%!TEX root = ceres-solver.tex
\chapter{Solving}
Effective use of Ceres requires some familiarity with the basic components of a nonlinear least squares solver, so before we describe how to configure the solver, we will begin by taking a brief look at how some of the core optimization algorithms in Ceres work and the various linear solvers and preconditioners that power it.

\section{Trust Region Methods}
Let $x \in \mathbb{R}^{n}$ be an $n$-dimensional vector of variables, and
$ F(x) = \left[f_1(x),   \hdots,  f_{m}(x) \right]^{\top}$ be a $m$-dimensional function of $x$.  We are interested in solving the following optimization problem~\footnote{At the level of the non-linear solver, the block and residual structure is not relevant, therefore our discussion here is in terms of an optimization problem defined over a state vector of size $n$.},
\begin{equation}
        \arg \min_x \frac{1}{2}\|F(x)\|^2\ .
        \label{eq:nonlinsq}
\end{equation}
Here, the Jacobian $J(x)$ of $F(x)$ is an $m\times n$ matrix, where $J_{ij}(x) = \partial_j f_i(x)$  and the gradient vector $g(x) = \nabla  \frac{1}{2}\|F(x)\|^2 = J(x)^\top F(x)$. Since the efficient global optimization of~\eqref{eq:nonlinsq} for general $F(x)$ is an intractable problem, we will have to settle for finding a local minimum.

The general strategy when solving non-linear optimization problems is to solve a sequence of approximations to the original problem~\cite{nocedal2000numerical}. At each iteration, the approximation is solved to determine a correction $\Delta x$ to the vector $x$. For non-linear least squares, an approximation can be constructed by using the linearization $F(x+\Delta x) \approx F(x) + J(x)\Delta x$, which leads to the following linear least squares  problem:
\begin{equation}
         \min_{\Delta x} \frac{1}{2}\|J(x)\Delta x + F(x)\|^2
        \label{eq:linearapprox}
\end{equation}
Unfortunately, na\"ively solving a sequence of these problems and updating $x \leftarrow x+ \Delta x$ leads to an algorithm that may not converge.  To get a convergent algorithm, we need to control the size of the step $\Delta x$. And this is where the idea of a trust-region comes in. The generic trust-region loop for non-linear least squares problems looks something like this


\begin{algorithmic}
\REQUIRE Initial point $x$ and a trust region radius $\mu$.
\LOOP
\STATE{Solve $\arg \min_{\Delta x} \frac{1}{2}\|J(x)\Delta x + F(x)\|^2$ s.t. $\|D(x)\Delta x\|^2 \le \mu$}
\STATE{$\rho = \frac{\displaystyle \|F(x + \Delta x)\|^2 - \|F(x)\|^2}{\displaystyle \|J(x)\Delta x + F(x)\|^2 - \|F(x)\|^2}$}
\IF {$\rho > \epsilon$}
\STATE{$x = x + \Delta x$}
\ENDIF
\IF {$\rho > \eta_1$}
\STATE{$\rho = 2 * \rho$}
\ELSE
\IF {$\rho < \eta_2$}
\STATE {$\rho = 0.5 * \rho$}
\ENDIF
\ENDIF
\ENDLOOP
\end{algorithmic}

Here, $\mu$ is the trust region radius, $D(x)$ is some matrix used to define a metric on the domain of $F(x)$ and $\rho$ measures the quality of the step $\Delta x$, i.e., how well did the linear model predict the decrease in the value of the non-linear objective. The idea is to increase or decrease the radius of the trust region depending on how well the linearization predicts the behavior of the non-linear objective, which in turn is reflected in the value of $\rho$.

The key computational step in a trust-region algorithm is the solution of the constrained optimization problem
\begin{align}
        \arg\min_{\Delta x}& \frac{1}{2}\|J(x)\Delta x + F(x)\|^2 \\
        \text{such that}&\quad  \|D(x)\Delta x\|^2 \le \mu
\label{eq:trp}
\end{align}

There are a number of different ways of solving this problem, each giving rise to a different concrete trust-region algorithm. Currently Ceres, implements two trust-region algorithms - Levenberg-Marquardt and Powell's Dogleg.

\subsection{Levenberg-Marquardt}
The Levenberg-Marquardt algorithm~\cite{levenberg1944method, marquardt1963algorithm} is the most popular algorithm for solving non-linear least squares problems.  It was also the first trust region algorithm to be developed~\cite{levenberg1944method,marquardt1963algorithm}. Ceres implements an exact step~\cite{madsen2004methods} and an inexact step variant of the Levenberg-Marquardt algorithm~\cite{wright1985inexact,nash1990assessing}.

