Main Content

`lsqlin`

Create pseudorandom data for the problem of minimizing the norm of ```
C*x –
d
```

subject to bounds and linear inequality constraints. Create a
problem for 15 variables, subject to the bounds `lb = –1`

and
`ub = 1`

and subject to 150 linear constraints ```
A*x
<= b
```

.

N = 15; % Number of variables rng default % For reproducibility A = randn([10*N,N]); b = 5*ones(size(A,1),1); Aeq = []; % No equality constraints beq = []; ub = ones(N,1); lb = -ub; C = 10*eye(N) + randn(N); C = (C + C.')/2; % Symmetrize the matrix d = 20*randn(N,1);

`lsqlin`

Code generation requires the `'active-set'`

algorithm, which
requires an initial point `x0`

. To solve the problem in MATLAB^{®} using the algorithm required by code generation, set options and an
initial point.

x0 = zeros(size(d)); options = optimoptions('lsqlin','Algorithm','active-set');

To solve the problem, call `lsqlin`

.

[x,fv,~,ef,output,lam] = lsqlin(C,d,A,b,Aeq,beq,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 value of the optimality tolerance, and constraints are satisfied to within the value of the constraint tolerance.

After `lsqlin`

solves this problem, look at the number of
nonzero Lagrange multipliers of each type. See how many solution components are
unconstrained by subtracting the total number of nonzero Lagrange
multipliers.

nl = nnz(lam.lower); nu = nnz(lam.upper); ni = nnz(lam.ineqlin); nunconstrained = N - nl - nu - ni; fprintf('Number of solution components at lower bounds: %g\n',nl); fprintf('Number of solution components at upper bounds: %g\n',nu); fprintf('Number of solution components at inequality: %g\n',ni); fprintf('Number of unconstrained solution components: %g\n',nunconstrained);

Number of solution components at lower bounds: 3 Number of solution components at upper bounds: 2 Number of solution components at inequality: 5 Number of unconstrained solution components: 5

To solve the same problem using code generation, complete the following steps.

Write a function that incorporates all of the preceding steps. To produce less output, set the

`Display`

option to`'off'`

.function [x,fv,lam] = solvelsqlin N = 15; % Number of variables rng default % For reproducibility A = randn([10*N,N]); b = 5*ones(size(A,1),1); Aeq = []; % No equality constraints beq = []; ub = ones(N,1); lb = -ub; C = 10*eye(N) + randn(N); C = (C + C.')/2; % Symmetrize the matrix d = 20*randn(N,1); x0 = zeros(size(d)); options = optimoptions('lsqlin','Algorithm','active-set',... 'Display','off'); [x,fv,~,ef,output,lam] = lsqlin(C,d,A,b,Aeq,beq,lb,ub,x0,options); nl = nnz(lam.lower); nu = nnz(lam.upper); ni = nnz(lam.ineqlin); nunconstrained = N - nl - nu - ni; fprintf('Number of solution components at lower bounds: %g\n',nl); fprintf('Number of solution components at upper bounds: %g\n',nu); fprintf('Number of solution components at inequality: %g\n',ni); fprintf('Number of unconstrained solution components: %g\n',nunconstrained); end

Create a configuration for code generation. In this case, use

`'mex'`

.`cfg = coder.config('mex');`

Generate code for the

`solvelsqlin`

function.codegen -config cfg solvelsqlin

Test the generated code by running the generated file, which is named

`solvelsqlin_mex.mexw64`

or similar.[x2,fv2,lam2] = solvelsqlin_mex;

Number of solution components at lower bounds: 1 Number of solution components at upper bounds: 5 Number of solution components at inequality: 6 Number of unconstrained solution components: 3

The number of solution components at various bounds has changed from the previous solution. To see whether these differences are important, compare the solution point differences and function value differences.

disp([norm(x - x2), abs(fv - fv2)])

1.0e-12 * 0.0007 0.2274

The differences between the two solutions are negligible.

`lsqlin`

| `optimoptions`

| `quadprog`

| `codegen`

(MATLAB Coder)