Linear and NonLinear Mixed Integer Optimization
Institute for Computational and Experimental Research in Mathematics (ICERM)
February 27, 2023  March 3, 2023
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Monday, February 27, 2023

8:50  9:00 am ESTWelcome11th Floor Lecture Hall
 Brendan Hassett, ICERM/Brown University

9:00  9:45 am ESTOn Constrained MixedInteger DRSubmodular Minimization11th Floor Lecture Hall
 Speaker
 Simge Küçükyavuz, Northwestern University
 Session Chair
 Jon Lee, University of Michigan
Abstract
Diminishing Returns (DR)submodular functions encompass a broad class of functions that are generally nonconvex and nonconcave. We study the problem of minimizing any DRsubmodular function, with continuous and general integer variables, under box constraints and possibly additional monotonicity constraints. We propose valid linear inequalities for the epigraph of any DRsubmodular function under the constraints. We further provide the complete convex hull of such an epigraph, which, surprisingly, turns out to be polyhedral. We propose a polynomialtime exact separation algorithm for our proposed valid inequalities, with which we first establish the polynomialtime solvability of this class of mixedinteger nonlinear optimization problems. This is joint work with Kim Yu.

10:00  10:30 am ESTCoffee Break11th Floor Collaborative Space

10:30  11:15 am ESTSemidefinite Optimization with Eigenvector Branching11th Floor Lecture Hall
 Speaker
 Kurt Anstreicher, University of Iowa
 Session Chair
 Jon Lee, University of Michigan
Abstract
Semidefinite programming (SDP) problems typically utilize the constraint that Xxx' is positive semidefinite to obtain a convex relaxation of the condition X=xx', where x is an nvector. We consider a new hyperplane branching method for SDP based on using an eigenvector of Xxx'. This branching technique is related to previous work of Saxeena, Bonami and Lee who used such an eigenvector to derive a disjunctive cut. We obtain excellent computational results applying the new branching technique to difficult instances of the twotrustregion subproblem.

11:30 am  12:15 pm ESTA Breakpoints Based Method for the Maximum Diversity and Dispersion Problems11th Floor Lecture Hall
 Speaker
 Dorit Hochbaum, University of California, Berkeley
 Session Chair
 Jon Lee, University of Michigan
Abstract
The maximum diversity, or dispersion, problem (MDP), is to select from a given set a subset of elements of given size (budget), so that the sum of pairwise distances, or utilities, between the selected elements is maximized. We introduce here a method, called the Breakpoints (BP) algorithm, based on a technique proposed in Hochbaum (2009), to generate the concave piecewise linear envelope of the solutions to the relaxation of the problem for all values of the budget. The breakpoints in this envelope are provably optimal for the respective budgets and are attained using a parametric cut procedure that is very efficient. The problem is then solved, for any given value of the budget, by applying a greedylike method to add or subtract nodes from adjacent breakpoints. This method works well if for the given budget there are breakpoints that are ``close". However, for many data sets and budgets this is not the case, and the breakpoints are sparse. We introduce a perturbation technique applied to the utility values in cases where there is paucity of breakpoints, and show that this results in denser collections of breakpoints. Furthermore, each optimal perturbed solution is quite close to an optimal nonperturbed solution. We compare the performance of our breakpoints algorithm to leading methods for these problems: The metaheuristic OBMA, that was shown recently to perform better than GRASP, Neighborhood search and Tabu Search, and Gurobian integer programming software. It is demonstrated that our method dominates the performance of these methods in terms of computation speed and in comparable or better solution quality.

12:30  2:30 pm ESTLunch/Free Time

2:30  3:15 pm ESTApproximating integer programs with monomial orders11th Floor Lecture Hall
 Virtual Speaker
 Akshay Gupte, University of Edinburgh
 Session Chair
 Dorit Hochbaum, University of California, Berkeley
Abstract
We consider the problem of maximizing a function over integer points in a compact set. Inner and outerapproximations of the integer feasible set are obtained using families of monomial orders over the integer lattice. The convex hull is characterized when the monomial orders satisfy some properties. When the objective function is submodular or subadditive, we provide a theoretical guarantee on the quality of the innerapproximations in terms of their gap to the optimal value. An algorithm is proposed to generate feasible solutions, and it is competitive with a commercial solver in numerical experiments on benchmark test instances for integer LPs.

