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By Daniel Axehill.

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A drawback with the dual algorithm is also mentioned. If the Hessian is ill-conditioned, numerical problems might occur since the dual algorithm starts from the unconstrained optimum. The numerical properties of the algorithm presented in [48] are further examined in [78], where an extension to handle ill-conditioned problems is presented and the algorithm is compared to two primal QP solvers QPSOL and VEO2A. In [31], the algorithm is extended to the positive semidefinite case. In [7], the QR factorization used in [48] is replaced by the use of a Schur complement (as in [61]) for block elimination of the KKT matrix.

They can also be used for state estimation, fault detection and verification, see for example [17]. 1 Quadratic Programming In many applications, even linear MPC is considered computationally expensive. 5) has to be solved. One way of speeding up the solution of the optimization problem is to use QP solvers tailored for MPC, where the special structure of the KKT system is used to decrease the complexity of the algorithm. Basically two approaches can be used. First, general methods utilizing the structure in block-banded equation systems can be used.

In a branch and bound method, there are several parameters and choices that may affect the performance drastically. Two important parameters are the choice of the next node to solve and the choice of the branch variable. The three most common criteria for node selection are • Depth first • Breadth first • Best first In depth first, the next node to solve is chosen as one of the child nodes of the current node. This process is continued until a node is pruned. After a node is pruned the socalled backtracking starts.

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