Disjunctive inequalities
Introduction
Disjunctive inequalities are a form of disjunctive constraints that can be applied to linear programming. Disjunctive constraints are applied in all disjunctive programming, which just refers to the use of logical constraints, which include the “Or” and “And” statements. In order to solve a disjunctive, the constraints have to be converted into multiple integer programming (MIP) constraints, which is called disjunction. Disjunction involves the implementation of a binary variable to create a new set of constraints that can be solved easily. Two common methods for disjunction are the Big-M Reformulation and the Convex-Hull Reformulation.
Method
General
When given a set of inequalities, such as , the disjunctive form is given by: . In order to turn the problem into a solvable MIP or MILP, logical constraints are created by using sufficiently large numbers, such as M1 and M2, and a binary variable y for each inequality. This is shown below by M1, M2, y1, and y1:
To set the binary variable to be mutually exclusive, the sum of the variables is set to 1 and the range is set to {0,1}.
Big-M Reformulation
Convex-Hull Reformulation
Examples
Applications
Conclusion
References
Authors: Daniel Ladron, Grant Logan, Matthew Dinh, Derek Moore (CBE/SysEn 6800, Fall 2021)