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Least-Squares (Model Fitting) Algorithms

Least Squares Definition

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:

minxF(x)22=minxiFi2(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:

SolverF(x)Constraints
mldivideC·xdNone
lsqnonnegC·xdx ≥ 0
lsqlinC·xdBound, linear
lsqnonlinGeneral F(x)Bound
lsqcurvefitF(x, xdata) – ydataBound

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

  • Trust-region-reflective

  • Levenberg-Marquardt

  • lsqlin active-set

  • lsqlin interior-point

  • The algorithm used by lsqnonneg

All the algorithms except the lsqlin active-set algorithm 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].

Trust-Region-Reflective Least Squares

Trust-Region-Reflective Least Squares Algorithm

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,

mins{q(s), sN}.(10-1)

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

min{12sTHs+sTg  such that  DsΔ},(10-2)

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 10-2 (see [48]); such algorithms typically involve the computation of a full eigensystem and a Newton process applied to the secular equation

1Δ1s=0.

Such algorithms provide an accurate solution to Equation 10-2. 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 10-2, 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 10-2 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 s1 and s2, where s1 is in the direction of the gradient g, and s2 is either an approximate Newton direction, i.e., a solution to

Hs2=g,(10-3)

or a direction of negative curvature,

s2THs2<0.(10-4)

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:

  1. Formulate the two-dimensional trust-region subproblem.

  2. Solve Equation 10-2 to determine the trial step s.

  3. If f(x + s) < f(x), then x = x + s.

  4. 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.

Large Scale Nonlinear Least Squares

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

minxifi2(x)=minxF(x)22,(10-5)

where F(x) is a vector-valued function with component i of F(x) equal to fi(x). The basic method used to solve this problem is the same as in the general case described in Trust-Region Methods for Nonlinear Minimization. However, the structure of the nonlinear least-squares problem is exploited to enhance efficiency. In particular, an approximate Gauss-Newton direction, i.e., a solution s to

minJs+F22,(10-6)

(where J is the Jacobian of F(x)) is used to help define the two-dimensional subspace S. Second derivatives of the component function fi(x) are not used.

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

JTJs=JTF,

although the normal equations are not explicitly formed.

Large Scale Linear Least Squares

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

f(x)=Cx+d22,

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.,

CTCx=CTd,

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.

Interior-Point Linear Least Squares

The lsqlin 'interior-point' algorithm uses the interior-point-convex quadprog Algorithm. The quadprog problem definition is to minimize a quadratic function

minx12xTHx+cTx

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

Cxd22=(Cxd)T(Cxd)=(xTCTdT)(Cxd)=(xTCTCx)(xTCTddTCx)+dTd=12xT(2CTC)x+(2CTd)Tx+dTd.

This fits into the quadprog framework by setting the H matrix to 2CTC and the c vector to (–2CTd). (The additive term dTd has no effect on the location of the minimum.) After this reformulation of the lsqlin problem, the quadprog 'interior-point-convex' algorithm calculates the solution.

Levenberg-Marquardt Method

In the least-squares problem a function f(x) is minimized that is a sum of squares.

minxf(x)=F(x)22=iFi2(x).(10-7)

Problems of this type occur in a large number of practical applications, especially when fitting model functions to data, i.e., nonlinear parameter estimation. They are also prevalent in control where you want the output, y(x,t), to follow some continuous model trajectory, φ(t), for vector x and scalar t. This problem can be expressed as

minxnt1t2(y(x,t)φ(t))2dt,(10-8)

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

When the integral is discretized using a suitable quadrature formula, the above can be formulated as a least-squares problem:

minxnf(x)=i=1m(y¯(x,ti)φ¯(ti))2,(10-9)

where y¯ and φ¯ include the weights of the quadrature scheme. Note that in this problem the vector F(x) is

F(x)=[y¯(x,t1)φ¯(t1)y¯(x,t2)φ¯(t2)...y¯(x,tm)φ¯(tm)].