It can be shown, that the solution to~\eqref{eq:trp} can be obtained by solving an unconstrained optimization of the form
\begin{align}
        \arg\min_{\Delta x}& \frac{1}{2}\|J(x)\Delta x + F(x)\|^2 +\lambda  \|D(x)\Delta x\|^2
\end{align}
Where, $\lambda$ is a Lagrange multiplier that is inverse related to $\mu$. In Ceres, we solve for
\begin{align}
        \arg\min_{\Delta x}& \frac{1}{2}\|J(x)\Delta x + F(x)\|^2 + \frac{1}{\mu} \|D(x)\Delta x\|^2
\label{eq:lsqr}
\end{align}
The matrix $D(x)$ is a non-negative diagonal matrix, typically the square root of the diagonal of the matrix $J(x)^\top J(x)$.

Before going further, let us make some notational simplifications. We will assume that the matrix $\sqrt{\mu} D$ has been concatenated at the bottom of the matrix $J$ and similarly a vector of zeros has been added to the bottom of the vector $f$ and the rest of our discussion will be in terms of $J$ and $f$, \ie the linear least squares problem.
\begin{align}
 \min_{\Delta x} \frac{1}{2} \|J(x)\Delta x + f(x)\|^2 .
 \label{eq:simple}
\end{align}
For all but the smallest problems the solution of~\eqref{eq:simple} in each iteration of the Levenberg-Marquardt algorithm is the dominant computational cost in Ceres. Ceres provides a number of different options for solving~\eqref{eq:simple}. There are two major classes of methods - factorization and iterative.

The factorization methods are based on computing an exact solution of~\eqref{eq:lsqr} using a Cholesky or a QR factorization and lead to an exact step Levenberg-Marquardt algorithm. But it is not clear if an exact solution of~\eqref{eq:lsqr} is necessary at each step of the LM algorithm to solve~\eqref{eq:nonlinsq}. In fact, we have already seen evidence that this may not be the case, as~\eqref{eq:lsqr} is itself a regularized version of~\eqref{eq:linearapprox}. Indeed, it is possible to construct non-linear optimization algorithms in which the linearized problem is solved approximately. These algorithms are known as inexact Newton or truncated Newton methods~\cite{nocedal2000numerical}.

An inexact Newton method requires two ingredients. First, a cheap method for approximately solving systems of linear equations. Typically an iterative linear solver like the Conjugate Gradients method is used for this purpose~\cite{nocedal2000numerical}. Second, a termination rule for the iterative solver. A typical termination rule is of the form
\begin{equation}
        \|H(x) \Delta x + g(x)\| \leq \eta_k \|g(x)\|. \label{eq:inexact}
\end{equation}
Here, $k$ indicates the Levenberg-Marquardt iteration number and $0 < \eta_k <1$ is known as the forcing sequence.  Wright \& Holt \cite{wright1985inexact} prove that a truncated Levenberg-Marquardt algorithm that uses an inexact Newton step based on~\eqref{eq:inexact} converges for any sequence $\eta_k \leq \eta_0 < 1$ and the rate of convergence depends on the choice of the forcing sequence $\eta_k$.

Ceres supports both exact and inexact step solution strategies. When the user chooses a factorization based linear solver, the exact step Levenberg-Marquardt algorithm is used. When the user chooses an iterative linear solver, the inexact step Levenberg-Marquardt algorithm is used.

\subsection{Powell's Dogleg}
Another strategy for solving the trust region problem~\eqref{eq:trp} was introduced by M. J. D. Powell. The key idea there is to compute two vectors
\begin{align}
        \Delta x^{\text{Gauss-Newton}} &= \arg \min_{\Delta x}\frac{1}{2} \|J(x)\Delta x + f(x)\|^2.\\
        \Delta x^{\text{Cauchy}} &= -\frac{\|g(x)\|^2}{\|J(x)g(x)\|^2}g(x).
\end{align}
Note that the vector $\Delta x^{\text{Gauss-Newton}}$ is the solution to~\eqref{eq:linearapprox} and $\Delta x^{\text{Cauchy}}$ is the vector that minimizes the linear approximation if we restrict ourselves to moving along the direction of the gradient.

Then Powell's Dogleg method finds a vector $\Delta x$ in the two dimensional subspace defined by $\Delta x^{\text{Gauss-Newton}}$ and $\Delta x^{\text{Cauchy}}$ that solves the trust region problem. For more details on the exact reasoning and computations, please see Madsen et al~\cite{madsen2004methods}.

The key advantage of the Dogleg over Levenberg Marquardt is that if the step computation for a particular choice of $\mu$ does not result in sufficient decrease in the value of the objective function, Levenberg-Marquardt solves the linear approximation from scratch with a small value of $\mu$. Dogleg on the other hand, only needs to compute the interpolation between the Gauss-Newton and the Cauchy vectors, as neither of them depend on the value of $\mu$.

