meisteroreo.blogg.se - Quadratic programming

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. 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). 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. This means that if there is a solution to the primal minimization problem, then there is a solution to the dual maximization problem, and the dual maximum value is equal to the primal minimum value.QP problems are encountered in many real-world applications. For quadratic optimization, strong duality holds if is positive semidefinite.

The relationship between the factored dual vector and the unfactored dual vector is.

With a factored quadratic objective, the dual problem may also be expressed as:.

The Lagrangian dual problem for quadratic optimization with objective is given by: ».

The dual maximizer provides information about the primal problem, including sensitivity of the minimum value to changes in the constraints. The dual maximum value is always less than or equal to the primal minimum value, so it provides a lower bound.

The primal minimization problem has a related maximization problem that is the Lagrangian dual problem.

The objective function may be specified in the following ways:.

With QuadraticOptimization, parameter equations of the form par  val, where par is not in vars and val is numerical or an array with numerical values, may be included in the constraints to define parameters used in f or cons.

The constraints cons can be specified by:.

Vector variable restricted to the geometric region Variable with name and dimensions inferred

The variable specification vars should be a list with elements giving variables in one of the following forms:.

When the objective function is real valued, QuadraticOptimization solves problems with by internally converting to real variables, where and.

Mixed-integer quadratic optimization finds and that solve the problem:.

The space consists of symmetric positive semidefinite matrices.

Quadratic optimization finds that solves the primal problem: ».Quadratic optimization is a convex optimization problem that can be solved globally and efficiently with real, integer or complex variables.Quadratic optimization is typically used in problems such as parameter fitting, portfolio optimization and geometric distance problems.Quadratic optimization is also known as quadratic programming (QP) or linearly constrained quadratic optimization.