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
A more elaborate switching technique could improve the convergence but for practical use, the result is satisfactory. 5 53 Numerical Evaluation changed. This information is obtained from the main algorithm. To obtain the solution times, the Matlab command cputime is used. Input to the solvers are the system matrices, so any existing preprocessing of the problem is included in the total solution time. In order to monitor the progress of the algorithm the residual for the search directions is calculated in each iteration in the iterative solver.
5d) is required to be satisfied in an infeasible method. 5d) defining the central-path. This results in A(∆x) − ∆S = −(A(x) + A0 − S) ∗ ∗ A (∆Z) = c − A (Z) ∆ZS + Z∆S + ∆Z∆S = τ In − ZS S + ∆S 0, Z + ∆Z 0. 7d) It is important to notice that an infeasible method normally tries to get feasible. 7d) is solved exactly. Although the use of an infeasible method simplifies the choice of an initial point, the choice is not without importance. This issue is a well-investigated problem, and there are trade-offs that have to be made.
4 51 Preconditioning To derive Preconditioner II, assume that Hi−1 Ni ≈ 0 is valid for all i = s. As an intermediate result, note that i NiT Hi−1 Ni ≈ NsT Hs−1 Ns . 24) svec(D2 ) D3 i=s −1 svec(D1i ) − Ni Hi svec(∆P ) ∆x , i=s . 19). 25) is the solution to H1 .. . Hs Fs Gs ∆ = Dvec . 26) . Hni FsT GTs This linear system of equations have a nice structure. ∆P , ∆x and ∆Zs can be found by solving a system of equations, as if we had a single constraint.