The Dogleg method can only be used with the exact factorization based linear solvers.

\section{\texttt{LinearSolver}}
Recall that in both of the trust-region methods described above, the key computational cost is the solution of a linear least squares problem of the form
\begin{align}
 \min_{\Delta x} \frac{1}{2} \|J(x)\Delta x + f(x)\|^2 .
 \label{eq:simple2}
\end{align}


Let $H(x)= J(x)^\top J(x)$ and $g(x) = -J(x)^\top  f(x)$. For notational convenience let us also drop the dependence on $x$. Then it is easy to see that solving~\eqref{eq:simple2} is equivalent to solving the {\em normal equations}
\begin{align}
H \Delta x  &= g \label{eq:normal}
\end{align}

Ceres provides a number of different options for solving~\eqref{eq:normal}.

\subsection{\texttt{DENSE\_QR}}
For small problems (a couple of hundred parameters and a few thousand residuals) with relatively dense Jacobians, \texttt{DENSE\_QR} is the method of choice~\cite{bjorck1996numerical}. Let $J = QR$ be the QR-decomposition of $J$, where $Q$ is an orthonormal matrix and $R$ is an upper triangular matrix~\cite{trefethen1997numerical}. Then it can be shown that the solution to~\eqref{eq:normal} is given by
\begin{align}
    \Delta x^* = -R^{-1}Q^\top f
\end{align}
Ceres uses \texttt{Eigen}'s dense QR decomposition routines.


\subsection{\texttt{SPARSE\_NORMAL\_CHOLESKY}}
Large non-linear least square problems are usually sparse. In such cases, using a dense QR factorization is inefficient. Let $H = R^\top R$ be the Cholesky factorization of the normal equations, where $R$ is an upper triangular matrix, then the  solution to ~\eqref{eq:normal} is given by
\begin{equation}
    \Delta x^* = R^{-1} R^{-\top} g.
\end{equation}
The observant reader will note that the $R$ in the Cholesky factorization of $H$ is the same upper triangular matrix $R$ in the QR factorization of $J$. Since $Q$ is an orthonormal matrix, $J=QR$ implies that $J^\top J = R^\top Q^\top Q R = R^\top R$.


There are two variants of Cholesky factorization -- sparse and dense. \texttt{SPARSE\_NORMAL\_CHOLESKY}, as the name implies performs a sparse Cholesky factorization of the normal equations. This leads to substantial savings in time and memory for large sparse problems. We use the Professor Tim Davis' \texttt{CHOLMOD} library (part of the \texttt{SuiteSparse} package) to perform the sparse cholesky~\cite{chen2006acs}.


\subsection{\texttt{DENSE\_SCHUR} \& \texttt{SPARSE\_SCHUR}}
While it is possible to use \texttt{SPARSE\_NORMAL\_CHOLESKY} to solve bundle adjustment problems, bundle adjustment problem have a special structure, and a more efficient scheme for solving~\eqref{eq:normal} can be constructed.

Suppose that the SfM problem consists of $p$ cameras and $q$ points and the variable vector $x$ has the  block structure $x = [y_{1},\hdots,y_{p},z_{1},\hdots,z_{q}]$. Where, $y$ and $z$ correspond to camera and point parameters, respectively.  Further, let the camera blocks be of size $c$ and the point blocks be of size $s$ (for most problems $c$ =  $6$--$9$ and $s = 3$). Ceres does not impose any constancy requirement on these block sizes, but choosing them to be constant simplifies the exposition.

