Least squares, in general, is the problem of finding a vector *x*
that is a local minimizer to a function that is a sum of squares, possibly subject
to some constraints:

$$\underset{x}{\mathrm{min}}{\Vert F(x)\Vert}_{2}^{2}=\underset{x}{\mathrm{min}}{\displaystyle \sum _{i}{F}_{i}^{2}(x)}$$

such that *A·x* ≤ *b*,
*Aeq·x* = *beq*,
*lb ≤ x ≤ ub*.

There are several Optimization Toolbox™ solvers available for various types of
*F*(*x*) and various types of constraints:

Solver | F(x) | Constraints |
---|---|---|

`mldivide` | C·x –
d | None |

`lsqnonneg` | C·x –
d | x ≥ 0 |

`lsqlin` | C·x –
d | Bound, linear |

`lsqnonlin` | General F(x) | Bound |

`lsqcurvefit` | F(x,
xdata) – ydata | Bound |

There are five least-squares algorithms in Optimization Toolbox solvers, in addition to the algorithms used in `mldivide`

:

All the algorithms except `lsqlin`

active-set are large-scale;
see Large-Scale vs. Medium-Scale Algorithms. For a general survey of
nonlinear least-squares methods, see Dennis [8]. Specific details on the Levenberg-Marquardt method can be found
in Moré [28].

The `lsqlin`

`'interior-point'`

algorithm uses the interior-point-convex quadprog Algorithm, and the `lsqlin`

`'active-set'`

algorithm uses the active-set
`quadprog`

algorithm. The `quadprog`

problem definition is to minimize a quadratic function

$$\underset{x}{\mathrm{min}}\frac{1}{2}{x}^{T}Hx+{c}^{T}x$$

subject to linear constraints and bound constraints. The
`lsqlin`

function minimizes the squared 2-norm of the vector
*Cx – d* subject to linear constraints and bound
constraints. In other words, `lsqlin`

minimizes

$$\begin{array}{c}{\Vert Cx-d\Vert}_{2}^{2}={\left(Cx-d\right)}^{T}\left(Cx-d\right)\\ =\left({x}^{T}{C}^{T}-{d}^{T}\right)\left(Cx-d\right)\\ =\left({x}^{T}{C}^{T}Cx\right)-\left({x}^{T}{C}^{T}d-{d}^{T}Cx\right)+{d}^{T}d\\ =\frac{1}{2}{x}^{T}\left(2{C}^{T}C\right)x+{\left(-2{C}^{T}d\right)}^{T}x+{d}^{T}d.\end{array}$$

This fits into the `quadprog`

framework by setting the
*H* matrix to
2*C ^{T}C* and the

`lsqlin`

problem, `quadprog`

calculates the solution.**Note**

The `quadprog`

`'interior-point-convex'`

algorithm has two code paths. It
takes one when the Hessian matrix *H* is an ordinary (full)
matrix of doubles, and it takes the other when *H* is a sparse
matrix. For details of the sparse data type, see Sparse Matrices.
Generally, the algorithm is faster for large problems that have relatively few
nonzero terms when you specify *H* as `sparse`

. Similarly, the algorithm
is faster for small or relatively dense problems when you specify
*H* as `full`

.

Many of the methods used in Optimization Toolbox solvers are based on *trust regions,* a simple
yet powerful concept in optimization.

To understand the trust-region approach to optimization, consider the
unconstrained minimization problem, minimize
*f*(*x*), where the function takes vector
arguments and returns scalars. Suppose you are at a point *x*
in *n*-space and you want to improve, i.e., move to a point
with a lower function value. The basic idea is to approximate
*f* with a simpler function *q*, which
reasonably reflects the behavior of function *f* in a
neighborhood *N* around the point *x*. This
neighborhood is the trust region. A trial step *s* is computed
by minimizing (or approximately minimizing) over *N*. This is
the trust-region subproblem,

$$\underset{s}{\mathrm{min}}\left\{q(s),\text{}s\in N\right\}.$$ | (1) |

The current point is updated to be
*x* + *s* if *f*(*x* + *s*) < *f*(*x*); otherwise, the current point remains unchanged and
*N*, the region of trust, is shrunk and the trial step
computation is repeated.

The key questions in defining a specific trust-region approach to minimizing
*f*(*x*) are how to choose and compute the
approximation *q* (defined at the current point
*x*), how to choose and modify the trust region
*N*, and how accurately to solve the trust-region
subproblem. This section focuses on the unconstrained problem. Later sections
discuss additional complications due to the presence of constraints on the
variables.

