# Tutorials

### Practical Guidelines for Solving Difficult Mixed Integer Programs, Version 2.0

The paper “Practical Guidelines for Solving Difficult Mixed Integer Programs” by Klotz and Newman was written in 2012, and significant parts of that material first appeared in 2007. Since then, mixed-integer solvers have evolved significantly. Two of the three models the paper describes were unsolvable at the time, but some solvers now handle them easily, as of the last five years. This tutorial will provide an update to the guidelines. We will consider additional functionality in the solvers and look at a new set of challenging models.

### Stochastic Failure Models for Network Analysis

Risk-awareness in the design and operation of networks need to systematically consider the performance of the network and consider investment to extend it to face the new challenge to provide their service. In general networks are subject to a trade-off between a low probability of unavailability of the service they provide and the very high cost associated with the impossibility of providing service in that event. Providers, users and regulators wish to understand how to better operate and redesign these networks, so researcher have developed techniques to evaluate their performance and study their stability. To analyze network failures attention must be paid to three important aspects: first to choose a failure model that is complex enough to capture the interaction between components but, at the same time simple to calibrate with the available information; second study the performance, this means that given a model we need to develop methodology to compute the performance of the network, and third to compare network designs and chose according to a budget constraint those which perform better.

In this tutorial we will discuss new failure models that are appearing to capture dependence in components and how to estimate performance metrics considering simulation techniques. We will finish with some comment on the network design problem.

### Mariel Lavieri

University of Michigan, USA

### Chronic Disease Management; The Past, the Present, and the Future

Chronic disease management often involves sequential decisions that have long-term implications. Those decisions are based on high dimensional information, which pose a problem for traditional modeling paradigms. In some instances, the disease dynamics might not be known, but instead are learned as new information becomes available.

In this tutorial I describe some of the ongoing research modeling medical decisions of patients with chronic conditions. Model conception and validation is described, as well as the role of multidisciplinary collaborations in ensuring practical impact of this type of work. I conclude this tutorial with a discussion of future research directions in the field.

### Andre Luiz Diniz

Eletrobras Cepel, Brazil

### OR in Energy Planning: Modeling Strategies, Solving Techniques and Practical Aspects

Energy planning is a fertile field for the application of operations research techniques and mathematical programming tools. There are a lot of interesting problems, ranging from long-term planning to day-ahead or intra-day scheduling tools. Despite the improvement in commercial packages to solve LP, MILP or MINLP problems, extensive expertise is still required to model, decompose and solve such problems in an efficient way, in order to derive tools that provide implementable results as closest as possible to reality, in reasonable CPU times.

One of the main challenges in these types of problems is how to handle uncertainty. In the past, this issue was mainly addressed in long-term problems, with the application of stochastic programming techniques such as SDDP to represent uncertainty in some crucial input variables such as hydro inflows to reservoirs. In modern power systems, the large increase in intermittent generation has forced unit commitment and short-term scheduling tools to also address this issue, leading to different ways to model here and now/wait and see decisions, and favoring the application of alternative techniques such as robust optimization. In addition, the modeling of hourly operation aspects in long-term planning tools is becoming more and more important, posing a big challenge in handling the trade-off between accuracy in system representation and computational time.

Besides providing an overview of decomposition techniques and solving strategies to tackle power generation planning and scheduling problems, this talk also briefly discusses alternative approaches to model different aspects of system operation, such as power flows in the electrical network, dynamic thermal unit commitment characteristics and nonconvex/time linking constraints for the operation of hydro and pumped storage plants, always with a look at practical applications in large-scale systems.

### Emilio Carrizosa

University of Seville, Spain

Abstract to come

### Models and Decomposition Methods for Large-Scale Network Optimization

Network optimization problems lie at the hearth of network design planning in transportation and telecommunications systems. This talk focuses on a challenging class of problems referred as general network design which offers a unified view of integrated facility location and network design. Their main difficulty stems from the inherent interrelation between two levels of the decision process. The first level considers design decisions such as the selection of nodes to locate facilities and the activation of links to connect nodes. The second level considers tactical decisions such as the assignment of nodes to facilities and the routing of flows through the network. We present an overview of some of the most prominent mathematical programming formulations that have been used in combination with decomposition methods to develop state-of-the-art exact solution algorithms capable of solving large-scale instances of three important classes of problems involving both linear and nonlinear cost functions: hub network design, multi-level facility location, and multi-commodity network design. We highlight the connections between Lagrangean relaxation, Dantzig-Wolfe decomposition, Benders decomposition, and related primal relaxations in the context of such problems. We also point out to several algorithmic refinements that have been used to accelerate the convergence of such decomposition methods. Practical implementation guidelines to improve their overall performance are discussed.

