What Is Optimization Toolbox?

3 Standard Algorithmsbe canceled, Lagrange multipliers () are necessary to balancethe deviations in magnitude of the objective function and constraint gradients.Because only active constraints are included in this canceling operation,constraints that are not active must not be included in this operation and soare given Lagrange multipliers equal to 0. This is stated implicitly in the lasttwo Kuhn-Tucker equations.The solution of the KT equations forms the basis to many nonlinearprogramming algorithms. These algorithms attempt to compute theLagrange multipliers directly. Constrained quasi-Newton methods guaranteesuperlinear convergence by accumulating second-order information regardingthe KT equations using a quasi-Newton updating procedure. These methodsare commonly referred to as Sequential Quadratic Programming (SQP)methods, since a QP subproblem is solved at each major iteration (also knownas Iterative Quadratic Programming, Recursive Quadratic Programming, andConstrained Variable Metric methods).Sequential Quadratic Programming (SQP)SQP methods represent the state of the art in nonlinear programmingmethods. Schittkowski [38], for example, has implemented and tested aversion that outperforms every other tested method in terms of efficiency,accuracy, and percentage of successful solutions, over a large number of testproblems.Based on the work of Biggs [1], Han [24], and Powell ([34] and [35]), themethod allows you to closely mimic Newton’s method for constrainedoptimization just as is done for unconstrained optimization. At each majoriteration, an approximation is made of the Hessian of the Lagrangian functionusing a quasi-Newton updating method. This is then used to generate a QPsubproblem whose solution is used to form a search direction for a line searchprocedure. An overview of SQP is found in Fletcher [15], Gill et. al. [21],Powell [37], and Schittkowski [25]. The general method, however, is statedhere.3-30