Solve constrained linear leastsquares problems
Linear leastsquares solver with bounds or linear constraints.
Solves leastsquares curve fitting problems of the form
$$\underset{x}{\mathrm{min}}\frac{1}{2}{\Vert C\cdot xd\Vert}_{2}^{2}\text{suchthat}\{\begin{array}{c}A\cdot x\le b,\\ Aeq\cdot x=beq,\\ lb\le x\le ub.\end{array}$$
finds
the minimum for x
= lsqlin(problem
)problem
, where problem
is
a structure. Create the problem
structure by exporting
a problem from Optimization app, as described in Exporting Your Work.
[
, for any input arguments described
above, returns:x
,resnorm
,residual
,exitflag
,output
,lambda
]
= lsqlin(___)
The squared 2norm of the residual resnorm =
$${\Vert C\cdot xd\Vert}_{2}^{2}$$
The residual residual = C*x  d
A value exitflag
describing the
exit condition
A structure output
containing information
about the optimization process
A structure lambda
containing the
Lagrange multipliers
The factor ½ in the definition of the problem affects the
values in the lambda
structure.
Find the x
that minimizes
the norm of C*x  d
for an overdetermined problem
with linear inequality constraints.
Specify the problem and constraints.
C = [0.9501 0.7620 0.6153 0.4057 0.2311 0.4564 0.7919 0.9354 0.6068 0.0185 0.9218 0.9169 0.4859 0.8214 0.7382 0.4102 0.8912 0.4447 0.1762 0.8936]; d = [0.0578 0.3528 0.8131 0.0098 0.1388]; A = [0.2027 0.2721 0.7467 0.4659 0.1987 0.1988 0.4450 0.4186 0.6037 0.0152 0.9318 0.8462]; b = [0.5251 0.2026 0.6721];
Call lsqlin
to solve the problem.
x = lsqlin(C,d,A,b)
Warning: The trustregionreflective algorithm can handle bound constraints only; using activeset algorithm instead. Optimization terminated. x = 0.1299 0.5757 0.4251 0.2438
Find the x
that minimizes
the norm of C*x  d
for an overdetermined problem
with linear inequality constraints and bounds.
Specify the problem and constraints.
C = [0.9501 0.7620 0.6153 0.4057 0.2311 0.4564 0.7919 0.9354 0.6068 0.0185 0.9218 0.9169 0.4859 0.8214 0.7382 0.4102 0.8912 0.4447 0.1762 0.8936]; d = [0.0578 0.3528 0.8131 0.0098 0.1388]; A =[0.2027 0.2721 0.7467 0.4659 0.1987 0.1988 0.4450 0.4186 0.6037 0.0152 0.9318 0.8462]; b =[0.5251 0.2026 0.6721]; Aeq = [3 5 7 9]; beq = 4; lb = 0.1*ones(4,1); ub = 2*ones(4,1);
Call lsqlin
to solve the problem.
x = lsqlin(C,d,A,b,Aeq,beq,lb,ub)
Warning: The trustregionreflective algorithm can handle bound constraints only; using activeset algorithm instead. Optimization terminated. x = 0.1000 0.1000 0.1599 0.4090
Find the x
that minimizes the norm of C*x  d
for an overdetermined problem with linear inequality constraints. Use a start point and nondefault options.
Specify the problem and constraints.
C = [0.9501 0.7620 0.6153 0.4057 0.2311 0.4564 0.7919 0.9354 0.6068 0.0185 0.9218 0.9169 0.4859 0.8214 0.7382 0.4102 0.8912 0.4447 0.1762 0.8936]; d = [0.0578 0.3528 0.8131 0.0098 0.1388]; A =[0.2027 0.2721 0.7467 0.4659 0.1987 0.1988 0.4450 0.4186 0.6037 0.0152 0.9318 0.8462]; b =[0.5251 0.2026 0.6721]; Aeq = []; beq = []; lb = []; ub = [];
Set a start point.
x0 = 0.1*ones(4,1);
Set options to choose the 'activeset'
algorithm, which is the only algorithm that uses a start point.
options = optimoptions('lsqlin','Algorithm','activeset');
Call lsqlin
to solve the problem.
x = lsqlin(C,d,A,b,Aeq,beq,lb,ub,x0,options)
Optimization terminated. x = 0.1299 0.5757 0.4251 0.2438
Obtain and interpret all lsqlin
outputs.
