Compact quasi-Newton representation

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 often used for large problems and constrained optimization.

The compact representation of a quasi-Newton matrix for the inverse Hessian ${\displaystyle H_{k}}$ or direct Hessian ${\displaystyle B_{k}}$ of a nonlinear objective function ${\displaystyle f(x):\mathbb {R} ^{n}\to \mathbb {R} }$ expresses a sequence of recursive rank-1 or rank-2 matrix updates as one rank-${\displaystyle k}$ or rank-${\displaystyle 2k}$ update of an initial matrix.^[1][2] Because it is derived from quasi-Newton updates, it uses differences of iterates and gradients ${\displaystyle \nabla f(x_{k})=g_{k}}$ in its definition ${\displaystyle \{s_{i-1}=x_{i}-x_{i-1},y_{i-1}=g_{i}-g_{i-1}\}_{i=1}^{k}}$. In particular, for ${\displaystyle r=k}$ or ${\displaystyle r=2k}$ the rectangular ${\displaystyle n\times r}$ matrices ${\displaystyle U_{k},J_{k}}$ and the ${\displaystyle r\times r}$ square symmetric systems ${\displaystyle M_{k},N_{k}}$ depend on the ${\displaystyle s_{i},y_{i}}$'s and define the quasi-Newton representations

${\displaystyle H_{k}=H_{0}+U_{k}M_{k}^{-1}U_{k}^{T},\quad {\text{ and }}\quad B_{k}=B_{0}+J_{k}N_{k}^{-1}J_{k}^{T}}$

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 ${\displaystyle g\in \mathbb {R} ^{n}}$ is:

${\displaystyle {\begin{aligned}p_{k}^{(0)}&=J_{k}^{T}g\\{\text{solve}}\quad N_{k}p_{k}^{(1)}&=p_{k}^{(0)}\quad \quad {\text{(}}N_{k}{\text{ is small)}}\\p_{k}^{(2)}&=J_{k}p_{k}^{(1)}\\p_{k}^{(3)}&=H_{0}g\\p_{k}^{\phantom {(4)}}&=p_{k}^{(2)}+p_{k}^{(3)}\end{aligned}}}$

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 ${\displaystyle k}$ identity plus rank-1 matrices.^[7] Specifically, for ${\textstyle S_{k}={\begin{bmatrix}s_{0}&s_{1}&\ldots s_{k-1}\end{bmatrix}},}$ ${\displaystyle ~Y_{k}={\begin{bmatrix}y_{0}&y_{1}&\ldots y_{k-1}\end{bmatrix}},}$ ${\displaystyle ~(R_{k})_{ij}=s_{i-1}^{T}y_{j-1},}$ ${\displaystyle ~\rho _{i-1}=1/s_{i-1}^{T}y_{i-1}}$ and ${\textstyle ~V_{i}=I-\rho _{i-1}y_{i-1}s_{i-1}^{T}}$ when ${\displaystyle 1\leq i\leq j\leq k}$ the product of ${\displaystyle k}$ rank-1 updates to the identity is $${\displaystyle \prod _{i=1}^{k}V_{i-1}=\left(I-\rho _{0}y_{0}s_{0}^{T}\right)\cdots \left(I-\rho _{k-1}y_{k-1}s_{k-1}^{T}\right)=I-Y_{k}R_{k}^{-1}S_{k}^{T}}$$ The BFGS update can be expressed in terms of products of the ${\displaystyle V_{i}}$'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 ${\displaystyle v_{k}}$ and ${\displaystyle c_{k}}$ such that ${\displaystyle v_{k}^{T}y_{k}\neq 0}$ and ${\displaystyle c_{k}^{T}s_{k}\neq 0}$ general recursive update formulas for the inverse and direct Hessian estimates are

By making specific choices for the parameter vectors ${\displaystyle v_{k}}$ and ${\displaystyle c_{k}}$ well known methods are recovered

Table 1: Quasi-Newton updates parametrized by vectors ${\displaystyle v_{k}}$ and ${\displaystyle c_{k}}$