A key characteristic of the bundle adjustment problem is that there is no term $f_{i}$ that includes two or more point blocks.  This in turn implies that the matrix $H$ is of the form
\begin{equation}
        H =  \left[
                \begin{matrix} B & E\\ E^\top & C
                \end{matrix}
                \right]\ ,
\label{eq:hblock}
\end{equation}
where, $B \in \reals^{pc\times pc}$ is a block sparse matrix with $p$ blocks of size $c\times c$ and  $C \in \reals^{qs\times qs}$ is a block diagonal matrix with $q$ blocks of size $s\times s$. $E \in \reals^{pc\times qs}$ is a general block sparse matrix, with a block of size $c\times s$ for each observation. Let us now block partition $\Delta x = [\Delta y,\Delta z]$ and $g=[v,w]$ to restate~\eqref{eq:normal} as the block structured linear system
\begin{equation}
        \left[
                \begin{matrix} B & E\\ E^\top & C
                \end{matrix}
                \right]\left[
                        \begin{matrix} \Delta y \\ \Delta z
                        \end{matrix}
                        \right]
                        =
                        \left[
                                \begin{matrix} v\\ w
                                \end{matrix}
                                \right]\ ,
\label{eq:linear2}
\end{equation}
and apply Gaussian elimination to it. As we noted above, $C$ is a block diagonal matrix, with small diagonal blocks of size $s\times s$.
Thus, calculating the inverse of $C$ by inverting each of these blocks is  cheap. This allows us to  eliminate $\Delta z$ by observing that $\Delta z = C^{-1}(w - E^\top \Delta y)$, giving us
\begin{equation}
        \left[B - EC^{-1}E^\top\right] \Delta y = v - EC^{-1}w\ .  \label{eq:schur}
\end{equation}
The matrix
\begin{equation}
S = B - EC^{-1}E^\top\ ,
\end{equation}
is the Schur complement of $C$ in $H$. It is also known as the {\em reduced camera matrix}, because the only variables participating in~\eqref{eq:schur} are the ones corresponding to the cameras. $S \in \reals^{pc\times pc}$ is a block structured symmetric positive definite matrix, with blocks of size $c\times c$. The block $S_{ij}$ corresponding to the pair of images $i$ and $j$ is non-zero if and only if the two images observe at least one common point.

Now, \eqref{eq:linear2}~can  be solved by first forming $S$, solving for $\Delta y$, and then back-substituting $\Delta y$ to obtain the value of $\Delta z$.
Thus, the solution of what was an $n\times n$, $n=pc+qs$ linear system is reduced to the inversion of the block diagonal matrix $C$, a few matrix-matrix and matrix-vector multiplies, and the solution of block sparse $pc\times pc$ linear system~\eqref{eq:schur}.  For almost all  problems, the number of cameras is much smaller than the number of points, $p \ll q$, thus solving~\eqref{eq:schur} is significantly cheaper than solving~\eqref{eq:linear2}. This is the {\em Schur complement trick}~\cite{brown-58}.

This still leaves open the question of solving~\eqref{eq:schur}. The
method of choice for solving symmetric positive definite systems
exactly is via the Cholesky
factorization~\cite{trefethen1997numerical} and depending upon the
structure of the matrix, there are, in general, two options. The first
is direct factorization, where we store and factor $S$ as a dense
matrix~\cite{trefethen1997numerical}. This method has $O(p^2)$ space complexity and $O(p^3)$ time
complexity and is only practical for problems with up to a few hundred
cameras. Ceres implements this strategy as the \texttt{DENSE\_SCHUR} solver.


 But, $S$ is typically a fairly sparse matrix, as most images
only see a small fraction of the scene. This leads us to the second
option: sparse direct methods. These methods store $S$ as a sparse
matrix, use row and column re-ordering algorithms to maximize the
sparsity of the Cholesky decomposition, and focus their compute effort
on the non-zero part of the factorization~\cite{chen2006acs}.
Sparse direct methods, depending on the exact sparsity structure of the Schur complement,
allow bundle adjustment algorithms to significantly scale up over those based on dense
factorization. Ceres implements this strategy as the \texttt{SPARSE\_SCHUR} solver.

\subsection{\texttt{CGNR}}
For general sparse problems, if the problem is too large for \texttt{CHOLMOD} or a sparse linear algebra library is not linked into Ceres, another option is the \texttt{CGNR} solver. This solver uses the Conjugate Gradients solver on the {\em normal equations}, but without forming the normal equations explicitly. It exploits the relation
\begin{align}
    H x = J^\top J x = J^\top(J x)
\end{align}
When the user chooses \texttt{ITERATIVE\_SCHUR} as the linear solver, Ceres automatically switches from the exact step algorithm to an inexact step algorithm.

%Currently only the \texttt{JACOBI} preconditioner is available for use with this solver. It uses the block diagonal of $H$ as a preconditioner.


\subsection{\texttt{ITERATIVE\_SCHUR}}
Another option for bundle adjustment problems is to apply PCG to the reduced camera matrix $S$ instead of $H$. One reason to do this is that $S$ is a much smaller matrix than $H$, but more importantly, it can be shown that $\kappa(S)\leq \kappa(H)$.  Ceres implements PCG on $S$ as the \texttt{ITERATIVE\_SCHUR} solver. When the user chooses \texttt{ITERATIVE\_SCHUR} as the linear solver, Ceres automatically switches from the exact step algorithm to an inexact step algorithm.