3:30  4:00 pm ESTCoffee Break11th Floor Collaborative Space

4:00  4:45 pm ESTSolving ACOPF problems11th Floor Lecture Hall
 Speaker
 Daniel Bienstock, Columbia University
 Session Chair
 Dorit Hochbaum, University of California, Berkeley
Abstract
In this talk we will detail our recent experience in solving the ACOPF problem, a notorious MINLP. We will do this from two perspectives. First, we will detail our experience in the recent, and ongoing GO competition for solving modern, largescale versions of ACOPF which include scenario constraints and integer variables. Second, we will outline challenges to stateoftheart MINLP solvers based on spatial branchandbound that arise in ACOPF instances. Finally we will discuss some fundamental issues related to numerical precision.

5:00  6:30 pm ESTReception11th Floor Collaborative Space
Tuesday, February 28, 2023

9:00  9:45 am ESTMaximal quadratic free sets: basic constructions and steps towards a full characterization11th Floor Lecture Hall
 Speaker
 Gonzalo Muñoz, Universidad de O'Higgins
 Session Chair
 Daniel Bienstock, Columbia University
Abstract
In 1971, Balas introduced intersection cuts as a method for generating cutting planes in integer optimization. These cuts are derived from convex Sfree sets, and inclusionwise maximal Sfree sets yield the strongest intersection cuts. When S is a lattice, maximal Sfree sets are wellstudied from theoretical and computational standpoints. In this talk, we focus on the case when S is defined by a general quadratic inequality and show how to construct basic maximal quadraticfree sets. Additionally, we explore how to generalize the basic procedure to construct a plethora of new maximal quadraticfree sets for homogeneous quadratics. Joint work with Joseph Paat and Felipe Serrano.

10:00  10:30 am ESTCoffee Break11th Floor Collaborative Space

10:30  11:15 am ESTFrom micro to macro structure: a journey in company of the Unit Commitment problem11th Floor Lecture Hall
 Speaker
 Antonio Frangioni, Università di Pisa
 Session Chair
 Daniel Bienstock, Columbia University
Abstract
The fact that "challenging problems motivate methodological advances", as obvious as it may seem, is nonetheless very true. I was drawn long ago to Unit Commitment problems because of a specific methodology, but studying it led us to interesting results for entirely different ones. This talk will summarise on (the current status of) a long journey of discovery that ebbed and flowed between different notions of structure, starting from the "macro" one of spatial decomposition and its algorithmic implications, descending to the "micro" one of the Perspective Reformulation of tiny fragments of the problem, putting both back together to fullproblem size with the definition of strong but large formulations (and the nontrivial tradeoffs they entail), and finally skyrocketing to large and hugescale problems (stochastic UC, stochastic reservoirs optimization, longterm energy system design) where UC (and its substructures) is but one of the multiple nested forms of structure. The talk will necessarily have to focus on a few of the results that hopefully have broader usefulness than just UC, among which a recent one on the Convex Hull of StarShaped MINLPs, but it will also try to give a broadbrush of the larger picture, with some time devoted to discussing the nontrivial implications of actually implementing solution methods for hugescale problems with multiple nested form of heterogeneous structure and the (surely partial and tentative) attempts at tackling these issues within the SMS++ modelling system.

11:30 am  12:15 pm ESTA fast combinatorial algorithm for the bilevel knapsack interdiction problem11th Floor Lecture Hall
 Speaker
 Ricardo Fukasawa, University of Waterloo
 Session Chair
 Daniel Bienstock, Columbia University
Abstract
The singlelevel (classical) knapsack problem consists of picking a subset of n items maximizing profit, subject to a budget (knapsack) constraint. In the bilevel knapsack interdiction problem, there are two levels of decisionmakers. The lowerlevel decision maker is attempting to solve the classical knapsack problem. The upperlevel decision maker has the goal of minimizing the profit of the lowerlevel decision maker, by picking items that will be then unavailable. The upperlevel has its own (independent) knapsack constraint to satisfy too. Previous approaches to this problem were based on using MIP solvers. We develop a branchandbound algorithm for the problem that relies on a strong lower bound and specialized branching, using combinatorial arguments. Our approach improves on the previous best approaches by up to two orders of magnitude in our test instances, significantly increasing the size of instances that can be consistently solved to optimality. This is joint work with Noah Weninger.

12:30  2:30 pm ESTLunch/Free Time

2:30  3:15 pm ESTNetwork Design Queueing MINLP: Models, Reformulations, and Algorithms11th Floor Lecture Hall
 Speaker
 Miguel Lejeune, George Washington University
 Session Chair
 Merve Bodur, University of Toronto
Abstract
We present several queueingbased optimization models to design networks in which the objective is to minimize the response time. The networks are modelled as collections of interdependent M/G/1 or M/G/K queueing systems with fixed and mobile servers. The optimization models take the form of nonconvex MINLP problems with fractional and bilinear terms. We derive a reformulation approach and propose a solution method that features a warmstart component, new optimalitybased bound tightening (OBBT) techniques, and an outer approximation algorithm. In particular, we propose new MILP and feasibility OBBT models that can derive multiple variable bounds at once. The proposed approach is applied to the dronebased delivery of automated external defibrillators to outofhospital cardiac arrests (OHCA) and naloxone to opioid overdoses. The computational experiments are based on reallife data from Virginia Beach, and ascertain the computational efficiency of the approach and its impact on the response time and the probability of survival of patients.