Define a problem with linear inequality constraints and bounds. The problem is overdetermined because there are four columns in the C
matrix but five rows. This means the problem has four unknowns and five conditions, even before including the linear constraints and bounds.
C = [0.9501 0.7620 0.6153 0.4057 0.2311 0.4564 0.7919 0.9354 0.6068 0.0185 0.9218 0.9169 0.4859 0.8214 0.7382 0.4102 0.8912 0.4447 0.1762 0.8936]; d = [0.0578 0.3528 0.8131 0.0098 0.1388]; A =[0.2027 0.2721 0.7467 0.4659 0.1987 0.1988 0.4450 0.4186 0.6037 0.0152 0.9318 0.8462]; b =[0.5251 0.2026 0.6721]; lb = 0.1*ones(4,1); ub = 2*ones(4,1);
Set options to use the 'interiorpoint'
algorithm.
options = optimoptions('lsqlin','Algorithm','interiorpoint');
The 'interiorpoint'
algorithm does not use an initial point, so set x0
to []
.
x0 = [];
Call lsqlin
with all outputs.
[x,resnorm,residual,exitflag,output,lambda] = ...
lsqlin(C,d,A,b,[],[],lb,ub,x0,options)
Minimum found that satisfies the constraints. Optimization completed because the objective function is nondecreasing in feasible directions, to within the default value of the optimality tolerance, and constraints are satisfied to within the default value of the constraint tolerance. x = 0.1000 0.1000 0.2152 0.3502 resnorm = 0.1672 residual = 0.0455 0.0764 0.3562 0.1620 0.0784 exitflag = 1 output = message: 'Minimum found that satisfies the constraints....' algorithm: 'interiorpoint' firstorderopt: 1.6296e08 constrviolation: 0 iterations: 6 cgiterations: [] lambda = ineqlin: [3x1 double] eqlin: [0x1 double] lower: [4x1 double] upper: [4x1 double]
Examine the nonzero Lagrange multiplier fields in more detail. First examine the Lagrange multipliers for the linear inequality constraint.
lambda.ineqlin
ans = 0.0000 0.2392 0.0000
Lagrange multipliers are nonzero exactly when the solution is on the corresponding constraint boundary. In other words, Lagrange multipliers are nonzero when the corresponding constraint is active. lambda.ineqlin(2)
is nonzero. This means that the second element in A*x
should equal the second element in b
, because the constraint is active.
[A(2,:)*x,b(2)]
ans = 0.2026 0.2026
Now examine the Lagrange multipliers for the lower and upper bound constraints.
lambda.lower
ans = 0.0409 0.2784 0.0000 0.0000
lambda.upper
ans = 1.0e10 * 0.4665 0.4751 0.5537 0.6247
The first two elements of lambda.lower
are nonzero. You see that x(1)
and x(2)
are at their lower bounds, 0.1
. All elements of lambda.upper
are essentially zero, and you see that all components of x
are less than their upper bound, 2
.
C
— Multiplier matrixreal matrixMultiplier matrix, specified as a matrix of doubles. C
represents
the multiplier of the solution x
in the expression C*x
 d
. C
is M
byN
,
where M
is the number of equations, and N
is
the number of elements of x
.
Example: C = [1,4;2,5;7,8]
Data Types: double
d
— Constant vectorreal vectorConstant vector, specified as a vector of doubles. d
represents
the additive constant term in the expression C*x  d
. d
is M
by1
,
where M
is the number of equations.
Example: d = [5;0;12]
Data Types: double
A
— Linear inequality constraint matrixreal matrixLinear inequality constraint matrix, specified as a matrix of
doubles. A
represents the linear coefficients in
the constraints A*x
≤ b
. A
has size Mineq
byN
,
where Mineq
is the number of constraints and N
is
the number of elements of x
. To save memory, pass A
as
a sparse matrix.
Example: A = [4,3;2,0;4,1];
means three linear
inequalities (three rows) for two decision variables (two columns).
