Which corresponds to the last row of equations 4.61 and 4.67, respectively.
#QUADRATIC PROGRAMMING FREE#
Variables that are free in sign can be easily treated by the method described in Section 8.2. Note also that the variables x are required to be non-negative in Eq. This is not an unreasonable assumption in practice, as many applications satisfy it. We will assume that the matrix H is at least positive semidefinite. Therefore, the problem has a unique global solution (if one exists). Further, if the matrix H is positive definite, the problem is strictly convex. Note also that if the matrix H is positive semidefinite, the QP problem is convex, so any solution (if one exists) represents a global minimum point (which need not be unique). Note that all of the linear inequality constraints are expressed in the “≤ form.” This is needed because we will use the KKT necessary conditions of Section 4.6, which require this form. Many other methods are available that can be considered as variations of that procedure in numerical implementation details. To give a flavor of the calculations needed to solve QP problems, we shall describe a method that is a simple extension of the Simplex method. Some of the available LP codes also have an option to solve QP problems (Schrage, 1981).
#QUADRATIC PROGRAMMING SOFTWARE#
Also, several commercially available software packages are available for solving QP problems, e.g., MATLAB, QPSOL (Gill et al., 1984), VE06A (Hopper, 1981), and E04NAF (NAG, 1984). Thus, it is not surprising that substantial research effort has been expended in developing and evaluating many algorithms for solving QP problems (Gill et al., 1981 Luenberger, 1984).
It is important to solve the QP subproblem efficiently so that large-scale problems can be treated. (10.25) and (10.26), the QP subproblem is obtained when a nonlinear problem is linearized and a quadratic step size constraint is imposed. In addition, many general nonlinear programming algorithms require solution of a quadratic programming subproblem at each iteration. Such problems are encountered in many real-world applications. Arora, in Introduction to Optimum Design (Second Edition), 2004 11.2 Quadratic Programming ProblemĪ quadratic programming (QP) problem has a quadratic cost function and linear constraints. We present such a procedure in Example 10.7. To aid the KKT solution process, we can use a graphical representation of the problem to identify the possible solution case and solve that case only. If the problem is simple, we can solve it using the KKT conditions of optimality given in Theorem 4.6. In the next chapter, we shall describe a method for solving general QP problems that is a simple extension of the Simplex method of linear programming. Also, many good programs have been developed to solve such problems. Therefore it is extremely important to solve a QP subproblem efficiently so that large-scale optimization problems can be treated. In addition, many general nonlinear programming algorithms require solution of a quadratic programming subproblem at each design cycle. QP problems are encountered in many real-world applications.
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