In the standard trust-region method ([48]), the quadratic approximation *q* is defined
by the first two terms of the Taylor approximation to *F* at
*x*; the neighborhood *N* is usually
spherical or ellipsoidal in shape. Mathematically the trust-region subproblem is
typically stated

$$\mathrm{min}\left\{\frac{1}{2}{s}^{T}Hs+{s}^{T}g\text{suchthat}\Vert Ds\Vert \le \Delta \right\},$$ | (2) |

where *g* is the gradient of *f* at the
current point *x*, *H* is the Hessian matrix
(the symmetric matrix of second derivatives), *D* is a diagonal
scaling matrix, Δ is a positive scalar, and ∥ . ∥ is the 2-norm. Good algorithms
exist for solving Equation 2 (see [48]); such algorithms typically involve the computation of all
eigenvalues of *H* and a Newton process applied to the secular equation

$$\frac{1}{\Delta}-\frac{1}{\Vert s\Vert}=0.$$

Such algorithms provide an accurate solution to Equation 2. However, they require time proportional
to several factorizations of *H*. Therefore, for trust-region
problems a different approach is needed. Several approximation and heuristic
strategies, based on Equation 2, have been
proposed in the literature ([42] and [50]). The approximation approach followed in
Optimization Toolbox solvers is to restrict the trust-region subproblem to a
two-dimensional subspace *S* ([39] and [42]). Once the subspace *S*
has been computed, the work to solve Equation 2 is trivial even if full
eigenvalue/eigenvector information is needed (since in the subspace, the problem
is only two-dimensional). The dominant work has now shifted to the determination
of the subspace.

The two-dimensional subspace *S* is determined with
the aid of a preconditioned conjugate gradient process described below. The
solver defines *S* as the linear space spanned by
*s*_{1} and
*s*_{2}, where
*s*_{1} is in the direction of the
gradient *g*, and *s*_{2}
is either an approximate Newton direction, i.e., a solution to

$$H\cdot {s}_{2}=-g,$$ | (3) |

or a direction of negative curvature,

$${s}_{2}^{T}\cdot H\cdot {s}_{2}<0.$$ | (4) |

The philosophy behind this choice of *S* is to force global
convergence (via the steepest descent direction or negative curvature direction)
and achieve fast local convergence (via the Newton step, when it exists).

A sketch of unconstrained minimization using trust-region ideas is now easy to give:

Formulate the two-dimensional trust-region subproblem.

Solve Equation 2 to determine the trial step

*s*.If

*f*(*x*+*s*) <*f*(*x*), then*x*=*x*+*s*.Adjust Δ.

These four steps are repeated until convergence. The trust-region dimension Δ
is adjusted according to standard rules. In particular, it is decreased if the
trial step is not accepted, i.e., *f*(*x* +
*s*) ≥
*f*(*x*). See [46] and [49] for a discussion of this aspect.

Optimization Toolbox solvers treat a few important special cases of
*f* with specialized functions: nonlinear least-squares,
quadratic functions, and linear least-squares. However, the underlying
algorithmic ideas are the same as for the general case. These special cases are
discussed in later sections.

An important special case for *f*(*x*) is
the nonlinear least-squares problem

$$\underset{x}{\mathrm{min}}{\displaystyle \sum _{i}{f}_{i}^{2}(x)}=\underset{x}{\mathrm{min}}{\Vert F(x)\Vert}_{2}^{2},$$ | (5) |

where *F*(*x*) is a vector-valued function
with component *i* of
*F*(*x*) equal to
*f _{i}*(

$$\mathrm{min}{\Vert Js+F\Vert}_{2}^{2},$$ | (6) |

(where *J* is the Jacobian of
*F*(*x*)) is used to help define the
two-dimensional subspace *S*. Second derivatives of the
component function
*f _{i}*(

In each iteration the method of preconditioned conjugate gradients is used to approximately solve the normal equations, i.e.,

$${J}^{T}Js=-{J}^{T}F,$$

although the normal equations are not explicitly formed.

In this case the function *f*(*x*)
to be solved is

$$f(x)={\Vert Cx+d\Vert}_{2}^{2},$$

possibly subject to linear constraints. The algorithm generates strictly
feasible iterates converging, in the limit, to a local solution. Each iteration
involves the approximate solution of a large linear system (of order
*n*, where *n* is the length of
*x*). The iteration matrices have the structure of the
matrix *C*. In particular, the method of preconditioned
conjugate gradients is used to approximately solve the normal equations,
i.e.,

$${C}^{T}Cx=-{C}^{T}d,$$

although the normal equations are not explicitly formed.

The subspace trust-region method is used to determine a search direction. However, instead of restricting the step to (possibly) one reflection step, as in the nonlinear minimization case, a piecewise reflective line search is conducted at each iteration, as in the quadratic case. See [45] for details of the line search. Ultimately, the linear systems represent a Newton approach capturing the first-order optimality conditions at the solution, resulting in strong local convergence rates.

