What Is Optimization Toolbox?

4 Large-Scale Algorithms(4-16)whichisreferredtoastheprimal problem: x consists of the primal variablesand s consists of the primal slack variables. The dual problem is(4-17)where y and w consist of the dual variables and z consists of the dual slacks.The optimality conditions for this linear program, i.e., the primal Equation4-16 and the dual Equation 4-17, arewhere and denote component-wise multiplication.(4-18)The quadratic equations and are called the complementarityconditions for the linear program; the other (linear) equations are called thefeasibility conditions. The quantityis the duality gap, which measures the residual of the complementarityportion of F when .The algorithm is a primal-dual algorithm, meaning that both theprimal and the dual programs are solved simultaneously. It can beconsidered a Newton-like method, applied to the linear-quadratic systemin Equation 4-18, while at the same time keeping theiterates x, z, w, ands positive, thus the name interior-point method. (Theiterates are in the strictly interior region represented by the inequalityconstraints in Equation 4-16.)4-16