The compact representation of a quasi-Newton matrix for the inverse Hessian or
direct Hessian of a nonlinear objective function expresses a sequence of recursive rank-1
or rank-2 matrix updates as one rank- or rank- update of an initial matrix.[1][2]
Because it is derived from quasi-Newton updates,
it uses differences of iterates and gradients in its definition
.
In particular, for or the rectangular matrices and the square symmetric systems depend on the 's and define the quasi-Newton representations
Because of the special matrix decomposition the compact representation is implemented in
state-of-the-art optimization software.[3][4][5][6]
When combined with limited-memory techniques it is a popular technique for constrained optimization with gradients.[7]
Linear algebra operations can be done efficiently, like matrix-vector products, solves or eigendecompositions. It can be combined
with line-search and trust region techniques, and the representation has been developed for many quasi-Newton
updates. For instance, the matrix vector product with the direct quasi-Newton Hessian and an arbitrary
vector is:
In the context of the GMRES method, Walker[8]
showed that a product of Householder transformations (an identity plus rank-1) can be expressed as
a compact matrix formula. This led to the derivation
of an explicit matrix expression for the product of identity plus rank-1 matrices.[7]
Specifically, for
and
when
the product of rank-1 updates to the identity is
The BFGS update can be expressed in terms of products of the 's, which
have a compact matrix formula. Therefore, the BFGS recursion can exploit these block matrix
representations
A parametric family of quasi-Newton updates includes many of the most known formulas.[9] For
arbitrary vectors and such that and
general recursive update formulas for the inverse and direct Hessian
estimates are
(2)
(3)
By making specific choices for the parameter vectors and well known
methods are recovered
Table 1: Quasi-Newton updates parametrized by vectors and
Collecting the updating vectors of the recursive formulas into matrices, define
upper triangular
lower triangular
and diagonal
With these definitions the compact representations of general rank-2 updates in (2) and (3) (including the well known quasi-Newton updates in Table 1)
have been developed in Brust:[11]
(4)
and the formula for the direct Hessian is
(5)
For instance, when the representation in (4) is
the compact formula for the BFGS recursion in (1).
Prior to the development of the compact representations of (2) and (3),
equivalent representations have been discovered for most known updates (see Table 1).
Along with the SR1 representation, the BFGS (Broyden-Fletcher-Goldfarb-Shanno) compact representation was the first compact formula known.[7] In particular, the inverse representation is
given by
The SR1 (Symmetric Rank-1) compact representation was first proposed in.[7] Using the definitions of
and from above, the inverse Hessian formula is given by
The direct Hessian is obtained by the Sherman-Morrison-Woodbury identity and has the form
The multipoint symmetric secant (MSS) method is a method that aims to satisfy multiple secant equations. The recursive
update formula was originally developed by Burdakov.[12] The compact representation for the direct Hessian was derived in [13]
Another equivalent compact representation for the MSS matrix is derived by rewriting
in terms of .[14]
The inverse representation can be obtained by application for the Sherman-Morrison-Woodbury identity.
Since the DFP (Davidon Fletcher Powell) update is the dual of the BFGS formula (i.e., swapping , and in the BFGS update), the compact representation for DFP can be immediately obtained from the one for BFGS.[15]
The PSB (Powell-Symmetric-Broyden) compact representation was developed for the direct Hessian approximation.[16] It is equivalent to substituting in (5)
For structured optimization problems in which the objective function can be decomposed into two parts
, where the gradients and Hessian of
are known but only the gradient of is known, structured BFGS formulas
exist. The compact representation of these methods has the general form of (5),
with specific and .[17]
The reduced compact representation (RCR) of BFGS is for linear equality constrained optimization
, where is underdetermined. In addition to the matrices
the RCR also stores the projections of the 's onto the nullspace of
For the compact representation of the BFGS matrix (with a multiple of the identity ) the (1,1) block of the inverse KKT matrix has the compact representation[18]
The most common use of the compact representations is for the limited-memory setting where denotes the memory parameter,
with typical values around (see e.g., [18][7]). Then, instead
of storing the history of all vectors one limits this to the most recent vectors and possibly or .
Further, typically the initialization is chosen as an adaptive multiple of the identity ,
with and . Limited-memory methods are frequently used for
large-scale problems with many variables (i.e., can be large), in which the limited-memory matrices
and (and possibly ) are tall and very skinny:
and .
^Zhu, C.; Byrd, R. H.; Lu, P.; Nocedal, J. (1997). "Algorithm 778: L-BFGS-B: Fortran subroutines for large-scale bound-constrained optimization". ACM Transactions on Mathematical Software (TOMS). 23 (4): 550-560. doi:10.1145/279232.279236.
^Brust, J.; Burdakov, O.; Erway, J.; Marcia, R. (2022). "Algorithm 1030: SC-SR1: MATLAB software for limited-memory SR1 trust-region methods". ACM Transactions on Mathematical Software (TOMS). 48 (4): 1-33. doi:10.1145/3550269.
^
Wächter, A.; Biegler, L. T. (2006). "On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming". Mathematical Programming. 106: 25-57. doi:10.1007/s10107-004-0559-y.
^ abcdeByrd, R. H.; Nocedal, J.; Schnabel, R. B. (1994). "Representations of Quasi-Newton Matrices and their use in Limited Memory Methods". Mathematical Programming. 63 (4): 129–156. doi:10.1007/BF01582063. S2CID5581219.
^Walker, H. F. (1988). "Implementation of the GMRES Method Using Householder Transformations". SIAM Journal on Scientific and Statistical Computing. 9 (1): 152–163. doi:10.1137/0909010.
^Dennis, Jr, J. E.; Moré, J. J. (1977). "Quasi-Newton methods, motivation and theory". SIAM Review. 19 (1): 46-89. doi:10.1137/1019005. hdl:1813/6056.{{cite journal}}: CS1 maint: multiple names: authors list (link)
^Brust, J. J. (2024). "Useful Compact Representations for Data-Fitting". arXiv:2403.12206 [math.OC].
^Burdakov, O. P. (1983). "Methods of the secant type for systems of equations with symmetric Jacobian matrix". Numerical Functional Analysis and Optimization. 6 (2): 1–18. doi:10.1080/01630568308816160.
^Burdakov, O. P.; Martínez, J. M.; Pilotta, E. A. (2002). "A limited-memory multipoint symmetric secant method for bound constrained optimization". Annals of Operations Research. 117: 51–70. doi:10.1023/A:1021561204463.
^Brust, J. J.; Erway, J. B.; Marcia, R. F. (2024). "Shape-changing trust-region methods using multipoint symmetric secant matrices". Optimization Methods and Software: 1–18. arXiv:2209.12057. doi:10.1080/10556788.2023.2296441.
^Erway, J. B.; Jain, V.; Marcia, R. F. (2013). Shifted limited-memory DFP systems. In 2013 Asilomar Conference on Signals, Systems and Computers. IEEE. pp. 1033–1037.
^Brust, J. J; Di, Z.; Leyffer, S.; Petra, C. G. (2021). "Compact representations of structured BFGS matrices". Computational Optimization and Applications. 80 (1): 55–88. doi:10.1007/s10589-021-00297-0.
^ abBrust, J. J; Marcia, R.F.; Petra, C.G.; Saunders, M. A. (2022). "Large-scale optimization with linear equality constraints using reduced compact representation". SIAM Journal on Scientific Computing. 44 (1): A103–A127. arXiv:2101.11048. Bibcode:2022SJSC...44A.103B. doi:10.1137/21M1393819.