Documentation

x = lsqlin(C,d,A,b) solves the linear system C*x = d in the least-squares sense, subject to A*x ≤ b.

x = lsqlin(C,d,A,b,Aeq,beq,lb,ub) adds linear equality constraints Aeq*x = beq and bounds lb ≤ x ≤ ub. If you do not need certain constraints such as Aeq and beq, set them to []. If x(i) is unbounded below, set lb(i) = -Inf, and if x(i) is unbounded above, set ub(i) = Inf.

x = lsqlin(C,d,A,b,Aeq,beq,lb,ub,x0,options) minimizes with an initial point x0 and the optimization options specified in options. Use optimoptions to set these options. If you do not want to include an initial point, set the x0 argument to [].

x = lsqlin(problem) finds the minimum for problem, where problem is a structure. Create the problem structure by exporting a problem from Optimization app, as described in Exporting Your Work.

[x,resnorm,residual,exitflag,output,lambda] = lsqlin(___), for any input arguments described above, returns:

The squared 2-norm of the residual resnorm = ${‖ C \cdot x - d ‖}_{2}^{2}$
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.

Examples

Least Squares with Linear Inequality Constraints

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)

Minimum found that satisfies the constraints.

Optimization completed because the objective function is non-decreasing 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.

Least Squares with Linear Constraints and Bounds

Find the x that minimizes the norm of C*x - d for an overdetermined problem with linear equality and 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)

Minimum found that satisfies the constraints.

Optimization completed because the objective function is non-decreasing 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.

Linear Least Squares with Nondefault Options

This example shows how to use nondefault options for linear least squares.

Set options to use the 'interior-point' algorithm and to give iterative display.

options = optimoptions('lsqlin','Algorithm','interior-point','Display','iter');

Set up a linear least-squares problem.

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];

Run the problem.

x = lsqlin(C,d,A,b,[],[],[],[],[],options)

 Iter            Fval  Primal Infeas    Dual Infeas  Complementarity  
    0   -7.687420e-02   1.600492e+00   6.150431e-01     1.000000e+00  
    1   -7.687419e-02   8.002458e-04   3.075216e-04     2.430833e-01  
    2   -3.162837e-01   4.001229e-07   1.537608e-07     5.945636e-02  
    3   -3.760545e-01   2.000617e-10   2.036997e-08     1.370933e-02  
    4   -3.912129e-01   1.000866e-13   1.006816e-08     2.548273e-03  
    5   -3.948062e-01   1.110223e-16   2.955102e-09     4.295807e-04  
    6   -3.953277e-01   0.000000e+00   1.237758e-09     3.102850e-05  
    7   -3.953581e-01   1.110223e-16   1.645865e-10     1.138719e-07  
    8   -3.953582e-01   1.110223e-16   2.399192e-13     5.693290e-11  

Minimum found that satisfies the constraints.

Optimization completed because the objective function is non-decreasing 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.

Return All Outputs

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 'interior-point' algorithm.

options = optimoptions('lsqlin','Algorithm','interior-point');

The 'interior-point' 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 non-decreasing 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.

resnorm = 0.1672

residual = 

    0.0455
    0.0764
   -0.3562
    0.1620
    0.0784

exitflag = 1

output = struct with fields:
            message: 'Minimum found that satisfies the constraints....'
          algorithm: 'interior-point'
      firstorderopt: 4.3374e-11
    constrviolation: 0
         iterations: 6
       cgiterations: []

lambda = struct with fields:
    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

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

lambda.upper

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.

Input Arguments

`C` — Multiplier matrix
real matrix

Multiplier matrix, specified as a matrix of doubles. C represents the multiplier of the solution x in the expression C*x - d. C is M-by-N, 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 vector
real vector

Constant vector, specified as a vector of doubles. d represents the additive constant term in the expression C*x - d. d is M-by-1, where M is the number of equations.

Example: d = [5;0;-12]

Data Types: double

`A` — Linear inequality constraint matrix
real matrix

Linear inequality constraint matrix, specified as a matrix of doubles. A represents the linear coefficients in the constraints A*x ≤ b. A has size Mineq-by-N, 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 vector
real vector

Linear 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-by-N.

Example: [4,0]

Data Types: double

`Aeq` — Linear equality constraint matrix
`[]` (default) | real matrix

Linear equality constraint matrix, specified as a matrix of doubles. Aeq represents the linear coefficients in the constraints Aeq*x = beq. Aeq has size Meq-by-N, 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 vector

Linear 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-by-N.

Example: [4,0]

Data Types: double

`lb` — Lower bounds
`[]` (default) | real vector or array

Lower 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 array

Upper 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 array

Initial point for the solution process, specified as a vector or array of doubles. x0 is used only by the 'trust-region-reflective' algorithm. Optional.

