What Is Optimization Toolbox?

3 Standard AlgorithmsPhase 1. In phase 1, the algorithm finds an initial basic feasible solution(see “Basic and Nonbasic Variables” on page 3-42 for a definition) by solvingan auxiliary piecewise linear programming problem. The objective function ofthe auxiliary problem is the linear penalty function ,where P j(x j)isdefinedbyP(x) measures how much a point x violates the lower and upper boundconditions. The auxiliary problem isThe original problem has a feasible basis point if and only if the auxiliaryproblem has minimum value 0.The algorithm finds an initial point for the auxiliary problem by a heuristicmethod that adds slack and artificial variables as necessary. The algorithmthen uses this initial point together with the simplex algorithm to solve theauxiliary problem. The optimal solution is the initial point for phase 2 ofthe main algorithm.Phase 2. In phase 2, the algorithm applies the simplex algorithm, starting atthe initial point from phase 1, to solve the original problem. At each iteration,the algorithm tests the optimality condition and stops if the current solutionis optimal. If the current solution is not optimal, the algorithm1 Chooses one variable, called the entering variable, from the nonbasicvariables and adds the corresponding column of the nonbasis to the basis(see “Basic and Nonbasic Variables” on page 3-42 for definitions).2 Chooses a variable, called the leaving variable, from the basic variables andremoves the corresponding column from the basis.3-40