ILP and LP with integer decision relaxation and Mult objective formulation

I was reading a research article, where they proposed ILP (say P1). Later on, they converted the ILP into LP by relaxing the integer decision variable (say P2). Further, they used Solver to find the solution to P2. I tried to find out whether Gurobi can solve the both problems. I have the following doubts in my mind:

1. If Gurobi can solve ILP, then what is the need to convert it into P2 manually using techniques such as Lagrange analysis/KKT condition? Then solve P2 by Gurobi. Does it have any benefit

2. If Gurobi can convert the P1 to P2, is it necessary to use manual conversion as stated in point 1?

2. Further, I have doubts about the formulation of multi multi-objective problem. A brief overview of the problem is given here..

V[1]=[1,2,3,6,7] #set 1

V[2]=[4,12,13,15] #set2

V[3]=[5,8,9,10,11,14] #set3

#A[i] is the first objective applied to each set

#B[i] is the second objective applied to each set

#C[i] is the third objective applied to each set

#W1, W2, W3 are weight factors for each objective

rs[g]= W1 sum(A[i] for i in V[g] ) - W2 sum ( B[i] for i in V[g] + W3 sum (C[i] for i in V[g])

The value of g can be 1 to 3

My objective is to max rs[g] for g in range(1,4)

In simple words, to maximize the rs[1], rs[2], rs[3] for each set.

Is it the correct way to represent this problem or not?

Maliheh Aramon

Gurobi Staff

April 29, 2025 01:18

Hi Ramesh,

Gurobi can directly solve ILPs, so there is no need to manually convert them to LPs. The underlying solution methodology for solving an ILP is called branch-and-bound which is an LP-based solver. For more background on this, you may find the article “Mixed-Integer Programming - A Primer on the Basics” helpful.

Additionally, the article “What types of models can Gurobi solve?” provides a comprehensive list of all supported model types.

Regarding your mention of a “multi multi-objective problem,” I'm not entirely sure what you mean by that. However, it seems you are dealing with a case where you have three final objectives—\(rs_1\), \(rs_2\), and \(rs_3\)—that you wish to maximize, subject to a common set of constraints defined over three corresponding sets of decision variables (sets 1, 2, and 3).

Gurobi supports solving multi-objective problems using both hierarchical and weighted approaches. I recommend reviewing Gurobi's documentation on multi-objective for details on how to model and configure these types of problems.

Best regards,

Maliheh

ILP and LP with integer decision relaxation and Mult objective formulation

Comments

Didn't find what you were looking for?