MIT, USA

### The JuMP Ecosystem for Mathematical Optimization

JuMP is a multi-award-winning domain-specific language for mathematical optimization. JuMP has already been successfully used in academic and industrial problems related to marketing, causal inference, daily fantasy sports, optimal control of aerial drones, machine learning, school bus routing, sustainable power systems expansion, and decarbonization of electrical networks. The JuMP ecosystem gives access to a wide range of highly-effective commercial and open-source optimization tools in a natural syntax that requires only a basic knowledge of mathematical optimization. JuMP provides this access with a performance that matches or exceeds those of commercial and open-source alternatives, as well as unparalleled versatility and extensibility allowed by the advanced features of the Julia programming language. In particular, JuMP and its infrastructure was used to develop the solver Pajarito.jl, which is currently the state-of-the-art for the mixed-integer conic optimization problems. In this tutorial, we begin with basic syntax and features of JuMP and associated packages assuming no previous knowledge of JuMP and only an elementary knowledge of mathematical optimization. We then cover more advanced features, give performance tips, and cover the recent improvements to JuMP. Finally, we demo some state-of-the-art features, including showing how various packages in the rich Julia ecosystem can be seamlessly combined to provide simple solutions to complicated problems in the optimal control of aerial drones.

### Martin Savelsbergh

Georgia Tech, USA

### Criterion Space Search Algorithms for Multi-objective Mixed Integer Programming

Multi-objective optimization problems are pervasive in practice. In contrast to single-objective optimization, the goal in multi-objective optimization is to generate a set of solutions that induces the Pareto front, i.e., the set of all nondominated points. A nondominated point is a vector of objective function values evaluated at a feasible solution, with the property that there exists no other feasible solution that is at least as good in all objective function values and better in one or more of them. Recently, criterion space search algorithms, in which the search for the Pareto front takes place in the space of the vectors of objective function values, i.e., the criterion space, have gained in popularity. These methods exploit the advances in single-objective optimization solvers, since they repeatedly solve single-objective optimization problems. We will introduce and discuss criterion space search algorithms for both pure and mixed multi-objective integer programs.

### Meisam Razaviyayn

University of Southern California, USA

### Solving “Nice” Non-Convex Optimization Problems and its Connections to Training Deep Neural Networks

When is solving a non-convex optimization problem easy? Despite significant research efforts to answer this question, most existing results are problem specific and cannot be applied even with simple changes in the objective function. In this talk, we provide theoretical insights to this question by answering two related questions: 1) Are all local optima of a given non-convex optimization problem globally optimal? 2) When can we compute a local optimum of a given non-convex constrained optimization problem efficiently?  In the first part of the talk, motivated by the non-convex training problem of deep neural networks, we provide simple sufficient conditions under which any local optimum of a given highly composite optimization problem is globally optimal. Unlike existing results in the literature, our sufficient condition applies to many non-convex optimization problems such as training problem of non-convex multi-linear neural networks and non-linear neural networks with pyramidal structures.

In the second part of the talk, we consider the problem of finding a local optimum of a constrained non-convex optimization problem under strict saddle point property. We show that, unlike the unconstrained scenario, the vanilla projected gradient descent algorithm fails to escape saddle points even in the presence of a single linear constraint. We then propose a trust region algorithm which converges to second order stationary points for optimization problems with small number of linear constraints. Our algorithm is the first optimization procedure, with polynomial per-iteration complexity, which converges to $\epsilon$-first order stationary points of a non-manifold constrained optimization problem in $O(\epsilon^{-3/2})$ iterations, and at the same time can escape saddle points under strict saddle property.