Solve Constrained zero-one Integer Linear Program Using Simulated Annealing

Question

Recently, I was reading about several techniques that solves Unconstrained Mixed Integer Linear Programs (UM-ILP) using a meta-heuristic algorithm called simulated annealing. I was thinking about the Constrained zero-one ILP. I have a linear objective function with a linear set of equality/inequality constraints and I'm thinking about reformulating the problem using the kkt/Lagrangian function. However, I'm not sure if it is even the right approach given that my optimization problem have binary variables and it is linear and hence, solutions such as the penalty method and barrier log would work best for me. My goal is to transform a constrained problem to non-constrained then apply an optimization method like simulated annealing to solve my new formulation of the problem but i'm not really sure which method is even the right approach? Also from what I read the problem will not be a simple minimization problem anymore but instead a saddle point problem. Thank you all for your time and replies.