${\displaystyle v_{k}}$	${\displaystyle {\text{method}}}$	${\displaystyle c_{k}}$	${\displaystyle {\text{method}}}$
${\displaystyle s_{k}}$	BFGS	${\displaystyle s_{k}}$	PSB (Powell Symmetric Broyden)
${\displaystyle y_{k}}$	${\displaystyle {\text{Greenstadt's}}}$	${\displaystyle y_{k}}$	DFP
${\displaystyle s_{k}-H_{k}y_{k}}$	SR1	${\displaystyle y_{k}-B_{k}s_{k}}$	SR1
		${\displaystyle P_{k}^{\text{S}}s_{k}}$[10]	MSS (Multipoint-Symmetric-Secant)

Collecting the updating vectors of the recursive formulas into matrices, define

$${\displaystyle S_{k}={\begin{bmatrix}s_{0}&s_{1}&\ldots &s_{k-1}\end{bmatrix}},}$$ $${\displaystyle Y_{k}={\begin{bmatrix}y_{0}&y_{1}&\ldots &y_{k-1}\end{bmatrix}},}$$ $${\displaystyle V_{k}={\begin{bmatrix}v_{0}&v_{1}&\ldots &v_{k-1}\end{bmatrix}},}$$ $${\displaystyle C_{k}={\begin{bmatrix}c_{0}&c_{1}&\ldots &c_{k-1}\end{bmatrix}},}$$

upper triangular

$${\displaystyle {\big (}R_{k}{\big )}_{ij}:={\big (}R_{k}^{\text{SY}}{\big )}_{ij}=s_{i-1}^{T}y_{j-1},\quad {\big (}R_{k}^{\text{VY}}{\big )}_{ij}=v_{i-1}^{T}y_{j-1},\quad {\big (}R_{k}^{\text{CS}}{\big )}_{ij}=c_{i-1}^{T}s_{j-1},\quad \quad {\text{ for }}1\leq i\leq j\leq k}$$

lower triangular

$${\displaystyle {\big (}L_{k}{\big )}_{ij}:={\big (}L_{k}^{\text{SY}}{\big )}_{ij}=s_{i-1}^{T}y_{j-1},\quad {\big (}L_{k}^{\text{VY}}{\big )}_{ij}=v_{i-1}^{T}y_{j-1},\quad {\big (}L_{k}^{\text{CS}}{\big )}_{ij}=c_{i-1}^{T}s_{j-1},\quad \quad {\text{ for }}1\leq j<i\leq k}$$

and diagonal

$${\displaystyle (D_{k})_{ij}:={\big (}D_{k}^{\text{SY}}{\big )}_{ij}=s_{i-1}^{T}y_{j-1},\quad \quad {\text{ for }}1\leq i=j\leq k}$$

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]

$${\displaystyle U_{k}={\begin{bmatrix}V_{k}&S_{k}-H_{0}Y_{k}\end{bmatrix}}}$$

$${\displaystyle M_{k}={\begin{bmatrix}0_{k\times k}&R_{k}^{\text{VY}}\\{\big (}R_{k}^{\text{VY}}{\big )}^{T}&R_{k}+R_{k}^{T}-(D_{k}+Y_{k}^{T}H_{0}Y_{k})\end{bmatrix}}}$$

and the formula for the direct Hessian is

$${\displaystyle J_{k}={\begin{bmatrix}C_{k}&Y_{k}-B_{0}S_{k}\end{bmatrix}}}$$

$${\displaystyle N_{k}={\begin{bmatrix}0_{k\times k}&R_{k}^{\text{CS}}\\{\big (}R_{k}^{\text{CS}}{\big )}^{T}&R_{k}+R_{k}^{T}-(D_{k}+S_{k}^{T}B_{0}S_{k})\end{bmatrix}}}$$

For instance, when ${\displaystyle V_{k}=S_{k}}$ 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