**Jacobian Multiply Function. **`lsqlin`

can solve the
linearly-constrained least-squares problem without using the matrix
*C* explicitly. Instead, it uses a Jacobian multiply
function `jmfun`

,

W = jmfun(Jinfo,Y,flag)

that you provide. The function must calculate the following products for a
matrix *Y*:

If

`flag == 0`

then`W = C'*(C*Y)`

.If

`flag > 0`

then`W = C*Y`

.If

`flag < 0`

then`W = C'*Y`

.

This can be useful if *C* is large, but contains enough
structure that you can write `jmfun`

without forming
*C* explicitly. For an example, see Jacobian Multiply Function with Linear Least Squares.

The least-squares problem minimizes a function
*f*(*x*) that is a sum of squares.

$$\underset{x}{\mathrm{min}}f(x)={\Vert F(x)\Vert}_{2}^{2}={\displaystyle \sum _{i}{F}_{i}^{2}(x)}.$$ | (7) |

Problems of this type occur in a large number of practical applications,
especially those that involve fitting model functions to data, such as nonlinear
parameter estimation. This problem type also appears in control systems, where the
objective is for the output *y*(*x,t*) to follow a
continuous model trajectory * φ*(*t*) for vector
*x* and scalar *t*. This problem can be
expressed as

$$\underset{x\in {\Re}^{n}}{\mathrm{min}}{\displaystyle \underset{{t}_{1}}{\overset{{t}_{2}}{\int}}{\left(y(x,t)-\phi (t)\right)}^{2}}dt,$$ | (8) |

where *y*(*x,t*) and *
φ*(*t*) are scalar functions.

Discretize the integral to obtain an approximation

$$\underset{x\in {\Re}^{n}}{\mathrm{min}}f(x)={\displaystyle \sum _{i=1}^{m}{\left(y(x,{t}_{i})-\phi ({t}_{i})\right)}^{2}},$$ | (9) |

where the *t _{i}* are evenly spaced. In this
problem, the vector

$$F(x)=\left[\begin{array}{c}y(x,{t}_{1})-\phi ({t}_{1})\\ y(x,{t}_{2})-\phi ({t}_{2})\\ \mathrm{...}\\ y(x,{t}_{m})-\phi ({t}_{m})\end{array}\right].$$

In this type of problem, the residual ∥*F*(*x*)∥ is likely to be small at the optimum, because the general practice
is to set realistically achievable target trajectories. Although you can minimize
the function in Equation 7 using a general, unconstrained minimization
technique, as described in Basics of Unconstrained Optimization, certain
characteristics of the problem can often be exploited to improve the iterative
efficiency of the solution procedure. The gradient and Hessian matrix of Equation 7 have a special structure.

Denoting the *m*-by-*n* Jacobian matrix of
*F*(*x*) as
*J*(*x*), gradient vector of
*f*(*x*) as
*G*(*x*), Hessian matrix of
*f*(*x*) as
*H*(*x*), and Hessian matrix of each
*F _{i}*(

$$\begin{array}{l}G(x)=2J{(}^{x}F(x)\\ H(x)=2J{(}^{x}J(x)+2Q(x),\end{array}$$ | (10) |

where

$$Q(x)={\displaystyle \sum _{i=1}^{m}{F}_{i}(x)\cdot {D}_{i}(x)}.$$

A property of the matrix *Q*(*x*) is that when
the residual ∥*F*(*x*)∥ tends towards zero as *x _{k}*
approaches the solution, then

At each major iteration *k*, the Gauss-Newton method obtains a
search direction *d _{k}* that is a solution of
the linear least-squares problem

$$\underset{{d}_{k}\in {\Re}^{n}}{\mathrm{min}}{\Vert J({x}_{k}){d}_{k}+F({x}_{k})\Vert}_{2}^{2}.$$ | (11) |

The direction derived from this method is equivalent to the Newton direction when
the terms of *Q*(*x*) = 0. The algorithm can use the search direction
*d _{k}* as part of a line search
strategy to ensure that the function

The Gauss-Newton method often encounters problems when the second-order term
*Q*(*x*) is nonnegligible. The
Levenberg-Marquardt method overcomes this problem.