The cost of forming and storing the Schur complement $S$ can be prohibitive for large problems. Indeed, for an inexact Newton solver that computes $S$ and runs PCG on it, almost all of its time is spent in constructing $S$; the time spent inside the PCG algorithm is negligible in comparison. Because  PCG only needs access to $S$ via its product with a vector, one way to evaluate $Sx$ is to observe that
\begin{align}
  x_1 &= E^\top x \notag \\
  x_2 &= C^{-1} x_1 \notag\\
  x_3 &= Ex_2 \notag\\
  x_4 &= Bx \notag\\
  Sx &= x_4 - x_3\ .\label{eq:schurtrick1}
\end{align}
Thus, we can run PCG on $S$ with the same computational effort per iteration as PCG on $H$, while reaping the benefits of a more powerful preconditioner. In fact, we do not even need to compute $H$, \eqref{eq:schurtrick1} can be implemented using just the columns of $J$.

Equation~\eqref{eq:schurtrick1} is closely related to {\em Domain Decomposition methods} for solving large linear systems that arise in structural engineering and partial differential equations. In the language of Domain Decomposition, each point in a bundle adjustment problem is a domain, and the cameras form the interface between these domains. The iterative solution of the Schur complement then falls within the sub-category of techniques known as Iterative Sub-structuring~\cite{saad2003iterative,mathew2008domain}.

\section{Preconditioner}
The convergence rate of Conjugate Gradients  for solving~\eqref{eq:normal} depends on the distribution of eigenvalues of $H$~\cite{saad2003iterative}. A useful upper bound is $\sqrt{\kappa(H)}$, where, $\kappa(H)$f is the condition number of the matrix $H$. For most bundle adjustment problems, $\kappa(H)$ is high and a direct application of Conjugate Gradients to~\eqref{eq:normal} results in extremely poor performance.

The solution to this problem is to replace~\eqref{eq:normal} with a {\em preconditioned} system.  Given a linear system, $Ax =b$ and a preconditioner $M$ the preconditioned system is given by $M^{-1}Ax = M^{-1}b$. The resulting algorithm is known as Preconditioned Conjugate Gradients algorithm (PCG) and its  worst case complexity now depends on the condition number of the {\em preconditioned} matrix $\kappa(M^{-1}A)$.

The computational cost of using a preconditioner $M$ is the cost of computing $M$ and evaluating the product $M^{-1}y$ for arbitrary vectors $y$. Thus, there are two competing factors to consider: How much of $H$'s structure is captured by $M$ so that the condition number $\kappa(HM^{-1})$ is low, and the computational cost of constructing and using $M$.  The ideal preconditioner would be one for which $\kappa(M^{-1}A) =1$. $M=A$ achieves this, but it is not a practical choice, as applying this preconditioner would require solving a linear system equivalent to the unpreconditioned problem.  It is usually the case that the more information $M$ has about $H$, the more expensive it is use. For example, Incomplete Cholesky factorization based preconditioners  have much better convergence behavior than the Jacobi preconditioner, but are also much more expensive.


The simplest of all preconditioners is the diagonal or Jacobi preconditioner, \ie,  $M=\operatorname{diag}(A)$, which for block structured matrices like $H$ can be generalized to the block Jacobi preconditioner.

For \texttt{ITERATIVE\_SCHUR} there are two obvious choices for block diagonal preconditioners for $S$. The block diagonal of the matrix $B$~\cite{mandel1990block} and the block diagonal $S$, \ie the block Jacobi preconditioner for $S$. Ceres's implements both of these preconditioners and refers to them as  \texttt{JACOBI} and \texttt{SCHUR\_JACOBI} respectively.

For bundle adjustment problems arising in reconstruction from community photo collections, more effective preconditioners can be constructed by analyzing and exploiting the camera-point visibility structure of the scene~\cite{kushal2012}. Ceres implements the two visibility based preconditioners described by Kushal \& Agarwal as \texttt{CLUSTER\_JACOBI} and \texttt{CLUSTER\_TRIDIAGONAL}. These are fairly new preconditioners and Ceres' implementation of them is in its early stages and is not as mature as the other preconditioners described above.

\section{Ordering}
All three of the Schur based solvers depend on the user indicating to the solver, which of the parameter blocks correspond to the points and which correspond to the cameras. Ceres refers to them as \texttt{e\_block}s and \texttt{f\_blocks}. The only constraint on \texttt{e\_block}s is that there should be no term in the objective function with two or more \texttt{e\_block}s.

As we saw in Section~\ref{chapter:tutorial:bundleadjustment}, there are two ways to indicate \texttt{e\_block}s to Ceres. The first is to explicitly create an ordering vector \texttt{Solver::Options::ordering} containing the parameter blocks such that all the \texttt{e\_block}s/points occur before the \texttt{f\_blocks}, and setting \texttt{Solver::Options::num\_eliminate\_blocks} to the number \texttt{e\_block}s.

