lsqlin Solves a linear quadratic problem

Parameters

C :

a matrix of double, represents the multiplier of the solution x in the expression C⋅x - d. Number of columns in C is equal to the number of elements in x.

d :

a vector of double, represents the additive constant term in the expression C⋅x - d. Number of elements in d is equal to the number of rows in C matrix.

A :

a matrix of double, represents the linear coefficients in the inequality constraints A⋅x ≤ b.

b :

a vector of double, represents the linear coefficients in the inequality constraints A⋅x ≤ b.

Aeq :

a matrix of double, represents the linear coefficients in the equality constraints Aeq⋅x = beq.

beq :

a vector of double, represents the linear coefficients in the equality constraints Aeq⋅x = beq.

lb :

a vector of double, contains lower bounds of the variables.

ub :

a vector of double, contains upper bounds of the variables.

x0 :

a vector of double, contains initial guess of variables.

param :

a list containing the parameters to be set.

xopt :

a vector of double, the computed solution of the optimization problem.

resnorm :

a double, objective value returned as the scalar value norm(C⋅x-d)^2.

residual :

a vector of double, solution residuals returned as the vector d-C⋅x.

exitflag :

The exit status. See below for details.

output :

The structure consist of statistics about the optimization. See below for details.

lambda :

The structure consist of the Lagrange multipliers at the solution of problem. See below for details.

Description

Search the minimum of a constrained linear least square problem specified by :

The routine calls Ipopt for solving the linear least square problem, Ipopt is a library written in C++.

The options allows the user to set various parameters of the Optimization problem. It should be defined as type "list" and contains the following fields.

• Syntax : options= list("MaxIter", [---], "CpuTime", [---]);
• MaxIter : a Scalar, containing the Maximum Number of Iteration that the solver should take.
• CpuTime : a Scalar, containing the Maximum amount of CPU Time that the solver should take.
• Default Values : options = list("MaxIter", [3000], "CpuTime", [600]);

The exitflag allows to know the status of the optimization which is given back by Ipopt.

• exitflag=0 : Optimal Solution Found
• exitflag=1 : Maximum Number of Iterations Exceeded. Output may not be optimal.
• exitflag=2 : Maximum CPU Time exceeded. Output may not be optimal.
• exitflag=3 : Stop at Tiny Step.
• exitflag=4 : Solved To Acceptable Level.
• exitflag=5 : Converged to a point of local infeasibility.

For more details on exitflag see the ipopt documentation, go to http://www.coin-or.org/Ipopt/documentation/

The output data structure contains detailed informations about the optimization process. It has type "struct" and contains the following fields.

• output.iterations: The number of iterations performed during the search
• output.constrviolation: The max-norm of the constraint violation.

The lambda data structure contains the Lagrange multipliers at the end of optimization. In the current version the values are returned only when the the solution is optimal. It has type "struct" and contains the following fields.

• lambda.lower: The Lagrange multipliers for the lower bound constraints.
• lambda.upper: The Lagrange multipliers for the upper bound constraints.
• lambda.eqlin: The Lagrange multipliers for the linear equality constraints.
• lambda.ineqlin: The Lagrange multipliers for the linear inequality constraints.

Examples

//A simple linear least square example
C = [ 2 0;
-1 1;
0 2]
d = [1
0
-1];
A = [10 -2;
-2 10];
b = [4
-4];
[xopt,resnorm,residual,exitflag,output,lambda] = lsqlin(C,d,A,b)
// Press ENTER to continue

Examples

//A basic example for equality, inequality constraints and variable bounds
C = [1 1 1;
1 1 0;
0 1 1;
1 0 0;
0 0 1]
d = [89;
67;
53;
35;
20;]
A = [3 2 1;
2 3 4;
1 2 3];
b = [191
209
162];
Aeq = [1 2 1];
beq = 10;
lb = repmat(0.1,3,1);
ub = repmat(4,3,1);
[xopt,resnorm,residual,exitflag,output,lambda] = lsqlin(C,d,A,b,Aeq,beq,lb,ub)

Authors

• Harpreet Singh