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intqpipopt

Solves a linear quadratic problem.

Calling Sequence

xopt = intqpipopt(H,f)
xopt = intqpipopt(H,f,intcon)
xopt = intqpipopt(H,f,intcon,A,b)
xopt = intqpipopt(H,f,intcon,A,b,Aeq,beq)
xopt = intqpipopt(H,f,intcon,A,b,Aeq,beq,lb,ub)
xopt = intqpipopt(H,f,intcon,A,b,Aeq,beq,lb,ub,x0)
xopt = intqpipopt(H,f,intcon,A,b,Aeq,beq,lb,ub,x0,options)
xopt = intqpipopt(H,f,intcon,A,b,Aeq,beq,lb,ub,x0,options,"file_path")
[xopt,fopt,exitflag,output] = intqpipopt( ... )

Input Parameters

H :

A symmetric matrix of doubles, representing the Hessian of the quadratic problem.

f :

A vector of doubles, representing coefficients of the linear terms in the quadratic problem.

intcon :

A vector of integers, representing the variables that are constrained to be integers.

A :

A matrix of doubles, containing the coefficients of linear inequality constraints of size (m X n) where 'm' is the number of linear inequality constraints.

b :

A vector of doubles, related to 'A' and represents the linear coefficients in the linear inequality constraints of size (m X 1).

Aeq :

A matrix of doubles, containing the coefficients of linear equality constraints of size (m1 X n) where 'm1' is the number of linear equality constraints.

beq :

A vector of double, vector of doubles, related to 'Aeq' and represents the linear coefficients in the equality constraints of size (m1 X 1).

lb :

A vector of doubles, containing the lower bounds of the variables of size (1 X n) or (n X 1) where 'n' is the number of variables.

ub :

A vector of doubles, containing the upper bounds of the variables of size (1 X n) or (n X 1) where 'n' is the number of variables.

x0 :

A vector of doubles, containing the starting values of variables of size (1 X n) or (n X 1) where 'n' is the number of variables.

options :

A list, containing the option for user to specify. See below for details.

file_path :

path to bonmin opt file if used.

Outputs

xopt :

A vector of doubles, containing the computed solution of the optimization problem.

fopt :

A double, containing the value of the function at xopt.

exitflag :

An integer, containing the flag which denotes the reason for termination of algorithm. See below for details.

output :

A structure, containing the information about the optimization. See below for details.

Description

Search the minimum of a constrained linear quadratic optimization problem specified by :

intqpipopt calls Bonmin, a library written in C++ to solve the quadratic problem.

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

For more details on exitflag, see the Bonmin documentation which can be found on http://www.coin-or.org/Bonmin

The output data structure contains detailed information about the optimization process. It is of type "struct" and contains the following fields.

A few examples displaying the various functionalities of intqpipopt have been provided below. You will find a series of problems and the appropriate code snippets to solve them.

Example

Here we solve a simple objective function.

Find x in R^6 such that it minimizes:

//Example 1:
//Minimize 0.5*x'*H*x + f'*x with
f=[1; 2; 3; 4; 5; 6]; H=eye(6,6);
//Integer Constraints
intcon = [2 ,4];
//Running intqpipopt
[xopt,fopt,exitflag,output,lambda]=qpipoptmat(H,f)

Example

We proceed to add simple linear inequality constraints. 

//Example 2:
f=[1; 2; 3; 4; 5; 6]; H=eye(6,6);
//Inequality constraints
A= [0,1,0,1,2,-1;
-1,0,2,1,1,0];
b = [-1; 2.5];
//Integer Constraints
intcon = [2 ,4];
//Running intqpipopt
[xopt,fopt,exitflag,output,lambda]=intqpipopt(H,f,intcon,A,b)

Example

Here we build up on the previous example by adding linear equality constraints. We add the following constraints to the problem specified above: 

//Example 3:
//Minimize 0.5*x'*H*x + f'*x with
f=[1; 2; 3; 4; 5; 6]; H=eye(6,6);
//Inequality constraints
A= [0,1,0,1,2,-1;
-1,0,2,1,1,0];
b = [-1; 2.5];
//Equality constraints
Aeq= [1,-1,1,0,3,1;
-1,0,-3,-4,5,6;
2,5,3,0,1,0];
beq=[1; 2; 3];
//Integer Constraints
intcon = [2 ,4];
//Running intqpipopt
[xopt,fopt,exitflag,output,lambda]=intqpipopt(H,f,intcon,A,b,Aeq,beq)

Example

In this example, we proceed to add the upper and lower bounds to the objective function.

