Orbital Conflict: Cutting Planes for Symmetric Integer Programs

Cutting planes have been an important factor in the impressive progress made by integer programming (IP) solvers in the past two decades. However, cutting planes have had little impact on improving performance for symmetric IPs. Rather, the main breakthroughs for solving symmetric IPs have been achieved by cleverly exploiting symmetry in the enumeration phase of … Read more

A Computationally Efficient Algorithm for Computing Convex Hull Prices

Electricity markets worldwide allow participants to bid non-convex production offers. While non-convex offers can more accurately reflect a resource’s capabilities, they create challenges for market clearing processes. For example, system operators may be required to execute side payments to participants whose costs are not covered through energy sales as determined via traditional locational marginal pricing … Read more

Decomposing the Train Scheduling Problem into Integer Optimal Polytopes

This paper presents conditions for which the linear relaxation for the train scheduling problem is integer-optimal. These conditions are then used to identify how to partition a general problem’s feasible region into integer-optimal polytopes. Such an approach yields an extended formulation that contains far fewer binary variables. Our computational experiments show that this approach results … Read more

Exploiting Identical Generators in Unit Commitment

We present sufficient conditions under which thermal generators can be aggregated in mixed-integer linear programming (MILP) formulations of the unit commitment (UC) problem, while maintaining feasibility and optimality for the original disaggregated problem. Aggregating thermal generators with identical characteristics (e.g., minimum/maximum power output, minimum up/down-time, and cost curves) into a single unit reduces redundancy in … Read more

A Novel Matching Formulation for Startup Costs in Unit Commitment

We present a novel formulation for startup cost computation in the unit commitment problem (UC). Both the proposed formulation and existing formulations in the literature are placed in a formal, theoretical dominance hierarchy based on their respective linear programming relaxations. The proposed formulation is tested empirically against existing formulations on large-scale unit commitment instances drawn … Read more

Detecting Almost Symmetries of Graphs

We present a branch-and-bound framework to solve the following problem: Given a graph G and an integer k, find a subgraph of G formed by removing no more than k edges that contains the most symmetry. We call symmetries on such a subgraph “almost symmetries” of G. We implement our branch-and-bound framework in PEBBL to … Read more

The Ramping Polytope and Cut Generation for the Unit Commitment Problem

We present a perfect formulation for a single generator in the unit commitment problem, inspired by the dynamic programming approach taken by Frangioni and Gentile. This generator can have characteristics such as ramping constraints, time-dependent start-up costs, and start-up/shut-down ramping. To develop this perfect formulation we extend the result of Balas on unions of polyhedra … Read more

Modified Orbital Branching with Applications to Orbitopes and to Unit Commitment

The past decade has seen advances in general methods for symmetry breaking in mixed-integer linear programming. These methods are advantageous for general problems with general symmetry groups. Some important classes of MILP problems, such as bin packing and graph coloring, contain highly structured symmetry groups. This observation has motivated the development of problem-specific techniques. In … Read more

Using Symmetry to Optimize Over the Sherali-Adams Relaxation

In this paper we examine the impact of using the Sherali-Adams procedure on highly symmetric integer programming problems. Linear relaxations of the extended formulations generated by Sherali-Adams can be very large, containing on the order of n choose d many variables for the level-d closure. When large amounts of symmetry are present in the problem … Read more