By Janne Harju Johansson.
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Extra resources for A structure utilizing inexact primal-dual interior-point method for analysis of linear differential inclusions
Then • X ≻ 0 if and only if A ≻ 0 and S ≻ 0. 2 0 if and only if S 0. Lyapunov Stability Around year 1890 Lyapunov presented the famous theory regarding stability of dynamical systems, which we now denote Lyapunov theory. 1) is stable if and only if there exist a positive definite matrix P ∈ Sn , such that AT P + P A ≺ 0. 2) This stability analysis has been extended to a class of more general system descriptions. 3) where Ω ⊆ Rn×n , one can search for a quadratic Lyapunov function, to prove stability.
3 29 Iterative Methods Using direct methods is often utilized in optimization algorithms and result in numerically stable and efficient implementations. However, there is one aspect when using direct solvers for linear systems of equations which can deteriorate performance. The factorization is a memory consuming process. If the number of variables n is large, the use of factorization might be impossible. Storage of factorization matrices and the intermediate steps require memory that is not always available on an ordinary work station.
2005). 14) T Rnt ZRnt = λ. 15) and 22 3 Primal-Dual Interior-Point Methods Algorithm 2 Computing the NT scaling matrix 1: Compute the Cholesky factorizations: S = L1 LT1 Z = L2 LT2 2: Compute the singular value decomposition (SVD): LT2 L1 = U λV T where λ is a positive definite and diagonal matrix and U and V are unitary matrices. 16c) for some residuals D1 , D2 and D3 . , infeasible or feasible method. Now an important lemma that makes the solution of the search directions well-defined is presented.