//Example 4:
//Minimize 0.5*x'*H*x + f'*x with
f=[1; 2; 3; 4; 5; 6]; H=eye(6,6);
//Inequality constraints
A= [0,1,0,1,2,-1;
-1,0,2,1,1,0];
b = [-1; 2.5];
//Equality constraints
Aeq= [1,-1,1,0,3,1;
-1,0,-3,-4,5,6;
2,5,3,0,1,0];
beq=[1; 2; 3];
//Variable bounds
lb=[-1000; -10000; 0; -1000; -1000; -1000];
ub=[10000; 100; 1.5; 100; 100; 1000];
//Integer Constraints
intcon = [2 ,4];
//Running intqpipopt
[xopt,fopt,exitflag,output,lambda]=intqpipopt(H,f,intcon,A,b,Aeq,beq,lb,ub)

Example

In this example, we initialize the values of x to speed up the computation. We further enhance the functionality of qpipoptmat by setting input options.

//Example 5:
//Minimize 0.5*x'*H*x + f'*x with
f=[1; 2; 3; 4; 5; 6]; H=eye(6,6);
//Inequality constraints
A= [0,1,0,1,2,-1;
-1,0,2,1,1,0];
b = [-1; 2.5];
//Equality constraints
Aeq= [1,-1,1,0,3,1;
-1,0,-3,-4,5,6;
2,5,3,0,1,0];
beq=[1; 2; 3];
//Variable bounds
lb=[-1000; -10000; 0; -1000; -1000; -1000];
ub=[10000; 100; 1.5; 100; 100; 1000];
//Initial guess and options
x0 = repmat(0,6,1);
options = list("MaxIter", 300, "CpuTime", 100);
//Integer Constraints
intcon = [2 ,4];
//Running intqpipopt
[xopt,fopt,exitflag,output,lambda]=intqpipopt(H,f,intcon,A,b,Aeq,beq,lb,ub,x0,options)

Example

Infeasible Problems: Find x in R^6 such that it minimizes the objective function used above under the following constraints: 

//Example 6:
//Minimize 0.5*x'*H*x + f'*x with
f=[1; 2; 3; 4; 5; 6]; H=eye(6,6);
f=[1; 2; 3; 4; 5; 6]; H=eye(6,6);
//Inequality constraints
A= [0,1,0,1,2,-1;
-1,0,2,1,1,0];
b = [-1; 2.5];
//Equality constraints
Aeq= [0,1,0,1,2,-1;
-1,0,-3,-4,5,6];
beq=[4; 2];
//Integer Constraints
intcon = [2 ,4];
//Running intqpipopt
[xopt,fopt,exitflag,output,lambda]=intqpipopt(H,f,intcon,A,b,Aeq,beq)

Example

Unbounded Problems: Find x in R^6 such that it minimizes the objective function used above under the following constraints:

//Example 7:
//Minimize 0.5*x'*H*x + f'*x with
f=[1; 2; 3; 4; 5; 6]; H=eye(6,6); H(1,1) = -1;
//Inequality constraints
A= [0,1,0,1,2,-1;
-1,0,2,1,1,0];
b = [-1; 2.5];
//Equality constraints
Aeq= [1,-1,1,0,3,1;
-1,0,-3,-4,5,6];
beq=[1; 2];
intcon = [2 ,4];
//Running intqpipopt
[xopt,fopt,exitflag,output,lambda]=intqpipopt(H,f,intcon,A,b,Aeq,beq)

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