In problems of this kind, the residual ∥F(x)∥ is likely to be small at the optimum since it is general practice to set realistically achievable target trajectories. Although the function in LS can be minimized 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 LS have a special structure.

Denoting the m-by-n Jacobian matrix of F(x) as J(x), the gradient vector of f(x) as G(x), the Hessian matrix of f(x) as H(x), and the Hessian matrix of each Fi(x) as Hi(x), you have

G(x)=2J(x)TF(x)H(x)=2J(x)TJ(x)+2Q(x),(10-10)

where

Q(x)=i=1mFi(x)Hi(x).

The matrix Q(x) has the property that when the residual ∥F(x)∥ tends to zero as xk approaches the solution, then Q(x) also tends to zero. Thus when ∥F(x)∥ is small at the solution, a very effective method is to use the Gauss-Newton direction as a basis for an optimization procedure.

In the Gauss-Newton method, a search direction, dk, is obtained at each major iteration, k, that is a solution of the linear least-squares problem:

minxnJ(xk)F(xk)22.(10-11)

The direction derived from this method is equivalent to the Newton direction when the terms of Q(x) can be ignored. The search direction dk can be used as part of a line search strategy to ensure that at each iteration the function f(x) decreases.

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

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

(J(xk)TJ(xk)+λkI)dk=J(xk)TF(xk),(10-12)

or, optionally, of the equations

(J(xk)TJ(xk)+λkdiag(J(xk)TJ(xk)))dk=J(xk)TF(xk),(10-13)

where the scalar λk controls both the magnitude and direction of dk. Set option ScaleProblem to 'none' to choose Equation 10-12, and set ScaleProblem to 'Jacobian' to choose Equation 10-13.

You set the initial value of the parameter λ0 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 than the default, perhaps 1e2.

When λk is zero, the direction dk is identical to that of the Gauss-Newton method. As λk tends to infinity, dk tends towards the steepest descent direction, with magnitude tending to zero. This implies that for some sufficiently large λk, the term F(xk + dk) <  F(xk) holds true. The term λk can therefore be controlled to ensure descent even when second-order terms, which restrict the efficiency of the Gauss-Newton method, are encountered. When the step is successful (gives a lower function value), the algorithm sets λk+1 = λk/10. When the step is unsuccessful, the algorithm sets λk+1 = λk*10.

The Levenberg-Marquardt method therefore uses a search direction that is a cross between the Gauss-Newton direction and the steepest descent direction. This is illustrated in Figure 10-1, Levenberg-Marquardt Method on Rosenbrock's Function. The solution for Rosenbrock's function converges after 90 function evaluations compared to 48 for the Gauss-Newton method. The poorer efficiency is partly because the Gauss-Newton method is generally more effective when the residual is zero at the solution. However, such information is not always available beforehand, and the increased robustness of the Levenberg-Marquardt method compensates for its occasional poorer efficiency.

Figure 10-1. 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.

lsqlin Active-Set Algorithm

The lsqlin solver addresses the minimization problem

minx12Cxd22

subject to bounds and linear constraints. A simple calculation turns this into a quadratic programming problem.

Recall the problem quadprog addresses:

minx12xTHx+cTx(10-14)

subject to bounds and linear constraints. But

12Cxd22=12(Cxd)T(Cxd)=12xT(CTC)xdTCx+12dTd.

So making the definitions H = CTC and c = –CTd, the lsqlin problem becomes a quadprog problem, with an additional constant 1/2 dTd.

Both solvers accept constraints of the form A·x ≤ b, Aeq·x = beq, and l ≤ x ≤ u. m is the total number of linear constraints, the sum of number of rows of A and of Aeq.

The The lsqlin active-set algorithm is an active-set strategy (also known as a projection method) similar to that of Gill et al., described in [18] and [17]. It has been modified for both Linear Programming (LP) and Quadratic Programming (QP) problems. The remainder of the algorithm description addresses the equivalent quadratic programming (QP) problem.

The solution procedure involves two phases. The first phase involves the calculation of a feasible point (if one exists). The second phase involves the generation of an iterative sequence of feasible points that converge to the solution.

