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The compact representation for quasi-Newton methods is a matrix decomposition, which is typically used in gradient based optimization algorithms or for solving nonlinear systems. The decomposition uses a low-rank representation for the direct and/or inverse Hessian or the Jacobian of a nonlinear system. Because of this, the compact representation is particularly suited for large problems and constrained optimization.
![The compact matrix decomposition of a dense Hessian approximation](http://upload.wikimedia.org/wikipedia/commons/thumb/0/0a/Main-figure0.pdf/page1-390px-Main-figure0.pdf.jpg)
Definition
editThe compact representation of a quasi-Newton matrix for the inverse Hessian or direct Hessian of a nonlinear objective function is based on expressing a sequence of recursive rank-1 or rank-2 matrix updates as one rank- or rank- update of an initial matrix [1]. 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
Applications
editBecause of the special matrix decomposition the compact representation is implemented in state-of-the-art optimization software [2][3][4][5]. When combined with limited-memory techniques it is a popular technique for constrained optimization with gradients [6]. Linear algebra operations can be done efficiently, like mat-vecs, 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:
Background
editWalker [7] showed that a product of Householder transformations (an identity plus rank-1) can be expressed as a compact matrix formula. This result led the authors in [6] to derive an explicit matrix expression for the product of identity plus rank-1 matrices. 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
(1) |
Recursive quasi-Newton updates
editA parametric family of quasi-Newton updates includes many of the most known formulas [8]. 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
BFGS | PSB (Powell Symmetric Broyden) | ||
DFP | |||
SR1 | SR1 | ||
[9] | MSS (Multipoint-Symmetric-Secant) |
Compact Representations
editSince many quasi-Newton methods follow from the general updates in (2) and (3) the compact representation of these updates consequently give the representation of most QN formulas. Define
upper triangular
lower triangular
and diagonal
With these definitions the compact representations of the general rank-2 updates in (2) and (3) have been developed in Brust [10]. The formula for the inverse update is
| (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). The validity of these expressions can be simply checked by numerically comparing the difference between, e.g., (2) and (4) for the same .
Prior to the development of the compact representations of (2) and (3), equivalent representations have been discovered for most known updates (see Table 1).
An equivalent compact representation for the BFGS (Broyden-Fletcher-Goldfarb-Shanno) was developed in [6]. Along with the SR1 these were the first compact formulas known. In particular, the inverse representation is given by
After simplification, this formula is exactly (4) with .
The direct Hessian approximation can be found by applying the Sherman-Morrison-Woodbury identity to the inverse Hessian:
The SR1 (Symmetric Rank-1) compact representation was first proposed in [6]. 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
MSS
editThe multipoint symmetric secant (MSS) method is a method that aims to satisfy multiple secant equations. The recursive update formula was originally developed by Burdakov [11]. The compact representation for the direct Hessian was derived in [12][1]
Another equivalent compact representation for the MSS matrix is derived by rewriting in terms of [13].
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 [14].
PSB
editThe PSB (Powell-Symmetric-Broyden) compact representation was developed for the direct Hessian approximation[15]. It is equivalent to substituting in (5)
Structured BFGS
editFor 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 [16].
Reduced BFGS
editThe 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[17]
Limited Memory
editThe 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., [17][6]). 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 .
Implementations
editOpen source implementations include:
- ACM TOMS algorithm 1030 implements a L-SR1 solver [18] [19]
- R's
optim
general-purpose optimizer routine uses the L-BFGS-B method. - SciPy's optimization module's minimize method also includes an option to use L-BFGS-B.
- IPOPT with first order information
Non open source implementations include:
- Artelys Knitro nonlinear programming (NLP) solver use compact quasi-Newton matrices [2]
- L-BFGS-B (ACM TOMS algorithm 778)[20]
Works cited
edit- ^ a b Brust, J. J. (2018). Large-Scale Quasi-Newton Trust-Region Methods: High-Accuracy Solvers, Dense Initializations, and Extensions (PhD thesis). University of California, Merced.
- ^ a b Byrd, R. H.; Nocedal, J; Waltz, R. A. (2006). KNITRO: An integrated package for nonlinear optimization. In: Di Pillo, G., Roma, M. (eds) Large-Scale Nonlinear Optimization. Nonconvex Optimization and Its Applications, vol 83.: Springer, Boston, MA. p. 35-59.
{{cite book}}
: CS1 maint: location (link) - ^ 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.
- ^ a b c d e Byrd, 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. S2CID 5581219.
- ^ Walker, H. F. (1988). "Implementation of the GMRES Method Using Householder Transformations". SIAM Journal on Scientific and Statistical Computing. 9 (1). 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.
{{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. 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.
- ^ Kanzow, C.; Steck, D. (2023). "Regularization of limited memory quasi-Newton methods for large-scale nonconvex minimization". Mathematical Programming Computation. 15 (3): 417–444. doi:10.1007/s12532-023-00238-4.
- ^ 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.
- ^ a b Brust, 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. doi:10.1137/21M1393819.
- ^ "Collected Algorithms of the ACM (CALGO)". calgo.acm.org.
- ^ "TOMS Alg. 1030". calgo.acm.org/1030.zip.
- ^ Zhu, C.; Byrd, Richard H.; Lu, Peihuang; Nocedal, Jorge (1997). "L-BFGS-B: Algorithm 778: L-BFGS-B, FORTRAN routines for large scale bound constrained optimization". ACM Transactions on Mathematical Software. 23 (4): 550–560. doi:10.1145/279232.279236. S2CID 207228122.