For some problems this is an easy thing to do and we recommend its use. In some problems though, this is onerous and it would be better if Ceres could automatically determine \texttt{e\_block}s. Setting \texttt{Solver::Options::ordering\_type} to \texttt{SCHUR} achieves this.

The \texttt{SCHUR} ordering algorithm is based on the observation that
the constraint that no two \texttt{e\_block} co-occur in a residual
block means that if we were to treat the sparsity structure of the
block matrix $H$ as a graph, then the set of \texttt{e\_block}s is an
independent set in this graph. The larger the number of
\texttt{e\_block}, the smaller is the size of the Schur complement $S$. Indeed the reason Schur based solvers are so efficient at solving bundle adjustment problems is because the number of points in a bundle adjustment problem is usually an order of magnitude or two larger than the number of cameras.

Thus, the aim of the \texttt{SCHUR} ordering algorithm is to identify the largest independent set in the graph of $H$. Unfortunately this is an NP-Hard problem. But there is a  greedy approximation algorithm that performs well~\cite{li2007miqr} and we use it to identify \texttt{e\_block}s in Ceres.

\section{\texttt{Solver::Options}}

\texttt{Solver::Options} controls the overall behavior of the solver. We list the various settings and their default values below.

\begin{enumerate}

\item{\texttt{trust\_region\_strategy\_type }} (\texttt{LEVENBERG\_MARQUARDT}) The  trust region step computation algorithm used by Ceres. Currently \texttt{LEVENBERG\_MARQUARDT } and \texttt{DOGLEG} are the two valid choices.

\item{\texttt{max\_num\_iterations }}(\texttt{50}) Maximum number of iterations for Levenberg-Marquardt.

\item{\texttt{max\_solver\_time\_in\_seconds }} ($10^9$) Maximum amount of time for which the solver should run.

\item{\texttt{num\_threads }}(\texttt{1})
Number of threads used by Ceres to evaluate the Jacobian.

\item{\texttt{initial\_trust\_region\_radius } ($10^4$)} The size of the initial trust region. When the \texttt{LEVENBERG\_MARQUARDT} strategy is used, the reciprocal of this number is the initial regularization parameter.

\item{\texttt{max\_trust\_region\_radius } ($10^{16}$)} The trust region radius is not allowed to grow beyond this value.
\item{\texttt{max\_trust\_region\_radius } ($10^{-32}$)} The solver terminates, when the trust region becomes smaller than this value.

\item{\texttt{min\_relative\_decrease }}($10^{-3}$) Lower threshold for relative decrease before a Levenberg-Marquardt step is acceped.

\item{\texttt{lm\_min\_diagonal } ($10^6$)} The \texttt{LEVENBERG\_MARQUARDT} strategy, uses a diagonal matrix to regularize the the trust region step. This is the lower bound on the values of this diagonal matrix.

\item{\texttt{lm\_max\_diagonal } ($10^{32}$)}  The \texttt{LEVENBERG\_MARQUARDT} strategy, uses a diagonal matrix to regularize the the trust region step. This is the upper bound on the values of this diagonal matrix.

\item{\texttt{max\_num\_consecutive\_invalid\_steps } (5)} The step returned by a trust region strategy can sometimes be numerically invalid, usually because of conditioning issues. Instead of crashing or stopping the optimization, the optimizer can go ahead and try solving with a smaller trust region/better conditioned problem. This parameter sets the number of consecutive retries before the minimizer gives up.

\item{\texttt{function\_tolerance }}($10^{-6}$) Solver terminates if
\begin{align}
\frac{|\Delta \text{cost}|}{\text{cost}} < \texttt{function\_tolerance}
\end{align}
where, $\Delta \text{cost}$ is the change in objective function value (up or down) in the current iteration of Levenberg-Marquardt.

\item \texttt{Solver::Options::gradient\_tolerance } Solver terminates if
\begin{equation}
    \frac{\|g(x)\|_\infty}{\|g(x_0)\|_\infty} < \texttt{gradient\_tolerance}
\end{equation}
where $\|\cdot\|_\infty$ refers to the max norm, and $x_0$ is the vector of initial parameter values.

\item{\texttt{parameter\_tolerance }}($10^{-8}$) Solver terminates if
\begin{equation}
    \frac{\|\Delta x\|}{\|x\| + \texttt{parameter\_tolerance}} < \texttt{parameter\_tolerance}
\end{equation}
where $\Delta x$ is the step computed by the linear solver in the current iteration of Levenberg-Marquardt.

