Equality-Constrained QP

The system (11.12) and (11.13) resulting from the KKT conditions in a QP problem has the following features  

2) W is also nonpositive definite when the Hessian G is positive definite and the matrix A is m-rank. 

The various methods used to solve QP problems differ in the following aspects  

2) The search for inequality constraints to be inserted into the system. 

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This approach operates in two phases. First, a sufficient number of elements is found in order to satisfy the linearization of all of the constraints at the initial point. In this way we guarantee that a feasible QP subproblem exists for (27). Second, to avoid convergence to a suboptimal solution with too few elements, we retain additional dummy elements in the formulation that are constrained to be less than or equal to a negligible element length. These elements can be placed at all nonzero element locations, but in practice they need only be associated with elements that have active error bounds at the QP solution. Now once the QP subproblem is solved, multipliers on the upper bounds of the dummy elements are checked for positive values. These indicate that the objective function can be further improved by relaxing the dummy element. After relaxation (which effectively adds another nonzero element to the problem), another dummy element is added in order to allow for any additional nonzero elements that may be needed. 

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. 

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