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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.

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

(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

(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

(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

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?

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?

(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