to the feasibility conditions, whichare all linear. The two directions are combined with a step length parameterand update v to obtain the new iteratewhere the step length parameteris chosen so thatsatisfiesIn solving for the preceding steps, the algorithm computes a (sparse) directfactorization on a modification of the Cholesky factors of If A hasdense columns, it instead uses the Sherman-Morrison formula, and if thatsolution is not adequate (the residual is too large), it uses preconditionedconjugate gradients to find a solution.4-17