7 June 1995 Semidefinite programming for quadratically constrained quadratic programs
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We consider the linear least squares problem subject to multiple quadratic constraints, which is motivated by a practical application in controller design. We use the techniques of convex optimization, in particluar, interior-point methods for semi-definite programming. We reduce a quasi-convex potential function. Each iteration requires calculating a primal and dual search direction and minimizing along the plane defined by these search directions. The primal search direction requires solving a least squares problem whose matrix is composed of a block- Toeplitz portion plus other structured matrices. We make use of Kronecker products and FFTs to greatly reduce the calculation. In addition, the matrix updates and matrix inverses in the plane search are actually low-rank updates to structured matrices so we are able to further reduce the flops required. Consequently, we can design controllers for problems of considerable size.
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Julia A. Olkin, Paul James Titterton, "Semidefinite programming for quadratically constrained quadratic programs", Proc. SPIE 2563, Advanced Signal Processing Algorithms, (7 June 1995); doi: 10.1117/12.211397; https://doi.org/10.1117/12.211397


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