Inequality quadratic programming

Reformulating the necessaiy conditions as a linear quadratic program has an interesting side effect. We can simply add linearizations of the inactive inequalities to the problem ana let the ac tive set be selected by the algorithm used to solve the linear quadratic program. 

Another method for solving nonlinear programming problems is based on quadratic programming (QP)1. Quadratic programming is an optimization procedure that minimizes a quadratic objective function subject to linear inequality or equality (or both types) of constraints. For example, a quadratic function of two variables x and X2 would be of the general form ... 

The vector x can contain slack variables, so the equality constraints (8.33) may contain some constraints that were originally inequalities but have been converted to equalities by inserting slacks. Codes for quadratic programming allow arbitrary upper and lower bounds on x we assume x>0 only for simplicity. 

For problems with only equality constraints, we could simply solve the linear equations (8.66)-(8.67) for (Ax, AX) and iterate. To accommodate both equalities and inequalities, an alternative viewpoint is useful. Consider the quadratic programming problem... 

The minimization of the quadratic performance index in Eq. (8-64), subject to the constraints in Eqs. (8-67) to (8-69) and the step response model in Eq. (8-61), can be formulated as a standard QP (quadratic programming) problem. Consequently, efficient QP solution techniques can be employed. When the inequality constraints in Eqs. (8-67) to (8-69) are omitted, the optimization problem has an analytical solution (Camacho and Bordons, Model Predictive Control, 2d ed., Springer-Verlag, New York, 2004 Maciejowski, Predictive Control with Constraints, Prentice-Hall, Upper Saddle River, N.J., 2002). If the quadratic terms in Eq. (8-64) are replaced by linear terms, an LP (linear programming) problem results that can also be solved by using standard methods. This MPC formulation for SISO control problems can easily be extended to MIMO problems. 

Gautam and Seider (25) implemented the quadratic programming algorithm of Wolfe (30, 31) because compositions satisfy inequality constraints (n. 0) without transformations. Moles of solid... 

Successive Quadratic Programming (SQP) The above approach to finding the optimum is called a feasible path method, as it attempts at all times to remain feasible with respect to the equality and inequality constraints as it moves to the optimum. A quite different method exists called the Successive Quadratic Programming (SQP) method, which only requires one be feasible at the final solution. Tests that compare the GRG and SQP methods generally favor the SQP method so it has the reputation of being one of the best methods known for nonlinear optimization for the type of problems considered here. 

Algorithms for the solution of quadratic programs, such as the Wolfe (1959) algorithm, are very reliable and readily available. Hence, these have been used in preference to the implementation of the Newton-Raphson method. For each iteration, the quadratic objective function is minimized subject to linearized equality and inequality constraints. Clearly, the most computationally expensive step in carrying out an iteration is in the evaluation of the Lapla-cian of the Lagrangian, V xL x , X which is also the Hessian matrix of the La-grangian that is, the matrix of second derivatives with respect to X . 

Another difficulty shared by all the methods that split inequality constraints into active and passive families, which has been already emphasized in linear and quadratic programming, is related to the zigzagging (jamming) phenomenon that arises from the insertion and removal of the same active constraints. 

A quadratic programming problem minimizes a quadratic function of n variables subject to m linear inequality or equality constraints. A convex QP is the simplest form of a nonlinear programming problem with inequality constraints. A number of practical optimization problems are naturally posed as a QP problem, such as constrained least squares and some model predictive control problems. 

The introduction of inequality constraints results in a constrained optimization problem that can be solved numerically using linear or quadratic programming techniques (Edgar et al., 2001). As an example, consider the addition of inequality constraints to the MFC design problem in the previous section. Suppose that it is desired to calculate the M-step control policy AU(k) that minimizes the quadratic objective function J in Eq. 20-54, while satisfying the constraints in Eqs. 20-59, 20-60, and 20-61. The output predictions are made using the step-response model in Eq. 20-36. This MFC... 

Because the set-point calculations are repeated as often as every minute, the steady-state optimization problem must be solved quickly and reliably. If the optimization problem is based on a linear process model, linear inequality constraints, and either a linear or a quadratic cost function, the hnear and quadratic programming techniques discussed in Chapter 19 can be employed. 

The augmented Lagrangian method is not the only approach to solving constrained optimization problems, yet a complete discussion of this subject is beyond the scope of this text. We briefly consider a popular, and efficient, class of methods, as it is used by fmincon, sequential quadratic programming (SQP). We wUl find it useful to introduce a common notation for the equality and inequality constraints using slack variables. 

It will be shown that the solution to any mathematical programming problem must satisfy not only the original constraints, but also certain side conditions. These new conditions take the form of inequalities in a new set of non-negative variables, and it will turn out that in quadratic... 

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