Data Types: double
b
— Linear inequality constraint vectorreal vectorLinear inequality constraint vector, specified as a vector of
doubles. b
represents the constant vector in the
constraints A*x
≤ b
. b
has length Mineq
,
where A
is Mineq
byN
.
Example: [4,0]
Data Types: double
Aeq
— Linear equality constraint matrix[]
(default)  real matrixLinear equality constraint matrix, specified as a matrix of
doubles. Aeq
represents the linear coefficients
in the constraints Aeq*x
= beq
. Aeq
has size Meq
byN
,
where Meq
is the number of constraints and N
is
the number of elements of x
. To save memory, pass Aeq
as
a sparse matrix.
Example: A = [4,3;2,0;4,1];
means three linear
inequalities (three rows) for two decision variables (two columns).
Data Types: double
beq
— Linear equality constraint vector[]
(default)  real vectorLinear equality constraint vector, specified as a vector of
doubles. beq
represents the constant vector in
the constraints Aeq*x
= beq
. beq
has length Meq
,
where Aeq
is Meq
byN
.
Example: [4,0]
Data Types: double
lb
— Lower bounds[]
(default)  real vector or arrayLower bounds, specified as a vector or array of doubles. lb
represents
the lower bounds elementwise in lb
≤ x
≤ ub
.
Internally, lsqlin
converts an array lb
to
the vector lb(:)
.
Example: lb = [0;Inf;4]
means x(1)
≥ 0
, x(3) ≥ 4
.
Data Types: double
ub
— Upper bounds[]
(default)  real vector or arrayUpper bounds, specified as a vector or array of doubles. ub
represents
the upper bounds elementwise in lb
≤ x
≤ ub
.
Internally, lsqlin
converts an array ub
to
the vector ub(:)
.
Example: ub = [Inf;4;10]
means x(2)
≤ 4
, x(3) ≤ 10
.
Data Types: double
x0
— Initial point[]
(default)  real vector or arrayInitial point for the solution process, specified as a vector
or array of doubles. x0
is used only by the 'activeset'
algorithm.
Optional.
If you do not provide an x0
for the 'activeset'
algorithm, lsqlin
sets x0
to
the zero vector. If any component of this zero vector x0
violates
the bounds, lsqlin
sets x0
to
a point in the interior of the box defined by the bounds.
Example: x0 = [4;3]
Data Types: double
options
— Options for lsqlin
options created using optimoptions
or the Optimization appOptions for lsqlin
, specified as the output
of the optimoptions
function
or the Optimization app.
Some options are absent from the optimoptions
display.
These options are listed in italics. For details, see View Options.
All Algorithms
 Choose the algorithm:
The The For more information on choosing the algorithm, see Choosing the Algorithm. 
Diagnostics  Display diagnostic information about the function to
be minimized or solved. The choices are 
Display  Level of display returned to the command line.
The

Use  Use the You cannot choose
the 
MaxIterations  Maximum number of iterations allowed, a positive integer.
The default value is 
trustregionreflective
Algorithm Options
FunctionTolerance  Termination tolerance on the function
value, a positive scalar. The default is  
JacobianMultiplyFcn  Function
handle for the Jacobian multiply function. For largescale structured
problems, this function should compute the Jacobian matrix product W = jmfun(Jinfo,Y,flag) where
In each case, See Jacobian Multiply Function with Linear Least Squares for an example.  
MaxPCGIter  Maximum number of PCG (preconditioned
conjugate gradient) iterations, a positive scalar. The default is  
OptimalityTolerance  Termination tolerance on the firstorder
optimality, a positive scalar. The default is  
PrecondBandWidth  Upper bandwidth of preconditioner
for PCG (preconditioned conjugate gradient). By default, diagonal
preconditioning is used (upper bandwidth of 0). For some problems,
increasing the bandwidth reduces the number of PCG iterations. Setting  
SubproblemAlgorithm  Determines how the iteration step
is calculated. The default,  
TolPCG  Termination tolerance on the PCG
(preconditioned conjugate gradient) iteration, a positive scalar.
The default is  
TypicalX  Typical 
interiorpoint
Algorithm Options
ConstraintTolerance  Tolerance on the constraint violation, a positive
scalar. The default is 
OptimalityTolerance  Termination tolerance on the firstorder
optimality, a positive scalar. The default is 
problem
— Optimization problemstructureOptimization problem, specified as a structure with the following fields.
