Iterative Solution of Augmented Systems Arising in Interior Methods
Summary (1 min read)
1.3. Notation and assumptions.
- It is often the case in practice that the equations and variables corresponding to unit rows of A are eliminated directly from the KKT system.
- This elimination creates no additional nonzero elements and provides a smaller "partially condensed" system with an Ω(1/μ) diagonal term added to H.
- It will be shown that preconditioners for both the full and partially condensed KKT systems depend on the eigenvalues of the same matrix (see Lemmas 3.4 and 3.5) .
- It follows that their analysis also applies to preconditioners defined for the partially condensed system.
By successively replacing H by
- Fixing μ defines a regularization of the problem, which allows the formulation of methods that do not require an assumption on the rank of the equality constraint Jacobian.
- With an appropriate choice of constraints, this feature can be used to guarantee that the nonlinear functions and their derivatives are well defined at all points generated by the interior method.
- Note that the authors cannot compute the condensed or doubly augmented system for these equation because of the zero block.
- Then, provided that the constraint preconditioner is applied exactly at every PCG step, the right-hand side of (5.3) will remain zero for all subsequent iterations.
6. Some numerical examples.
- To illustrate the numerical performance of the proposed preconditioners, a PCG method was applied to a collection of illustrative large sparse KKT systems.
- The test matrices were generated from a number of realistic KKT systems arising in the context of primal-dual interior methods.
- The authors conclude with some randomly generated problems that illustrate some of the properties of the preconditioned matrices.
6.2. Results from randomly generated problems.
- The authors expect that the proposed preconditioners would asymptotically give a cluster of 700 unit eigenvalues.
- Table 6 .3 was generated with the same data used for Table 6 .2, with the one exception that strict complementarity was assumed not to hold.
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Cites background from "Iterative Solution of Augmented Sys..."
...Forsgren, Gill and Griffin [10] extend constraint-based preconditioners to deal with regularized saddle point systems using an approximation of the (1, 1) block coupled with an augmenting term (related to a product with the constraint matrix and regularized (2, 2) block)....
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Frequently Asked Questions (16)
Q2. What is the definition of a linearized equality constraint?
zero elements of G are associated with linearized equality constraints, where the corresponding subset of equations (5.1) are the Newton equations for a zero of the constraint residual.
Q3. What is the simplest way to solve the interior-point method?
The interior-point method requires the solution of systems with a KKT matrix of the form(6.1)( H −JT−J −Γ) ,where H is the n × n Hessian of the Lagrangian, J is the m × n Jacobian matrix of constraint gradients, and Γ is a positive-definite diagonal with some large and smallCopyright © by SIAM.
Q4. What is the simplest way to solve the first system?
If the zero elements of G are associated with linear constraints, and the system (5.3) is solved exactly, it suffices to compute the special step y only once, when solving the first system.
Q5. Why do the authors prefer to do the analysis in terms of the doubly augmented system?
The authors prefer to do the analysis in terms of the doubly augmented system because it provides the parameterization based on the scalar parameter ν.Copyright © by SIAM.
Q6. What is the right-hand side of (5.3)?
provided that the constraint preconditioner is applied exactly at every PCG step, the right-hand side of (5.3) will remain zero for all subsequent iterations.
Q7. What is the main reason for the choice of C and B?
It should be emphasized that the choice of C and B affects only the efficiency of the active-set constraint preconditioners and not the definition of the linear equations that need to be solved.
Q8. What is the advantage of using preconditioning in conjunction with the doubly augmented system?
An advantage of using preconditioning in conjunction with the doubly augmented system is that the linear equations used to apply the preconditioner need not be solved exactly.
Q9. What is the preconditioner for P 2P()?
The preconditioner (4.5) has the factorization P 1P(ν) = RPP 2 P(ν)R T P , where RPis the upper-triangular matrix (4.2a) and P 2P(ν) is given by(4.6) P 2P(ν) = ⎛⎝M + (1 + ν)ATCD−1C AC νATCνAC νDC νDB ⎞⎠ .
Q10. How many eigenvalues would be given to the preconditioners?
In this strict-complementarity case, the authors expect that the proposed preconditioners would asymptotically give a cluster of 700 unit eigenvalues.
Q11. What is the PCG method applied to a generic symmetric system?
Consider the PCG method applied to a generic symmetric system Ax = b with symmetric positive-definite preconditioner P and initial iterate x0 = 0.
Q12. What is the simplest way to generate the test matrices?
The data for the test matrices was generated using a primaldual trust-region method (see, e.g., [16, 20, 32]) applied to eight problems, Camshape, Channel, Gasoil, Marine, Methanol, Pinene, Polygon, and Tetra, from the COPS 3.0 test collection [6, 8, 9, 10].
Q13. How many eigenvalues of the preconditioned matrix will cluster close to unity?
Theorems 3.6 and 4.1 predict that for the preconditioners P (1) and P 1P(1), 700 (= m + rank(AS)) eigenvalues of the preconditioned matrix will cluster close to unity, with 600 of these eigenvalues exactly equal to one.
Q14. What is the difference between the eigenvalues of SC and S?
Since SC is independent of ν, the spectrum of P 1P(ν)−1B(ν) is also independent of ν. Lemma 3.5 implies that SC has at least rank(AS) eigenvalues that are 1 + O ( μ1/2 ) , which establishes part (b).To establish part (c), the authors need to estimate the difference between the eigenvalues of SC and S, where S is given by (3.3a).
Q15. How many variables would need to be factored instead of a matrix of dimension 100,?
For their example with 100 variables and 100,000 inequality constraints, a matrix of dimension 150 would need to be factored instead of a matrix of dimension 100,100.
Q16. What is the simplest way to show that the m1 row indices in S?
Without loss of generality it may be assumed that the rows of A are ordered so that the m1 row indices in S corresponding to linearlyCopyright © by SIAM.