2017 Winter School on Optimization and Operations Research

Prof. Michael Ferris Prof. Andrea Lodi

Optimization and Equilibrium in Applications

On Mixed-Integer Programming and its connection with Data Science

Michael C. Ferris

Andrea Lodi

Slides lecture 1Slides lecture 1
Slides lecture 2Slides lectures 2 and 3
Slides lecture 3
Slides lecture 4Slides lecture 4
Slides lecture 5Slides lecture 5

We consider models built up from a collection of optimizations within an interacting physical, economic or virtual system. We show how optimization and equilibrium concepts can be deployed and resulting models solved within an extended mathematical programming framework. Examples are drawn from sustainable land use modeling, power system design and economic operation, discrete Nash equilibria and risk analysis. The interplay between stochasticity, complementarity and hierarchical optimization will be highlighted.

We look at models of competition and risk within the context of power system markets. We demonstrate when social optima exist, what properties on risk measures and contracts are needed, and how to solve these problems in large scale practical settings. We look at specific structures that give rise to significant enhancements in computational performance. We position these models within a general setting of Nash Games that include linking equilibrium constraints and situtations where players solve multi-period stochastic programs.

Specific topics to be covered include:

  1. Extended Mathematical Programming, including Quadratic Support Functions
  2. Integrated Modeling with Optimization Problems
  3. Optimization and Equilibrium in Energy Economics
  4. Algorithms for Large Scale Problems involving Complementarity
  5. Environmental Applications

We consider optimization problems in which some of the decisions have to be taken within a discrete set. Depending on the modeling paradigm, this gives rise to either Mixed-Integer Linear or Mixed-Integer Nonlinear Programming problems, both defining a rich variety models whose impact in real-world applications is ubiquitous. In many relevant cases, those applications are solved by general-purpose solvers, which, especially in the linear case, have reached a very mature state, being routinely applied with relatively little tuning. However, these solvers are complex pieces of software and their development and improvement is far from being steady.

In this short course, we will describe the main characteristics and the basic components of Mixed-Integer Linear Programming solvers, we will discuss the reasons of their success, the challenges and show the direction they are following to be able to take into account the nonlinear setting, too. Finally, we will discuss that Mixed-Integer Programming can have in Data Science, the discipline integrating learning and optimization to take intelligent decisions by exploiting massive quantities of data.