The Sherman-Morrison Formula for the Determinant and its Application for Optimizing Quadratic Functions on Condition Sets Given by Extreme Generators
First a short survey is made of formulas, which deal with either the inverse, or the determinant of perturbed matrices, when a given matrix is modified with a scalar multiple of a dyad or a finite sum of dyads. By applying these formulas, an algorithmic solution will be developed for optimizing general (i. e. nonconcave, nonconvex) quadratic functions on condition sets given by extreme generators. (In other words: the condition set is given by its internal representation.) The main idea of our algorithm is testing copositivity of parametral matrices.
Keywordscopositive matrix determinant calculus extreme generator matrix inversion matrix perturbation optimization parametral matrix quadratic programming
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- 1.E. Bodewig, Matrix calculus, North-Holland Publishing Co., Amsterdam, 1956.Google Scholar
- 8.W. J. Duncan, Some devices for the solution of large sets of simultaneous linear equations, Phil. Mag., 35:660–670,1944.Google Scholar
- 11.G. Kéri, On a class of quadratic forms, in: A. Prékopa, ed., Survey of mathematical programming, 231–247, Akadémiai Kiadó, Budapest, 1979.Google Scholar
- 14.J. Sherman and W. J. Morrison, Adjustment of an inverse matrix corresponding to changes in the elements of a given column or a given row of the original matrix, Ann. Math. Statist., 20:621,1949.Google Scholar
- 17.M. A. Woodbury, Inverting modified matrices, Statist. Res. Group, Mem. Rep., No. 42., Princeton Univ., Princeton, N. J., 1950.Google Scholar