# Branch and bound method for solving non-convex integer non-linear multi-objective optimization problem?

Following are the characteristics of my problem:

Objective function: two non-linear functions and one linear function

Decision variable: two integer variables ($$X_1$$ and $$X_2$$)

Constraint: three (two bounding constraint and one relationship constraint)

• Constraint 1: $$0 \le X_1 \le A_1$$

• Constraint 2: $$0 \le X_2 \le A_2$$

• Constraint 3: $$X_1 + X_2 \le A_3$$

$$A_1,A_2$$ and $$A_3$$ are positive constant numbers.

Problem type: non-convex

Solution required: Global optimum

Can we solve this problem using brand-and-bound method?

• Really you again? With the same problem? OK. Oct 8 '21 at 18:00

However multi objective branch and bound methods are harder to come by, i am still not familiar with any solver that does it directly. I would consider an $$\varepsilon$$-constraint methodology to turn the multi objective problem into a series of single objective problems which can be solved to global optimality by standard MINLP solvers. In general the notion of solving a Multi-objective problem is ill defined for anything but linear multi-objective programs.