${\displaystyle H_{k}=H_{0}+U_{k}M_{k}^{-1}U_{k}^{T},\quad U_{k}={\begin{bmatrix}S_{k}&H_{0}Y_{k}\end{bmatrix}},\quad M_{k}^{-1}=\left[{\begin{smallmatrix}R_{k}^{-T}(D_{k}+Y_{k}^{T}H_{0}Y_{k})R_{k}^{-1}&-R_{k}^{-T}\\-R_{k}^{-1}&0\end{smallmatrix}}\right]}$ The direct Hessian approximation can be found by applying the Sherman-Morrison-Woodbury identity to the inverse Hessian:

${\displaystyle B_{k}=B_{0}+J_{k}N_{k}^{-1}J_{k}^{T},\quad J_{k}={\begin{bmatrix}B_{0}S_{k}&Y_{k}\end{bmatrix}},\quad N_{k}=\left[{\begin{smallmatrix}S^{T}B_{0}S_{k}&L_{k}\\L_{k}^{T}&-D_{k}\end{smallmatrix}}\right]}$

The SR1 (Symmetric Rank-1) compact representation was first proposed in.^[7] Using the definitions of ${\displaystyle D_{k},L_{k}}$ and ${\displaystyle R_{k}}$ from above, the inverse Hessian formula is given by

${\displaystyle H_{k}=H_{0}+U_{k}M_{k}^{-1}U_{k}^{T},\quad U_{k}=S_{k}-H_{0}Y_{k},\quad M_{k}=R_{k}+R_{k}^{T}-D_{k}-Y_{k}^{T}H_{0}Y_{k}}$

The direct Hessian is obtained by the Sherman-Morrison-Woodbury identity and has the form

${\displaystyle B_{k}=B_{0}+J_{k}N_{k}^{-1}J_{k}^{T},\quad J_{k}=Y_{k}-B_{0}S_{k},\quad N_{k}=D_{k}+L_{k}+L_{k}^{T}-S_{k}^{T}B_{0}S_{k}}$

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]

${\displaystyle B_{k}=B_{0}+J_{k}N_{k}^{-1}J_{k}^{T},\quad J_{k}={\begin{bmatrix}S_{k}&Y_{k}-B_{0}S_{k}\end{bmatrix}},\quad N_{k}=\left[{\begin{smallmatrix}W_{k}(S_{k}^{T}B_{0}S_{k}-(R_{k}-D_{k}+R_{k}^{T}))W_{k}&W_{k}\\W_{k}&0\end{smallmatrix}}\right]^{-1},\quad W_{k}=(S_{k}^{T}S_{k})^{-1}}$

Another equivalent compact representation for the MSS matrix is derived by rewriting ${\displaystyle J_{k}}$ in terms of ${\displaystyle J_{k}={\begin{bmatrix}S_{k}&B_{0}Y_{k}\end{bmatrix}}}$.^[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 ${\displaystyle H_{k}\leftrightarrow B_{k}}$, ${\displaystyle H_{0}\leftrightarrow B_{0}}$ and ${\displaystyle y_{k}\leftrightarrow s_{k}}$ 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 ${\displaystyle C_{k}=S_{k}}$ in (5)

${\displaystyle B_{k}=B_{0}+J_{k}N_{k}^{-1}J_{k}^{T},\quad J_{k}={\begin{bmatrix}S_{k}&Y_{k}-B_{0}S_{k}\end{bmatrix}},\quad N_{k}=\left[{\begin{smallmatrix}0&R_{k}^{\text{SS}}\\(R_{k}^{\text{SS}})^{T}&R_{k}+R_{k}^{T}-(D_{k}+S_{k}^{T}B_{0}S_{k})\end{smallmatrix}}\right]}$

For structured optimization problems in which the objective function can be decomposed into two parts ${\displaystyle f(x)={\widehat {k}}(x)+{\widehat {u}}(x)}$, where the gradients and Hessian of ${\displaystyle {\widehat {k}}(x)}$ are known but only the gradient of ${\displaystyle {\widehat {u}}(x)}$ is known, structured BFGS formulas exist. The compact representation of these methods has the general form of (5), with specific ${\displaystyle J_{k}}$ and ${\displaystyle N_{k}}$.^[17]