Active Set Iterations

In this method an active set matrix, Sk, is maintained that is an estimate of the active constraints (i.e., those that are on the constraint boundaries) at the solution point. Specifically, the active set Sk consists of the rows of Aeq, and a subset of the rows of A. Sk is updated at each iteration k, and is used to form a basis for a search direction dk. Equality constraints always remain in the active set Sk. The search direction dk is calculated and minimizes the objective function while remaining on active constraint boundaries. The feasible subspace for dk is formed from a basis Zk whose columns are orthogonal to the estimate of the active set Sk (i.e., SkZk = 0). Thus a search direction, which is formed from a linear summation of any combination of the columns of Zk, is guaranteed to remain on the boundaries of the active constraints.

The matrix Zk is formed from the last m – l columns of the QR decomposition of the matrix SkT, where l is the number of active constraints and l < m. That is, Zk is given by

Zk=Q[:,l+1:m],(10-15)

where

QTSkT=[R0].

Once Zk is found, a search direction dk is sought that minimizes the objective function at dk, where dk is in the null space of the active constraints. That is, dk is a linear combination of the columns of Zk: dk = Zkp for some vector p.

Then if you view the quadratic objective function as a function of p, by substituting for dk, the result is

q(p)=12pTZkTHZkp+cTZkp.(10-16)

Differentiating this with respect to p yields

q(p)=ZkTHZkp+ZkTc.(10-17)

q(p) is referred to as the projected gradient of the quadratic function because it is the gradient projected in the subspace defined by Zk. The term ZkTHZk is called the projected Hessian. Assuming the Hessian matrix H is positive definite, the minimum of the function q(p) in the subspace defined by Zk occurs when ∇q(p) = 0, which is the solution of the system of linear equations

ZkTHZkp=ZkTc.(10-18)

The next step is

xk+1=xk+αdk,  where dk=Zkp.(10-19)

At each iteration, because of the quadratic nature of the objective function, there are only two choices of step length α. A step of unity along dk is the exact step to the minimum of the function restricted to the null space of Sk. If such a step can be taken, without violation of the constraints, then this is the solution to QP (Equation 10-14). Otherwise, the step along dk to the nearest constraint is less than unity and a new constraint is included in the active set at the next iteration. The distance to the constraint boundaries in any direction dk is given by

α=mini{1,...,m}{(Aixkbi)Aidk},(10-20)

which is defined for constraints not in the active set, and where the direction dk is towards the constraint boundary, i.e., Aidk>0, i=1,...,m.

Lagrange multipliers, λk, are calculated that satisfy the nonsingular set of linear equations

SkTλk=c.(10-21)

If all elements of λk are positive, xk is the optimal solution of QP (Equation 10-14). However, if any component of λk is negative, and the component does not correspond to an equality constraint, then the corresponding element is deleted from the active set and a new iterate is sought.

Initialization

The algorithm requires a feasible point to start. If the initial point is not feasible, then you can find a feasible point by solving the linear programming problem

minγ, xnγ  such thatAix=bi,      i=1,...,me (the rows of Aeq)Aixγbi,  i=me+1,...,m  (the rows of A).(10-22)

The notation Ai indicates the ith row of the matrix A. You can find a feasible point (if one exists) to Equation 10-22 by setting x to a value that satisfies the equality constraints. You can determine this value by solving an under- or overdetermined set of linear equations formed from the set of equality constraints. If there is a solution to this problem, the slack variable γ is set to the maximum inequality constraint at this point.

You can modify the preceding QP algorithm for LP problems by setting the search direction d to the steepest descent direction at each iteration, where gk is the gradient of the objective function (equal to the coefficients of the linear objective function):

d=ZkZkTgk.(10-23)

If a feasible point is found using the preceding LP method, the main QP phase is entered. The search direction dk is initialized with a search direction d1 found from solving the set of linear equations

Hd1=gk,(10-24)

where gk is the gradient of the objective function at the current iterate xk (i.e., Hxk + c).

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