intfmincon

Calling Sequence

xopt = intfmincon(f,x0,intcon,A,b)
xopt = intfmincon(f,x0,intcon,A,b,Aeq,beq)
xopt = intfmincon(f,x0,intcon,A,b,Aeq,beq,lb,ub)
xopt = intfmincon(f,x0,intcon,A,b,Aeq,beq,lb,ub,nlc)
xopt = intfmincon(f,x0,intcon,A,b,Aeq,beq,lb,ub,nlc,options)
[xopt,fopt] = intfmincon(.....)
[xopt,fopt,exitflag]= intfmincon(.....)
[xopt,fopt,exitflag,gradient]=intfmincon(.....)
[xopt,fopt,exitflag,gradient,hessian]=intfmincon(.....)

Input Parameters

f :: A function, representing the objective function of the problem.
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.
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.
nlc :: A function, representing the Non-linear Constraints functions(both Equality and Inequality) of the problem. It is declared in such a way that non-linear inequality constraints (c), and the non-linear equality constraints (ceq) are defined as separate single row vectors.
options :: A list, containing the option for user to specify. See below for details.

Outputs

xopt :: A vector of doubles, containing the 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.
gradient :: a vector of doubles, containing the Objective's gradient of the solution.
hessian :: a matrix of doubles, containing the Objective's hessian of the solution.

Description

Search the minimum of a mixed integer constrained optimization problem specified by : Find the minimum of f(x) such that

intfmincon calls Bonmin, an optimization library written in C++, to solve the Constrained Optimization problem.

Options

The options allow the user to set various parameters of the Optimization problem. The syntax for the options is given by:

options= list("IntegerTolerance", [---], "MaxNodes",[---], "MaxIter", [---], "AllowableGap",[---] "CpuTime", [---],"gradobj", "off", "hessian", "off" );

IntegerTolerance : A Scalar, a number with that value of an integer is considered integer.
MaxNodes : A Scalar, containing the maximum number of nodes that the solver should search.
CpuTime : A scalar, specifying the maximum amount of CPU Time in seconds that the solver should take.
AllowableGap : A scalar, that specifies the gap between the computed solution and the the objective value of the best known solution stop, at which the tree search can be stopped.
MaxIter : A scalar, specifying the maximum number of iterations that the solver should take.
gradobj : A string, to turn on or off the user supplied objective gradient.
hessian : A scalar, to turn on or off the user supplied objective hessian.

The default values for the various items are given as:

options = list('integertolerance',1d-06,'maxnodes',2147483647,'cputime',1d10,'allowablegap',0,'maxiter',2147483647,'gradobj',"off",'hessian',"off")

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

0 : Optimal Solution Found
1 : InFeasible Solution.
2 : Objective Function is Continuous Unbounded.
3 : Limit Exceeded.
4 : User Interrupt.
5 : MINLP Error.

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

A few examples displaying the various functionalities of intfmincon 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, subjected to three linear inequality constraints.

Find x in R^2 such that it minimizes:

//Example 1:
//Objective function to be minimised
function [y, dy]=f(x)
y=-x(1)-x(2)/3;
dy= [-1,-1/3];
endfunction
//Starting point
x0=[0 , 0];
//Integer constraints
intcon = [1];
//Initializing the linear inequality constraints
A=[1,1 ; 1,1/4 ; 1,-1 ;];
b=[2;1;2];
//Calling Bonmin
[x,fval,exitflag,grad,hessian] =intfmincon(f, x0,intcon,A,b)
// Press ENTER to continue

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 2:
//Objective function to be minimised
function [y, dy]=f(x)
y=-x(1)-x(2)/3;
dy= [-1,-1/3];
endfunction
//Starting point
x0=[0 , 0];
//Integer constraints
intcon = [1];
//Initializing the linear inequality constraints
A=[1,1 ; 1,1/4 ; 1,-1 ;];
b=[2;1;2];
//Linear equality constraints
Aeq=[1,-1;2,1];
beq=[1,2];
//Calling Bonmin
[x,fval,exitflag,grad,hessian] =intfmincon(f, x0,intcon,A,b,Aeq,beq)
// Press ENTER to continue

Example

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

Find x in R^2 such that it minimizes:

//Example 3:
//Objective function to be minimised
function [y, dy]=f(x)
y=-x(1)-x(2)/3;
dy= [-1,-1/3];
endfunction
//Starting point
x0=[0 , 0];
//Integer constraints
intcon = [1];
//Initializing the linear inequality constraints
A=[1,1 ; 1,1/4 ; 1,-1 ;];
b=[2;1;2];
//Linear equality constraints
Aeq=[1,-1;2,1];
beq=[1,2];
//Adding the variable bounds  
lb=[-1, -%inf];
ub=[%inf, 1];
//Calling Bonmin
[x,fval,exitflag,grad,hessian] =intfmincon(f, x0,intcon,A,b,Aeq,beq,lb,ub)
// Press ENTER to continue

Example

Finally, we add the non-linear constraints to the problem. Note that there is a notable difference in the way this is done as compared to defining the linear constraints.