The reduced compact representation (RCR) of BFGS is for linear equality constrained optimization ${\displaystyle {\text{ minimize }}f(x){\text{ subject to: }}Ax=b}$, where ${\displaystyle A}$ is underdetermined. In addition to the matrices ${\displaystyle S_{k},Y_{k}}$ the RCR also stores the projections of the ${\displaystyle y_{i}}$'s onto the nullspace of ${\displaystyle A}$

$${\displaystyle Z_{k}={\begin{bmatrix}z_{0}&z_{1}&\cdots z_{k-1}\end{bmatrix}},\quad z_{i}=Py_{i},\quad P=I-A(A^{T}A)^{-1}A^{T},\quad 0\leq i\leq k-1}$$

For ${\displaystyle B_{k}}$ the compact representation of the BFGS matrix (with a multiple of the identity ${\displaystyle B_{0}}$) the (1,1) block of the inverse KKT matrix has the compact representation^[18]

${\displaystyle K_{k}={\begin{bmatrix}B_{k}&A^{T}\\A&0\end{bmatrix}},\quad B_{0}={\frac {1}{\gamma _{k}}}I,\quad H_{0}=\gamma _{k}I,\quad \gamma _{k}>0}$

${\displaystyle {\big (}K_{k}^{-1}{\big )}_{11}=H_{0}+U_{k}M_{k}^{-1}U_{k}^{T},\quad U_{k}={\begin{bmatrix}A^{T}&S_{k}&Z_{k}\end{bmatrix}},\quad M_{k}=\left[{\begin{smallmatrix}-AA^{T}/\gamma _{k}&\\&G_{k}\end{smallmatrix}}\right],\quad G_{k}=\left[{\begin{smallmatrix}R_{k}^{-T}(D_{k}+Y_{k}^{T}H_{0}Y_{k})R_{k}^{-1}&-H_{0}R_{k}^{-T}\\-H_{0}R_{k}^{-1}&0\end{smallmatrix}}\right]^{-1}}$

The most common use of the compact representations is for the limited-memory setting where ${\displaystyle m\ll n}$ denotes the memory parameter, with typical values around ${\displaystyle m\in [5,12]}$ (see e.g.,^[18][7]). Then, instead of storing the history of all vectors one limits this to the ${\displaystyle m}$ most recent vectors ${\displaystyle \{(s_{i},y_{i}\}_{i=k-m}^{k-1}}$ and possibly ${\displaystyle \{v_{i}\}_{i=k-m}^{k-1}}$ or ${\displaystyle \{c_{i}\}_{i=k-m}^{k-1}}$. Further, typically the initialization is chosen as an adaptive multiple of the identity ${\displaystyle H_{k}^{(0)}=\gamma _{k}I}$, with ${\displaystyle \gamma _{k}=y_{k-1}^{T}s_{k-1}/y_{k-1}^{T}y_{k-1}}$ and ${\displaystyle B_{k}^{(0)}={\frac {1}{\gamma _{k}}}I}$. Limited-memory methods are frequently used for large-scale problems with many variables (i.e., ${\displaystyle n}$ can be large), in which the limited-memory matrices ${\displaystyle S_{k}\in \mathbb {R} ^{n\times m}}$ and ${\displaystyle Y_{k}\in \mathbb {R} ^{n\times m}}$ (and possibly ${\displaystyle V_{k},C_{k}}$) are tall and very skinny: ${\displaystyle S_{k}={\begin{bmatrix}s_{k-l-1}&\ldots &s_{k-1}\end{bmatrix}}}$ and ${\displaystyle Y_{k}={\begin{bmatrix}y_{k-l-1}&\ldots &y_{k-1}\end{bmatrix}}}$.

Open source implementations include:

• ACM TOMS algorithm 1030 implements a L-SR1 solver^[19][20]
• 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) solvers use compact quasi-Newton matrices ^[3]
• L-BFGS-B (ACM TOMS algorithm 778)^[21]