Ismael de Farias The set partitioning problem and its decision variant (i.e., the exact cover problem) are combinatorial optimization problems that were historically crucial in the quantum optimization community. This problem is also employed in the main problem of the branch-and-price approach in many real-world optimization problems, including, but not limited to, redistricting and scheduling. Motivated by recent … Read more
We report and analyze the results of our computational testing of branch-and-cut for the complementarity-constrained optimization problem (CCOP). Besides the MIP cuts commonly present in commercial optimization software, we used inequalities that explore complementarity constraints. To do so, we generalized two families of cuts proposed earlier by de Farias, Johnson, and Nemhauser that had never … Read more
We study the convex hull of the feasible set of the semi-continuous knapsack problem, in which the variables belong to the union of two intervals. Besides being important in its own right, the semi-continuous knapsack problem arises in a number of other contexts, e.g. it is a relaxation of general mixed-integer programming. We show how … Read more
We give new facets and valid inequalities for the separable piecewise linear optimization knapsack polytope. We also extend the inequalities to the case in which some of the variables are semi-continuous. In a companion paper (de Farias, Gupta, Kozyreff, Zhao, 2011) we demonstrate the efficiency of the inequalities when used as cuts in a branch-and-cut … Read more
We report and analyze the results of our extensive computational testing of branch-and-cut for piecewise linear optimization using the cutting planes given recently by Zhao and de Farias. Besides analysis of the performance of the cuts, we also analyze the effect of formulation on the performance of branch-and-cut. Finally, we report and analyze initial results … Read more
We study the set $S = \{(x, y) \in \Re_{+} \times Z^{n}: x + B_{j} y_{j} \geq b_{j}, j = 1, \ldots, n\}$, where $B_{j}, b_{j} \in \Re_{+} – \{0\}$, $j = 1, \ldots, n$, and $B_{1} | \cdots | B_{n}$. The set $S$ generalizes the mixed-integer rounding (MIR) set of Nemhauser and Wolsey and … Read more
Piecewise linear functions (PLFs) are commonly used to approximate nonlinear functions. They are also of interest in their own, arising for example in problems with economies of scale. Early approaches to piecewise linear optimization (PLO) assumed continuous PLFs. They include the incremental cost MIP model of Markowitz and Manne and the convex combination MIP model … Read more
We study the convex hull $P$ of the set $S = \{(x, y) \in \Re_{+} \times Z^{n}: x + B_{i} y_{ij} \geq b_{ij}, j \in N_{i}, i \in M\}$, where $M = \{1, \ldots, m\}$, $N_{i} = \{1, \ldots, n_{i}\}$ $\forall i \in M$, $\sum_{i = 1}^{m}n_{i} = n$, and $B_{1} | \cdots | B_{m}$. … Read more
We study the convex hull of the feasible set of the semi-continuous knapsack problem, in which the variables belong to the union of two intervals. Besides being important in its own right, the semi-continuous knapsack problem is a relaxation of general mixed-integer programming. We show how strong inequalities valid for the semi-continuous knapsack polyhedron can … Read more
A cardinality constrained knapsack problem is a continuous knapsack problem in which no more than a specified number of nonnegative variables are allowed to be positive. This structure occurs, for example, in areas as finance, location, and scheduling. Traditionally, cardinality constraints are modeled by introducing auxiliary 0-1 variables and additional constraints that relate the continuous … Read more