Mixed Integer Quadratic Programming with linear constraints in Matlab calling Gurobi

Question

I have some troubles to understand how to implement the following MIQP (Mixed Integer Quadratic Programming) with linear constraints in Matlab calling Gurobi.

Let me explain in a schematic way my setting.

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(1) x is the unknown and it is a column vector with size 225x1.

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(2) The objective function (which should be minimised wrto x) looks like

which can be rewritten as

I have a Matlab script computing alpha, Q,c (Q,c sparse) when some_known_parameters1 are given:

function [alpha, Q,c]=matrix_objective_function(some_known_parameters1)

%...

end

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(3) The constraints are linear in x, include equalities and inequalities, and are written in the form

I have a Matlab script computing Aeq,beq,Aineq,bineq (Aeq,Aineq sparse) when some_known_parameters2 is given:

function [Aeq,beq,Aineq,bineq]=constraints(some_known_parameters2)

%...

end

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(4) Some components of x are restricted to be in {0,1}. I have a Matlab script producing a string of letters B (binary), C (continous) when some_known_parameters3 is given:

function type=binary_continuous(some_known_parameters3)

%...

end

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Now, I need to put together (1)-(4) using Gurobi. I am struggling to understand how. I found this example but it looks very cryptic to me. Below I report some lines I have attempted to write, but they are incomplete and I would like your help to complete them.

clear 
rng default

%Define some_known_parameters1, 
 some_known_parameters2,some_known_parameters3 [...]

%1) generate alpha,Q,c,Aeq,beq,Aineq,bineq,type with Q,c,Aeq, Aineq sparse
[alpha, Q,c]=matrix_objective_function(some_known_parameters1)
[Aeq,beq,Aineq,bineq]=constraints(some_known_parameters2)
type=binary_continuous(some_known_parameters3)



%2) Set up Gurobi
clear model;
model.A=[Aineq; Aeq];
model.rhs=full([bineq(:); beq(:)]); 
model.sense=[repmat('<', size(Aineq,1),1); repmat('=', size(Aeq,1),1)];
model.Q=Q; %not sure?
model.alpha=alpha; %not sure?
model.c=c; %not sure?
model.vtype=type;
result=gurobi(model); %how do I get just the objective function here without the minimiser?

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Questions:

(1) I'm not sure about

model.Q=Q; 
model.alpha=alpha; 
model.c=c;

I'm just trying to set the matrices of the objective function using the letters provided here but it gives me error. The example here seems to me doing

model.Q=Q; 
model.obj=c; 

But then how do I set alpha? Is it ignoring it because it does not change the set of solutions?

(2) How do I get as output stored in a matrix just the minimum value of the objective function without the corresponding x?

Answer 1

(1) You're right, there's no need to pass the constant alpha since it doesn't affect the optimal solution. Gurobi's MATLAB API only accepts sparse matrices. Furthermore model.obj is always the c vector in the problem statement:

model.Q = sparse(Q); 
model.obj = c;

(2) To get the optimal objective value, you first need to pass your model to gurobi and solve it. Then you can access it via the objval attribute:

results = gurobi(model);
val = results.objval + alpha