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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_1(x)\ \le\ 0\ \And \ f_2(x)\ \le\ 0 } , the disjunctive form is given by: . The constraints would be created by using sufficiently large numbers, such as M1 and M2, and a binary variable y (y1 & y2):
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_1(x)\ \le\ M_1\ast(1-y_1)}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_2(x)\ \le\ M_2\ast(1-y_2)}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_1\ +y_2\ =\ 1 }
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_1\ ,\ y_2\epsilon\ {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)