If you do not provide an x0 for the 'trust-region-reflective' 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 app

Options 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

`Algorithm`	Choose the algorithm: `'interior-point'` (default) `'trust-region-reflective'` The `'trust-region-reflective'` algorithm allows only upper and lower bounds, meaning no linear inequalities or equalities. If you specify both the `'trust-region-reflective'` and linear constraints, `lsqlin` uses the `'interior-point'` algorithm. The `'trust-region-reflective'` algorithm does not allow equal upper and lower bounds. 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 `'on'` or the default `'off'`.
`Display`	Level of display returned to the command line. `'off'` or `'none'` displays no output. `'final'` displays just the final output (default). The `'interior-point'` algorithm allows additional values: `'iter'` gives iterative display. `'iter-detailed'` gives iterative display with a detailed exit message. `'final-detailed'` displays just the final output, with a detailed exit message.
`MaxIterations`	Maximum number of iterations allowed, a positive integer. The default value is `200`.

trust-region-reflective Algorithm Options

`FunctionTolerance`	Termination tolerance on the function value, a positive scalar. The default is `100*eps`, about `2.2204e-14`.
`JacobianMultiplyFcn`	Function handle for the Jacobian multiply function. For large-scale structured problems, this function should compute the Jacobian matrix product `CY`, `C'Y`, or `C'(CY)` without actually forming `C`. Write the function in the form W = jmfun(Jinfo,Y,flag) where `Jinfo` contains a matrix used to compute `CY` (or `C'Y`, or `C'(CY)`). `jmfun` must compute one of three different products, depending on the value of `flag` that `lsqlin` passes: If `flag == 0` then `W = C'(CY)`. If `flag > 0` then `W = CY`. If `flag < 0` then `W = C'Y`. In each case, `jmfun` need not form `C` explicitly. `lsqlin` uses `Jinfo` to compute the preconditioner. See Passing Extra Parameters for information on how to supply extra parameters if necessary. 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 `max(1,floor(numberOfVariables/2))`. For more information, see Trust-Region-Reflective Algorithm.
`OptimalityTolerance`	Termination tolerance on the first-order optimality, a positive scalar. The default is `100*eps`, about `2.2204e-14`. See First-Order Optimality Measure.
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 `PrecondBandWidth` to `Inf` uses a direct factorization (Cholesky) rather than the conjugate gradients (CG). The direct factorization is computationally more expensive than CG, but produces a better quality step toward the solution. For more information, see Trust-Region-Reflective Algorithm.
`SubproblemAlgorithm`	Determines how the iteration step is calculated. The default, `'cg'`, takes a faster but less accurate step than `'factorization'`. See Trust-Region-Reflective Least Squares.
TolPCG	Termination tolerance on the PCG (preconditioned conjugate gradient) iteration, a positive scalar. The default is `0.1`.
`TypicalX`	Typical `x` values. The number of elements in `TypicalX` is equal to the number of variables. The default value is `ones(numberofvariables,1)`. `lsqlin` uses `TypicalX` internally for scaling. `TypicalX` has an effect only when `x` has unbounded components, and when a `TypicalX` value for an unbounded component is larger than `1`.

interior-point Algorithm Options

`ConstraintTolerance`	Tolerance on the constraint violation, a positive scalar. The default is `1e-8`.
`OptimalityTolerance`	Termination tolerance on the first-order optimality, a positive scalar. The default is `1e-8`. See First-Order Optimality Measure.
`StepTolerance`	Termination tolerance on `x`, a positive scalar. The default is `1e-12`.

`problem` — Optimization problem
structure

Optimization problem, specified as a structure with the following fields.

`C`	Matrix multiplier in the term `C*x - d`
`d`	Additive constant in the term `C*x - d`
`Aineq`	Matrix for linear inequality constraints
`bineq`	Vector for linear inequality constraints
`Aeq`	Matrix for linear equality constraints
`beq`	Vector for linear equality constraints
`lb`	Vector of lower bounds
`ub`	Vector of upper bounds
`x0`	Initial point for `x`
`solver`	`'lsqlin'`
`options`	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

Output Arguments

`x` — Solution
real vector

Solution, returned as a vector that minimizes the norm of C*x-d subject to all bounds and linear constraints.

`resnorm` — Objective value
real scalar

Objective value, returned as the scalar value norm(C*x-d)^2.

`residual` — Solution residuals
real vector

Solution residuals, returned as the vector C*x-d.

`exitflag` — Algorithm stopping condition
integer

Algorithm 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.

`1`	Function converged to a solution `x`.
`2`	Step size smaller than `options.StepTolerance`, constraints satisfied. (`interior-point` algorithm)
`3`	Change in the residual was smaller than the specified tolerance.
`0`	Number of iterations exceeded `options.MaxIterations`.
`-2`	The problem is infeasible. Or, for the `interior-point` algorithm, step size smaller than `options.StepTolerance`, but constraints are not satisfied.
`-4`	Ill-conditioning prevents further optimization.
`-7`	Magnitude of search direction became too small. No further progress could be made.

The exit message for the interior-point algorithm can give more details on the reason lsqlin stopped, such as exceeding a tolerance. See Exit Flags and Exit Messages.

`output` — Solution process summary
structure

Solution process summary, returned as a structure containing information about the optimization process.

`iterations`	Number of iterations the solver took.
`algorithm`	One of these algorithms: `'interior-point'` `'trust-region-reflective'`
`constrviolation`	Constraint violation that is positive for violated constraints (not returned for the `'trust-region-reflective'` algorithm). `constrviolation = max([0;norm(Aeqx-beq, inf);(lb-x);(x-ub);(Ax-b)])`
`message`	Exit message.
`firstorderopt`	First-order optimality at the solution. See First-Order Optimality Measure.
`cgiterations`	Number of conjugate gradient iterations the solver performed. Nonempty only for the `'trust-region-reflective'` algorithm.

See Output Structures.

`lambda` — Lagrange multipliers
structure

Lagrange multipliers, returned as a structure with the following fields.

`lower`	Lower bounds `lb`
`upper`	Upper bounds `ub`
`ineqlin`	Linear inequalities
`eqlin`	Linear equalities

See Lagrange Multiplier Structures.

Tips

For problems with no constraints, you can use mldivide (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 trust-region-reflective 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 trust-region-reflective algorithm with a Jacobian multiply function. For information, see trust-region-reflective Algorithm Options.

Algorithms