Mixed-integer programming

Wednesday, August 23, 2017 - 9:00am - 9:45am
Jean-Paul Watson (Sandia National Laboratories)
Pyomo ( is a mature and widely used Python library for expressing and solving a wide range of mathematical programs, i.e., algebraic optimization models. Pyomo allows users to specify optimization models with linear, non-linear (including differential algebraic and ordinary differential equations), mixed discrete-continuous, and stochastic components. These models can then be solved with a wide range of commercial and open source solvers, with varying capabilities.
Thursday, August 11, 2016 - 2:00pm - 3:30pm
Jim Luedtke (University of Wisconsin, Madison)
In this lecture, we present two related methods, Lagrangian relaxation and Dantzig-Wolfe reformulation for exploiting structure of mixed-integer programming models to obtain better relaxations or solve large-scale instances. A Lagrangian relaxation is obtained by relaxing some constraints, ideally leaving only constraints that have special structure that can be exploited computationally. The problem of finding the best such relaxation is known as the Lagrangian dual, and we discuss the strength of this dual bound and how to compute it.
Wednesday, August 10, 2016 - 11:00am - 12:30pm
Jim Luedtke (University of Wisconsin, Madison)
In this lecture we describe the branch-and-cut algorithm, which is currently the state-of-the-art approach for solving mixed-integer optimization problems. We first describe the basic branch-and-bound framework, where efficiently solvable linear programming relaxations are used to eliminate parts of the search space. We then discuss the impact of formulation choice on such a solution procedure, and finally introduce the concept of valid inequalities (or cutting planes) as a mechanism for automatically improving a formulation.
Wednesday, August 10, 2016 - 9:00am - 10:30am
Jeff Linderoth (University of Wisconsin, Madison)
In this lecture, we demonstrate how to model logical
implications between decisions using integer variables. We will
introduce an algorithmic mechanism for converting logical implications
between constraints into equivalent algebraic constraints. We also
will show how to model the graphs of (non-convex) piecewise linear
functions using integer variables.
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