Polyhedral results for a class of cardinality constrained submodular minimization problems
Motivated by concave cost combinatorial optimization problems, we study the following mixed integer nonlinear set: P = {(w,x) : w >= f(a’x), e’x
(Mixed) Integer Linear Programming
An SDP approach for multiperiod mixed 0–1 linear programming models with stochastic dominance constraints for risk management
In this paper we consider multiperiod mixed 0–1 linear programming models under uncertainty. We propose a risk averse strategy using stochastic dominance constraints (SDC) induced by mixed-integer linear recourse as the risk measure. The SDC strategy extends the existing literature to the multistage case and includes both first-order and second-order constraints. We propose a stochastic … Read more
An Exact Extended Formulation for the Unrelated Parallel Machine Total Weighted Completion Time Problem
The plethora of research on NP-hard parallel machine scheduling problems is focused on heuristics due to the theoretically and practically challenging nature of these problems. Only a handful of exact approaches are available in the literature, and most of these suffer from scalability issues. Moreover, the majority of the papers on the subject are restricted … Read more
On the exact separation of rank inequalities for the maximum stable set problem
When addressing the maximum stable set problem on a graph G = (V,E), rank inequalities prescribe that, for any subgraph G[U] induced by U ⊆ V , at most as many vertices as the stability number of G[U] can be part of a stable set of G. These inequalities are very general, as many of … Read more
On Augmentation Algorithms for Linear and Integer-Linear Programming: From Edmonds-Karp to Bland and Beyond
Motivated by Bland’s linear-programming generalization of the renowned Edmonds-Karp efficient refinement of the Ford-Fulkerson maximum-flow algorithm, we discuss three closely-related natural augmentation rules for linear and integer-linear optimization. In several nice situations, we show that polynomially-many augmentation steps suffice to reach an optimum. In particular, when using “discrete steepest-descent augmentations” (i.e., directions with the best … Read more
A Generalization of Benders’ Algorithm for Two-Stage Stochastic Optimization Problems With Mixed Integer Recourse
We describe a generalization of Benders’ method for solving two-stage stochastic linear optimization problems in which there are both continuous and integer variables in the first and second stages. Benders’ method relies on finding effective lower approximations for the value function of the second-stage problem. In this setting, the value function is a discontinuous, non-convex, … Read more
On the Value Function of a Mixed Integer Linear Optimization Problem and an Algorithm for its Construction
This paper addresses the value function of a general mixed integer linear optimization problem (MILP). The value function describes the change in optimal objective value as the right-hand side is varied and understanding its structure is central to solving a variety of important classes of optimization problems. We propose a discrete representation of the MILP … Read more
New symmetries in mixed-integer linear optimization
We present two novel applications of symmetries for mixed-integer linear programming. First we propose two variants of a new heuristic to improve the objective value of a feasible solution using symmetries. These heuristics can use either the actual permutations or the orbits of the variables to find better feasible solutions. Then we introduce a new … Read more
Solving Bilevel Mixed Integer Program by Reformulations and Decomposition
In this paper, we study bilevel mixed integer programming (MIP) problem and present a novel computing scheme based on reformulations and decomposition strategy. By converting bilevel MIP into a constrained mathematical program, we present its single-level reformulations that are friendly to perform analysis and build insights. Then, we develop a decomposition algorithm based on column-and-constraint … Read more
Tight MIP Formulations of the Power-Based Unit Commitment Problem
This paper provides the convex hull description for the basic operation of slow- and quick-start units in power-based unit commitment (UC) problems. The basic operating constraints that are modeled for both types of units are: 1) generation limits and 2) minimum up and down times. Apart from this, the startup and shutdown processes are also … Read more