3:30  4:00 pm ESTCoffee Break11th Floor Collaborative Space

4:00  4:45 pm ESTIntegrated Computational and Experimental Research in Mixed Integer Programming with SageMath11th Floor Lecture Hall
 Virtual Speaker
 Matthias Köppe, UC Davis
 Session Chair
 Merve Bodur, University of Toronto
Abstract
I will introduce SageMath, a generalpurpose, Pythonbased, opensource mathematics system in development since 2005, from the viewpoint of its use in Mixed Integer Programming research. Originally conceived as a computer algebra system focused on algebraic number theory, SageMath has grown to a major integrating force in the mathematical software world that provides convenient and unified access to the best libraries and systems for polyhedral geometry, mixed integer linear programming, lattices, graph theory, commutative algebra, symbolic computation, computational group theory, and more. I will demonstrate some of these capabilities. The SageMath community welcomes contributions in the form of expository mathematical writing, constructions, algorithm development and implementation, cataloging, mathematical software integration, and more. I will highlight some possible entry points for making such contributions.
Wednesday, March 1, 2023

9:00  9:45 am ESTMinimizing quadratics over integers11th Floor Lecture Hall
 Speaker
 Alberto Del Pia, University of WisconsinMadison
 Session Chair
 Nick Sahinidis, Georgia Institute of Technology
Abstract
Mixed integer quadratic programming is the problem of minimizing a quadratic polynomial over points in a polyhedral region with some integer components. It is a natural extension of mixed integer linear programming and it has a wide array of applications. In this talk, I will survey some recent theoretical developments in mixed integer quadratic programming, with a focus on complexity, algorithms, and fundamental properties.

10:00  10:30 am ESTCoffee Break11th Floor Collaborative Space

10:30  11:15 am ESTOptimizing for Equity in Urban Planning11th Floor Lecture Hall
 Speaker
 Emily Speakman, University of Colorado  Denver
 Session Chair
 Nick Sahinidis, Georgia Institute of Technology
Abstract
In the Environmental Justice literature, the KolmPollak Equally Distributed Equivalent (EDE) is the preferred metric for quantifying the experience of a population. The metric incorporates both the center and the spread of the distribution of the individual experiences, and therefore, captures the experience of an “average” individual more accurately than the population mean. In particular, the mean is unable to measure the equity of a distribution, while the KolmPollak EDE is designed to penalize for inequity. In this talk, we explore the problem of finding an optimal distribution from various alternatives using the KolmPollak EDE to quantify optimal. Unfortunately, optimizing over the KolmPollak EDE in a mathematical programming model is not trivial because of the nonlinearity of the function. We discuss methods to overcome this difficulty and present computational results for practical applications. Our results demonstrate that optimizing over the KolmPollak EDE in a standard facility location model has the same computational burden as optimizing over the population mean. Moreover, it often results in solutions that are significantly more equitable while having little impact on the mean of the distribution, versus optimizing over the mean directly. Joint work with Drew Horton, Tom Logan, and Daphne Skipper

11:30 am  12:15 pm ESTExplicit convex hull description of bivariate quadratic sets with indicator variables11th Floor Lecture Hall
 Speaker
 Aida Khajavirad, Lehigh University
 Session Chair
 Nick Sahinidis, Georgia Institute of Technology
Abstract
We obtain an explicit description for the closure of the convex hull of bivariate quadratic sets with indicator variables the space of original variables. We present a simple separation algorithm that can be incorporated in branchandcut based solvers to enhance the quality of existing relaxations.

