What are some techniques for generating valid inequalities and cutting planes for integer programming?

The inequality x+y <= 3 does not hold for all integer values of x and y. You might want to consider a better example to help explain what a valid cut or inequality is...

For example in workforce scheduling optimization, valid inequalities are crucial constraints derived from real-world considerations. For instance, a logical constraint may prevent nurses from consecutive night and morning shifts: xij+xi(j+1)<=1. Such valid inequalities strengthen the integer programming formulation, ensuring adherence to regulations and operational constraints. Beyond improving the accuracy of feasible solutions, these inequalities also serve to validate solution optimality, checking if generated schedules violate derived constraints. This real-life example showcases how valid inequalities enhance optimization models, aligning workforce schedules with practical and regulatory standards. Hope this helps

Lets consider a distribution network optimization scenario, the linear relaxation of an integer programming problem transforms it into a linear programming problem, allowing non-integer solutions. For instance, if 60% of a facility is selected, a cutting plane algorithm introduces the inequality x+y<=1, where x and y are binary variables. This valid inequality refines the relaxation, ensuring a discrete facility selection, enhancing the model's accuracy for feasible integer solutions in real-world distribution network optimization.

"A linear relaxation of an IP problem is obtained by replacing the integrality constraints on the variables with non-negativity constraints, which results in a linear programming (LP) problem that is easier to solve." - That should be: A linear relaxation of an IP is obtained by removing the integrality constraints of the variables.

It'll be helpful to add figures how feasible solution sets are cut out by cutting planes. Also it's worth mentioning that "repeating until an integer solution" is not always a practical approach.

The choice of cutting plane generation techniques significantly influences the method's performance. Advanced strategies like lift-and-project cuts and mixed-integer rounding cuts offer tighter bounds but may require more computational resources. Additionally, balancing between adding cutting planes and solving the relaxed LP efficiently is crucial for overall algorithmic efficiency.

I was struggling with my IP problem containing binary variables. Adding valid inequalities by exploiting the structure of the model, it really helped my CPLEX solver to converge to better gap (98% to 32% in <2 mins than 98% to 50% in > 15 mins) quickly than without adding these inequalities although I would liked it much better if the optimality gap convergent was more :) .

Benefits: Improved solution quality, faster convergence, enhanced robustness, and facilitated model understanding. Challenges: Identifying strong valid inequalities, managing computational complexity, balancing tightening and solving efforts, and integrating with heuristics and branching. Addressing these enhances IP problem-solving efficacy.

Unless users are deep into these approaches, or unless their problems have specific structures, it might be better to make the solver do its job. Modern optimization solvers are already capable of generating and applying those cuts.

I think, the dynamic way of adding cutting planes to the branch and bound tree, might be beneficial only if the cutting planes are not general (i.e., general in the sense are the ones the solvers already implement). If the cutting planes are problem dependant and specific, it is a good idea to implement selectively using solver callback features.

In vehicle routing optimization, valid inequalities and cutting planes play a crucial role in optimizing delivery routes to minimize transportation costs while ensuring timely deliveries. For example, in a courier company's operations, valid inequalities can be generated to enforce capacity constraints on vehicles (e.g., limiting the number of packages a vehicle can carry) and to model the requirement of visiting specific locations (e.g., ensuring all parcels are delivered to their respective destinations). Cutting planes such as cover cuts can be applied to exploit the network structure of the road network, identifying optimal routes by considering subsets of nodes that form cliques representing potential delivery routes or subroutes.