Parameters

H :

a symmetric matrix of double, represents coefficients of quadratic in the quadratic problem.

f :

a vector of double, represents coefficients of linear in the quadratic problem

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.

fopt :

a double, the value of the function at 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 quadratic optimization problem specified by :

The routine calls Ipopt for solving the quadratic 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 https://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

//Ref : example 14 :
//https://www.me.utexas.edu/~jensen/ORMM/supplements/methods/nlpmethod/S2_quadratic.pdf
// min. -8*x1*x1 -16*x2*x2 + x1 + 4*x2
// such that
//    x1 + x2 <= 5,
//    x1 <= 3,
//    x1 >= 0,
//    x2 >= 0
H = [2 0
0 8];
f = [-8; -16];
A = [1 1;1 0];
b = [5;3];
lb = [0; 0];
ub = [%inf; %inf];
[xopt,fopt,exitflag,output,lambda] = qpipoptmat(H,f,A,b,[],[],lb,ub)
// Press ENTER to continue

Examples

//Find x in R^6 such that:
Aeq= [1,-1,1,0,3,1;
-1,0,-3,-4,5,6;
2,5,3,0,1,0];
beq=[1; 2; 3];
A= [0,1,0,1,2,-1;
-1,0,2,1,1,0];
b = [-1; 2.5];
lb=[-1000; -10000; 0; -1000; -1000; -1000];
ub=[10000; 100; 1.5; 100; 100; 1000];
x0 = repmat(0,6,1);
param = list("MaxIter", 300, "CpuTime", 100);
//and minimize 0.5*x'*H*x + f'*x with
f=[1; 2; 3; 4; 5; 6]; H=eye(6,6);
[xopt,fopt,exitflag,output,lambda]=qpipoptmat(H,f,A,b,Aeq,beq,lb,ub,x0,param)

Authors

• Keyur Joshi, Saikiran, Iswarya, Harpreet Singh