//Example 4:
//Objective function to be minimised
function [y, dy]=f(x)
y=x(1)*x(2)+x(2)*x(3);
dy= [x(2),x(1)+x(3),x(2)];
endfunction
//Starting point, linear constraints and variable bounds
x0=[0.1 , 0.1 , 0.1];
intcon = [2]
A=[];
b=[];
Aeq=[];
beq=[];
lb=[];
ub=[];
//Nonlinear constraints
function [c, ceq, cg, cgeq]=nlc(x)
c = [x(1)^2 - x(2)^2 + x(3)^2 - 2 , x(1)^2 + x(2)^2 + x(3)^2 - 10];
ceq = [];
cg=[2*x(1) , -2*x(2) , 2*x(3) ; 2*x(1) , 2*x(2) , 2*x(3)];
cgeq=[];
endfunction
//Options
options=list("MaxIter", [1500], "CpuTime", [500], "GradObj", "on","GradCon", "on");
//Calling Ipopt
[x,fval,exitflag,output] =intfmincon(f, x0,intcon,A,b,Aeq,beq,lb,ub,nlc,options)
// Press ENTER to continue

Example

We can further enhance the functionality of intfmincon by setting input options. We can pre-define the gradient of the objective function and/or the hessian of the lagrange function and thereby improve the speed of computation. This is elaborated on in example 5. We take the following problem and add simple non-linear constraints, specify the gradients and the hessian of the Lagrange Function. We also set solver parameters using the options.

//Example 5:
//Objective function to be minimised
function [y, dy]=f(x)
y=x(1)*x(2)+x(2)*x(3);
dy= [x(2),x(1)+x(3),x(2)];
endfunction
//Starting point, linear constraints and variable bounds
x0=[0.1 , 0.1 , 0.1];
intcon = [2]
A=[];
b=[];
Aeq=[];
beq=[];
lb=[];
ub=[];
//Nonlinear constraints
function [c, ceq, cg, cgeq]=nlc(x)
c = [x(1)^2 - x(2)^2 + x(3)^2 - 2 , x(1)^2 + x(2)^2 + x(3)^2 - 10];
ceq = [];
cg=[2*x(1) , -2*x(2) , 2*x(3) ; 2*x(1) , 2*x(2) , 2*x(3)];
cgeq=[];
endfunction
//Options
options=list("MaxIter", [1500], "CpuTime", [500], "GradObj", "on","GradCon", "on");
//Calling Ipopt
[x,fval,exitflag,output] =intfmincon(f, x0,intcon,A,b,Aeq,beq,lb,ub,nlc,options)
// Press ENTER to continue

Example

Infeasible Problems: Find x in R^3 such that it minimizes:

//Example 6:
//Objective function to be minimised
function [y, dy]=f(x)
y=x(1)*x(2)+x(2)*x(3);
dy= [x(2),x(1)+x(3),x(2)];
endfunction
//Starting point, linear constraints and variable bounds
x0=[1,1,1];
intcon = [2]
A=[];
b=[];
Aeq=[];
beq=[];
lb=[0 0.2,-%inf];
ub=[0.6 %inf,1];
//Nonlinear constraints
function [c, ceq, cg, cgeq]=nlc(x)
c=[x(1)^2-1,x(1)^2+x(2)^2-1,x(3)^2-1];
ceq=[x(1)^3-0.5,x(2)^2+x(3)^2-0.75];
cg = [2*x(1),0,0;2*x(1),2*x(2),0;0,0,2*x(3)];
cgeq = [3*x(1)^2,0,0;0,2*x(2),2*x(3)];
endfunction
//Options
options=list("MaxIter", [1500], "CpuTime", [500], "GradObj", "on","GradCon", "on");
//Calling Bonmin
[x,fval,exitflag,grad,hessian] =intfmincon(f, x0,intcon,A,b,Aeq,beq,lb,ub,nlc,options)
// Press ENTER to continue

Example

Unbounded Problems: Find x in R^3 such that it minimizes:

//Example 7:
//The below problem is an unbounded problem:
//Find x in R^3 such that it minimizes:
//f(x)= -(x1^2 + x2^2 + x3^2)
//x0=[0.1 , 0.1 , 0.1]
//  x1 <= 0
//  x2 <= 0
//  x3 <= 0
//Objective function to be minimised
function y=f(x)
y=-(x(1)^2+x(2)^2+x(3)^2);
endfunction
//Starting point, linear constraints and variable bounds
x0=[0.1 , 0.1 , 0.1];
intcon = [3]
A=[];
b=[];
Aeq=[];
beq=[];
lb=[];
ub=[0,0,0];
//Options
options=list("MaxIter", [1500], "CpuTime", [500]);
//Calling Bonmin
[x,fval,exitflag,grad,hessian] =intfmincon(f, x0,intcon,A,b,Aeq,beq,lb,ub,[],options)
// Press ENTER to continue

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intfmincon

Calling Sequence

Input Parameters

Outputs

Description

Options

Example

Example

Example

Example

Example

Example

Example

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