The Levenberg-Marquardt method (see [25] and [27]) uses a search direction that is a solution of the linear set of equations

$$\left(J{\left({x}_{k}\right)}^{T}J\left({x}_{k}\right)+{\lambda}_{k}I\right){d}_{k}=-J{\left({x}_{k}\right)}^{T}F\left({x}_{k}\right),$$ | (12) |

or, optionally, of the equations

$$\left(J{\left({x}_{k}\right)}^{T}J\left({x}_{k}\right)+{\lambda}_{k}\text{diag}\left(J{\left({x}_{k}\right)}^{T}J\left({x}_{k}\right)\right)\right){d}_{k}=-J{\left({x}_{k}\right)}^{T}F\left({x}_{k}\right),$$ | (13) |

where the scalar *λ _{k}* controls both the
magnitude and direction of

`diag(A)`

means the matrix of diagonal terms in
`A`

. Set the option `ScaleProblem`

to
`'none'`

to choose Equation 12, or set `ScaleProblem`

to
`'Jacobian'`

to choose Equation 13.You set the initial value of the parameter
*λ*_{0} using the
`InitDamping`

option. Occasionally, the `0.01`

default value of this option can be unsuitable. If you find that the
Levenberg-Marquardt algorithm makes little initial progress, try setting
`InitDamping`

to a different value from the default, such as
`1e2`

.

When *λ _{k}* is zero, the direction

Internally, the Levenberg-Marquardt algorithm uses an optimality tolerance
(stopping criterion) of `1e-4`

times the function tolerance.

The Levenberg-Marquardt method, therefore, uses a search direction that is a cross between the Gauss-Newton direction and the steepest descent direction.

Another advantage to the Levenberg-Marquardt method is when the Jacobian
*J* is rank-deficient. In this case, the Gauss-Newton method
can have numerical issues because the minimization problem in Equation 11
is ill-posed. In contrast, the Levenberg-Marquardt method has full rank at each
iteration, and, therefore, avoids these issues.

The following figure shows the iterations of the Levenberg-Marquardt method when minimizing Rosenbrock's function, a notoriously difficult minimization problem that is in least-squares form.

**Levenberg-Marquardt Method on Rosenbrock's Function**

For a more complete description of this figure, including scripts that generate the iterative points, see Banana Function Minimization.

When the problem contains bound constraints, `lsqcurvefit`

and `lsqnonlin`

modify the Levenberg-Marquardt iterations. If
a proposed iterative point *x* lies outside of the bounds, the
algorithm projects the step onto the nearest feasible point. In other words,
with *P* defined as the *projection*
operator that projects infeasible points onto the feasible region, the algorithm
modifies the proposed point *x* to
*P*(*x*). By definition, the projection
operator *P* operates on each component
*x _{i}* independently according
to

$$P({x}_{i})=\{\begin{array}{ll}{\text{lb}}_{i}\hfill & \text{if}{x}_{i}{\text{lb}}_{i}\hfill \\ {\text{ub}}_{i}\hfill & \text{if}{x}_{i}{\text{ub}}_{i}\hfill \\ {x}_{i}\hfill & \text{otherwise}\hfill \end{array}$$

or, equivalently,

$$P({x}_{i})=\mathrm{min}(\mathrm{max}({x}_{i},{\text{lb}}_{i}),{\text{ub}}_{i}).$$

The algorithm modifies the stopping criterion for the first-order optimality
measure. Let *x* be a proposed iterative point. In the
unconstrained case, the stopping criterion is

$${\Vert \nabla f(x)\Vert}_{\infty}\le \text{tol,}$$ | (14) |

where tol is the optimality tolerance value, which is
`1e-4*FunctionTolerance`

. In the bounded case, the stopping
criterion is

$${\Vert x-P\left(x-\nabla f(x)\right)\Vert}_{\infty}^{2}\le \text{tol}{\Vert \nabla f(x)\Vert}_{\infty}.$$ | (15) |

To understand this criterion, first note that if *x* is in
the interior of the feasible region, then the operator *P* has
no effect. So, the stopping criterion becomes

$${\Vert x-P\left(x-\nabla f(x)\right)\Vert}_{\infty}^{2}={\Vert \nabla f(x)\Vert}_{\infty}^{2}\le \text{tol}{\Vert \nabla f(x)\Vert}_{\infty},$$

which is the same as the original unconstrained stopping criterion, $${\Vert \nabla f(x)\Vert}_{\infty}\le \text{tol}$$. If the boundary constraint is active, meaning *x* –
∇*f*(*x*) is not feasible, then at a point where the algorithm should
stop, the gradient at a point on the boundary is perpendicular to the boundary.
Therefore, the point *x* is equal to *P*(*x* –
∇*f*(*x*)), the projection of the steepest descent step, as shown in the
following figure.

**Levenberg-Marquardt Stopping Condition**

`lsqcurvefit`

| `lsqlin`

| `lsqnonlin`

| `lsqnonneg`

| `quadprog`