Master of Science / Department of Industrial and Manufacturing Systems Engineering / Todd Easton / Integer programming (IP) is a class of mathematical models useful for modeling and optimizing many theoretical and industrial problems. Unfortunately, IPs are NP-complete, and many integer programs cannot currently be solved. Valid inequalities and their respective cuts are commonly used to reduce the effort required to solve IPs. This thesis poses the questions, do valid equality cuts exist and can they be useful for solving IPs? Several theoretical results related to valid equalities are presented in this thesis. It is shown that equality cuts exist if and only if the convex hull is not full dimensional. Furthermore, the addition of an equality cut can arbitrarily reduce the dimension of the linear relaxation. In addition to the theory on equality cuts, the idea of infeasibility conditions are presented. Infeasibility conditions introduce a set of valid inequalities whose intersection is the empty set. infeasibility conditions can be used to rapidly terminate a branch and cut algorithm. Applying the idea of equality cuts to the multi-demand multidimensional knapsack problem resulted in a new class of cutting planes named anticover cover equality (ACE) cuts. A simple algorithm, FACEBT, is presented for finding ACE cuts in a branching tree with complexity O(m n log n). A brief computational study shows that using ACE cuts exist frequently in the MDMKP instances studied. Every instance had at least one equality cut, while one instance had over 500,000. Additionally, computationally challenging instances saw an 11% improvement in computational effort. Therefore, equality cuts are a new topic of research in IP that is beneficial for solving some IP instances.

Master of Science / Department of Industrial and Manufacturing Systems Engineering / Todd W. Easton / Integer programming is an important discipline in operation research that positively impacts society. Unfortunately, no algorithm currently exists to solve IP's in polynomial time. Researchers are constantly developing new techniques, such as cutting planes, to help solve IPs faster. For example, DeLissa discovered the existence of equality cuts limited to zero and one coefficients for the multiple knapsack equality problem (MKEP). An equality cut is an improper cut because every feasible point satisfies the equality. However, such a cut always reduces the dimension of the linear relaxation space by at least one. This thesis introduces lifted equality cuts, which can have coefficients greater than or equal to two. Two main theorems provide the conditions for the existence of lifted equalities. These theorems provide the foundation for The Algorithm of Lifted Equality Cuts (ALEC), which finds lifted equality cuts in quadratic time. The computational study verifies the benefit of lifted equality cuts in random MKEP instances. ALEC generated millions of lifted equality cuts and reduced the solution time by an average of 15%. To the best of the author's knowledge, ALEC is the first algorithm that has found over 30.7 million cuts on a single problem, while reducing the solving time by 18%.