\item{\texttt{linear\_solver\_type }(\texttt{SPARSE\_NORMAL\_CHOLESKY})}

\item{\texttt{linear\_solver\_type }}(\texttt{SPARSE\_NORMAL\_CHOLESKY}/\texttt{DENSE\_QR}) Type of linear solver used to compute the solution to the linear least squares problem in each iteration of the Levenberg-Marquardt algorithm. If Ceres is build with \suitesparse linked in  then the default is \texttt{SPARSE\_NORMAL\_CHOLESKY}, it is \texttt{DENSE\_QR} otherwise.

\item{\texttt{preconditioner\_type }}(\texttt{JACOBI}) The preconditioner used by the iterative linear solver. The default is the block Jacobi preconditioner. Valid values are (in increasing order of complexity) \texttt{IDENTITY},\texttt{JACOBI}, \texttt{SCHUR\_JACOBI}, \texttt{CLUSTER\_JACOBI} and \texttt{CLUSTER\_TRIDIAGONAL}.

\item{\texttt{sparse\_linear\_algebra\_library } (\texttt{SUITE\_SPARSE})} Ceres supports the use of two sparse linear algebra libraries, \texttt{SuiteSparse}, which is enabled by setting this parameter to \texttt{SUITE\_SPARSE} and \texttt{CXSparse}, which can be selected by setting this parameter to $\texttt{CX\_SPARSE}$. \texttt{SuiteSparse} is a sophisticated and complex sparse linear algebra library and should be used in general. If your needs/platforms prevent you from using \texttt{SuiteSparse}, consider using \texttt{CXSparse}, which is a much smaller, easier to build library. As can be expected, its performance on large problems is not comparable to that of \texttt{SuiteSparse}.


\item{\texttt{num\_linear\_solver\_threads }}(\texttt{1}) Number of threads used by the linear solver.

\item{\texttt{num\_eliminate\_blocks }}(\texttt{0})
For Schur reduction based methods, the first 0 to num blocks are
    eliminated using the Schur reduction. For example, when solving
     traditional structure from motion problems where the parameters are in
     two classes (cameras and points) then \texttt{num\_eliminate\_blocks} would be the
     number of points.

\item{\texttt{ordering\_type }}(\texttt{NATURAL})
 Internally Ceres reorders the parameter blocks to help the
 various linear solvers. This parameter allows the user to
     influence the re-ordering strategy used. For structure from
     motion problems use \texttt{SCHUR}, for other problems \texttt{NATURAL} (default)
     is a good choice. In case you wish to specify your own ordering
     scheme, for example in conjunction with \texttt{num\_eliminate\_blocks},
     use \texttt{USER}.

\item{\texttt{ordering }} The ordering of the parameter blocks. The solver pays attention
    to it if the \texttt{ordering\_type} is set to \texttt{USER} and the ordering vector is
    non-empty.

\item{\texttt{use\_block\_amd } (\texttt{true})} By virtue of the modeling layer in Ceres being block oriented,
all the matrices used by Ceres are also block oriented.
When doing sparse direct factorization of these matrices, the
fill-reducing ordering algorithms can either be run on the
block or the scalar form of these matrices. Running it on the
block form exposes more of the super-nodal structure of the
matrix to the Cholesky factorization routines. This leads to
substantial gains in factorization performance. Setting this parameter to true, enables the use of a block oriented Approximate Minimum Degree ordering algorithm. Settings it to \texttt{false}, uses a scalar AMD algorithm. This option only makes sense when using \texttt{sparse\_linear\_algebra\_library = SUITE\_SPARSE} as it uses the \texttt{AMD} package that is part of \texttt{SuiteSparse}.

\item{\texttt{linear\_solver\_min\_num\_iterations }}(\texttt{1}) Minimum number of iterations used by the linear solver. This only makes sense when the linear solver is an iterative solver, e.g., \texttt{ITERATIVE\_SCHUR}.

\item{\texttt{linear\_solver\_max\_num\_iterations }}(\texttt{500}) Minimum number of iterations used by the linear solver. This only makes sense when the linear solver is an iterative solver, e.g., \texttt{ITERATIVE\_SCHUR}.

\item{\texttt{eta }} ($10^{-1}$)
 Forcing sequence parameter. The truncated Newton solver uses
    this number to control the relative accuracy with which the
     Newton step is computed. This constant is passed to ConjugateGradientsSolver which uses
     it to terminate the iterations when
\begin{equation}
      \frac{Q_i - Q_{i-1}}{Q_i} < \frac{\eta}{i}
\end{equation}

\item{\texttt{jacobi\_scaling }}(\texttt{true}) \texttt{true} means that the Jacobian is scaled by the norm of its columns before being passed to the linear solver. This improves the numerical conditioning of the normal equations.