 Matrix multiplier in the term C*x
 d 
 Additive constant in the term C*x
 d 
 Matrix for linear inequality constraints 
 Vector for linear inequality constraints 
 Matrix for linear equality constraints 
 Vector for linear equality constraints 
lb  Vector of lower bounds 
ub  Vector of upper bounds 
 Initial point for x 
 'lsqlin' 
 Options created with optimoptions 
Create the problem
structure by exporting
a problem from the Optimization app, as described in Exporting Your Work.
Data Types: struct
x
— Solutionreal vectorSolution, returned as a vector that minimizes the norm of C*xd
subject
to all bounds and linear constraints.
resnorm
— Objective valuereal scalarObjective value, returned as the scalar value norm(C*xd)^2
.
residual
— Solution residualsreal vectorSolution residuals, returned as the vector C*xd
.
exitflag
— Algorithm stopping conditionintegerAlgorithm stopping condition, returned as an integer identifying
the reason the algorithm stopped. The following lists the values of exitflag
and
the corresponding reasons lsqlin
stopped.
 Function converged to a solution 
 Change in the residual was smaller than the specified tolerance. 
 Number of iterations exceeded 
 The problem is infeasible. 
 Illconditioning prevents further optimization. 
 Magnitude of search direction became too small. No further progress could be made. 
The exit message for the interiorpoint
algorithm
can give more details on the reason lsqlin
stopped,
such as exceeding a tolerance. See Exit Flags and Exit Messages.
output
— Solution process summarystructureSolution process summary, returned as a structure containing information about the optimization process.
 Number of iterations the solver took. 
 One of these algorithms:

 Constraint violation that is positive
for violated constraints (not returned for the

 Exit message. 
 Firstorder optimality at the solution. See FirstOrder Optimality Measure. 
 Number of conjugate gradient iterations
the solver performed. Nonempty only for the 
See Output Structures.
lambda
— Lagrange multipliersstructureLagrange multipliers, returned as a structure with the following fields.
 Lower bounds 
 Upper bounds 
 Linear inequalities 
 Linear equalities 
For problems with no constraints, you can use \
(matrix
left division). When you have no constraints, lsqlin
returns x = C\d
.
Because the problem being solved is always convex, lsqlin
finds
a global, although not necessarily unique, solution.
Better numerical results are likely if you specify
equalities explicitly, using Aeq
and beq
,
instead of implicitly, using lb
and ub
.
The trustregionreflective
algorithm
does not allow equal upper and lower bounds. Use another algorithm
for this case.
If the specified input bounds for a problem are inconsistent,
the output x
is x0
and the outputs resnorm
and residual
are []
.
You can solve some large structured problems, including
those where the C
matrix is too large to fit in
memory, using the trustregionreflective
algorithm
with a Jacobian multiply function. For information, see trustregionreflective
Algorithm Options.
When the problem given to lsqlin
has only upper
and lower bounds; that is, no linear inequalities or equalities are
specified, and the matrix C
has at least as many
rows as columns, the default algorithm is trustregionreflective
.
This method is a subspace trustregion method based on the interiorreflective
Newton method described in [1].
Each iteration involves the approximate solution of a large linear
system using the method of preconditioned conjugate gradients (PCG).
See TrustRegionReflective Least Squares,
and in particular Large Scale Linear Least Squares.
lsqlin
uses the activeset
algorithm
when you specify it with optimoptions
, or when
you give linear inequalities or equalities. The algorithm is based
on quadprog
, which uses an active
set method similar to that described in [2]. It finds an initial feasible solution by first solving
a linear programming problem. See activeset
quadprog
Algorithm.
The 'interiorpoint'
algorithm is based on
the quadprog
'interiorpointconvex'
algorithm.
See InteriorPoint Linear Least Squares.
[1] Coleman, T. F. and Y. Li. "A Reflective Newton Method for Minimizing a Quadratic Function Subject to Bounds on Some of the Variables," SIAM Journal on Optimization, Vol. 6, Number 4, pp. 1040–1058, 1996.
[2] Gill, P. E., W. Murray, and M. H. Wright. Practical Optimization, Academic Press, London, UK, 1981.
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