A Two-Level Approach to Large Mixed-Integer Programs with Application to Cogeneration in Energy-Efficient Buildings

We study a two-stage mixed-integer linear program (MILP) with more than 1 million binary variables in the second stage. We develop a two-level approach by constructing a semi-coarse model (coarsened with respect to variables) and a coarse model (coarsened with respect to both variables and constraints). We coarsen binary variables by selecting a small number of pre-specified daily on/off profiles. We aggregate constraints by partitioning them into groups and summing over each group. With an appropriate choice of coarsened profiles, the semi-coarse model is guaranteed to find a feasible solution of the original problem and hence provides an upper bound on the optimal solution. We show that solving a sequence of coarse models converges to the same upper bound with proven finite steps. This is achieved by adding violated constraints to coarse models until all constraints in the semi-coarse model are satisfied. We demonstrate the effectiveness of our approach in cogeneration for buildings. The coarsened models allow us to obtain good approximate solutions at a fraction of the time required by solving the original problem. Extensive numerical experiments show that the two-level approach scales to large problems that are beyond the capacity of state-of-the-art commercial MILP solvers

Reformulation and decomposition of integer programs

In this survey we examine ways to reformulate integer and mixed integer programs. Typically, but not exclusively, one reformulates so as to obtain stronger linear programming relaxations, and hence better bounds for use in a branch-and-bound based algorithm. First we cover in detail reformulations based on decomposition, such as Lagrangean relaxation, Dantzig-Wolfe column generation and the resulting branch-and-price algorithms. This is followed by an examination of Benders’ type algorithms based on projection. Finally we discuss in detail extended formulations involving additional variables that are based on problem structure. These can often be used to provide strengthened a priori formulations. Reformulations obtained by adding cutting planes in the original variables are not treated here.Integer program, Lagrangean relaxation, column generation, branch-and-price, extended formulation, Benders' algorithm

A Compiler and Runtime Infrastructure for Automatic Program Distribution

This paper presents the design and the implementation of a compiler and runtime infrastructure for automatic program distribution. We are building a research infrastructure that enables experimentation with various program partitioning and mapping strategies and the study of automatic distribution's effect on resource consumption (e.g., CPU, memory, communication). Since many optimization techniques are faced with conflicting optimization targets (e.g., memory and communication), we believe that it is important to be able to study their interaction. We present a set of techniques that enable flexible resource modeling and program distribution. These are: dependence analysis, weighted graph partitioning, code and communication generation, and profiling. We have developed these ideas in the context of the Java language. We present in detail the design and implementation of each of the techniques as part of our compiler and runtime infrastructure. Then, we evaluate our design and present preliminary experimental data for each component, as well as for the entire system