\item{\texttt{logging\_type }}(\texttt{PER\_MINIMIZER\_ITERATION})


\item{\texttt{minimizer\_progress\_to\_stdout }}(\texttt{false})
By default the Minimizer progress is logged to \texttt{STDERR} depending on the \texttt{vlog} level. If this flag is
set to true, and \texttt{logging\_type } is not \texttt{SILENT}, the logging output
is sent to \texttt{STDOUT}.

\item{\texttt{return\_initial\_residuals }}(\texttt{false})
\item{\texttt{return\_final\_residuals }}(\texttt{false})


\item{\texttt{lsqp\_iterations\_to\_dump }}
 List of iterations at which the optimizer should dump the
     linear least squares problem to disk. Useful for testing and
     benchmarking. If empty (default), no problems are dumped.

\item{\texttt{lsqp\_dump\_directory }} (\texttt{/tmp})
 If \texttt{lsqp\_iterations\_to\_dump} is non-empty, then this setting determines the directory to which the files containing the linear least squares problems are written to.


\item{\texttt{lsqp\_dump\_format }}(\texttt{TEXTFILE}) The format in which linear least squares problems should be logged
when \texttt{lsqp\_iterations\_to\_dump} is non-empty.  There are three options
\begin{itemize}
\item{\texttt{CONSOLE }} prints the linear least squares problem in a human readable format
  to \texttt{stderr}. The Jacobian is printed as a dense matrix. The vectors
   $D$, $x$ and $f$ are printed as dense vectors. This should only be used
   for small problems.
\item{\texttt{PROTOBUF }}
   Write out the linear least squares problem to the directory
   pointed to by \texttt{lsqp\_dump\_directory} as a protocol
   buffer. \texttt{linear\_least\_squares\_problems.h/cc} contains routines for
   loading these problems. For details on the on disk format used,
   see \texttt{matrix.proto}. The files are named \texttt{lm\_iteration\_???.lsqp}. This requires that \texttt{protobuf} be linked into Ceres Solver.
\item{\texttt{TEXTFILE }}
   Write out the linear least squares problem to the directory
   pointed to by \texttt{lsqp\_dump\_directory} as text files
   which can be read into \texttt{MATLAB/Octave}. The Jacobian is dumped as a
   text file containing $(i,j,s)$ triplets, the vectors $D$, $x$ and $f$ are
   dumped as text files containing a list of their values.

   A \texttt{MATLAB/Octave} script called \texttt{lm\_iteration\_???.m} is also output,
   which can be used to parse and load the problem into memory.
\end{itemize}


\item{\texttt{check\_gradients }}(\texttt{false})
 Check all Jacobians computed by each residual block with finite
     differences. This is expensive since it involves computing the
     derivative by normal means (e.g. user specified, autodiff,
     etc), then also computing it using finite differences. The
     results are compared, and if they differ substantially, details
     are printed to the log.

\item{\texttt{gradient\_check\_relative\_precision }} ($10^{-8}$)
  Relative precision to check for in the gradient checker. If the
  relative difference between an element in a Jacobian exceeds
  this number, then the Jacobian for that cost term is dumped.

\item{\texttt{numeric\_derivative\_relative\_step\_size }} ($10^{-6}$)
 Relative shift used for taking numeric derivatives. For finite
     differencing, each dimension is evaluated at slightly shifted
     values, \eg for forward differences, the numerical derivative is

\begin{align}
       \delta &= \texttt{numeric\_derivative\_relative\_step\_size}\\
       \Delta f &= \frac{f((1 + \delta)  x) - f(x)}{\delta x}
\end{align}


     The finite differencing is done along each dimension. The
     reason to use a relative (rather than absolute) step size is
     that this way, numeric differentiation works for functions where
     the arguments are typically large (e.g. $10^9$) and when the
     values are small (e.g. $10^{-5}$). It is possible to construct
     "torture cases" which break this finite difference heuristic,
     but they do not come up often in practice.

\item{\texttt{callbacks }}
  Callbacks that are executed at the end of each iteration of the
     \texttt{Minimizer}. They are executed in the order that they are
     specified in this vector. By default, parameter blocks are
     updated only at the end of the optimization, i.e when the
     \texttt{Minimizer} terminates. This behavior is controlled by
     \texttt{update\_state\_every\_variable}. If the user wishes to have access
     to the update parameter blocks when his/her callbacks are
     executed, then set \texttt{update\_state\_every\_iteration} to true.

     The solver does NOT take ownership of these pointers.

\item{\texttt{update\_state\_every\_iteration }}(\texttt{false})
Normally the parameter blocks are only updated when the solver terminates. Setting this to true update them in every iteration. This setting is useful when building an interactive application using Ceres and using an \texttt{IterationCallback}.
\end{enumerate}

\section{\texttt{Solver::Summary}}
TBD