12:25  12:30 pm ESTGroup Photo (Immediately After Talk)11th Floor Lecture Hall

12:30  2:30 pm ESTNetworking LunchLunch/Free Time  11th Floor Collaborative Space

2:30  3:30 pm EST

3:30  4:00 pm ESTCoffee Break10th Floor Collaborative Space

4:00  4:45 pm ESTMatrix Completion over GF(2) with Applications to Index Coding11th Floor Lecture Hall
 Speaker
 Jeff Linderoth, University of WisconsinMadison
 Session Chair
 Kurt Anstreicher, University of Iowa
Abstract
We discuss integerprogrammingbased approaches to doing lowrank matrix completion over the finite field of two elements. We are able to derive an explicit description for the convex hull of an individual matrix element in the decomposition, using this as the basis of a new formulation. Computational results showing the superiority of the new formulation over a natural formulation based on McCormick inequalities with integervalued variables, and an extended disjunctive formulation arising from the parity polytope are given in the context of linear index coding.
Thursday, March 2, 2023

9:00  9:45 am ESTDantzigWolfe Bound by Cutting Planes11th Floor Lecture Hall
 Speaker
 Oktay Gunluk, Cornell University
 Session Chair
 Yuan Zhou, University of Kentucky
Abstract
DantzigWolfe (DW) decomposition is a wellknown technique in mixedinteger programming for decomposing and convexifying constraints to obtain potentially strong dual bounds. We investigate Fenchel cuts that can be derived using the DW decomposition algorithm and show that these cuts can provide the same dual bounds as DW decomposition. We show that these cuts, in essence, decompose the objective function cut one can simply write using the DW bound. Compared to the objective function cut, these Fenchel cuts lead to a formulation with lower dual degeneracy, and consequently a better computational performance under the standard branchandcut framework in the original space. We also discuss how to strengthen these cuts to improve the computational performance further. We test our approach on the Multiple Knapsack Assignment Problem and show that the proposed cuts are helpful in accelerating the solution time without the need to implement branch and price.

10:00  10:30 am ESTCoffee Break11th Floor Collaborative Space

10:30  11:15 am ESTNumber of inequalities in integerprogramming descriptions of a set11th Floor Lecture Hall
 Virtual Speaker
 Gennady Averkov, Brandeburg Technical University
 Session Chair
 Yuan Zhou, University of Kentucky
Abstract
I am going to present results obtained jointly with Manuel Aprile, Marco Di Summa, Christopher Hojny and Matthias Schymura. Assume you want to describe a set X of integer points as the set of integer solutions of a linear system of inequalities and you want to use a system for X with the minimum number of inequalities. Can you compute this number algorithmically? The answer is not known in general! Does the choice of the coefficient field (like the field of real and the fiel of rational numbers) have any influence on the number you get as the answer? Surprisingly, it does! On a philosophical level, should we do integer programming over the rationals or real coefficients? That's actually not quite clear, but for some aspects there is a difference so that it might be interesting to reflect on this point and weigh pros and cons.

11:30 am  12:15 pm EST2x2Convexifications for Convex Quadratic Optimization with Indicator Variables11th Floor Lecture Hall
 Speaker
 Alper Atamturk, University of California  Berkeley
 Session Chair
 Yuan Zhou, University of Kentucky
Abstract
In this talk, we present new strong relaxations for the convex quadratic optimization problem with indicator variables. For the bivariate case, we describe the convex hull of the epigraph in the original space of variables, and also give a conic quadratic extended formulation. Then, using the convex hull description for the bivariate case as a building block, we derive an extended SDP relaxation for the general case. This new formulation is stronger than other SDP relaxations proposed in the literature for the problem, including Shor's SDP relaxation, the optimal perspective relaxation as well as the optimal rankone relaxation. Computational experiments indicate that the proposed formulations are quite effective in reducing the integrality gap of the optimization problems. This is a joint work with Shaoning Han and Andres Gomez.

12:30  2:30 pm ESTLunch/Free Time

2:30  3:15 pm ESTInteger Semidefinite Programming  a New Perspective11th Floor Lecture Hall
 Speaker
 Renata Sotirov, Tilburg University
 Session Chair
 Fatma KılınçKarzan, Carnegie Mellon University
Abstract
Integer semidefinite programming can be viewed as a generalization of integer programming where the vector variables are replaced by positive semidefinite integer matrix variables. The combination of positive semidefiniteness and integrality allows to formulate various optimization problems as integer semidefinite programs (ISDPs). Nevertheless, ISDPs have received attention only very recently. In this talk we show how to extend the Chv\'atalGomory (CG) cuttingplane procedure to ISDPs. We also show how to exploit CG cuts in a branchandcut framework for ISDPs. Finally, we demonstrate the practical strength of the CG cuts in our branchandcut algorithm. Our results provide a new perspective on ISDPs.

3:30  4:00 pm ESTCoffee Break11th Floor Collaborative Space

4:00  4:45 pm ESTReciprocity between tree ensemble optimization and multilinear optimization11th Floor Lecture Hall
 Speaker
 Mohit Tawarmalani, Purdue University
 Session Chair
 Fatma KılınçKarzan, Carnegie Mellon University
Abstract
We establish a polynomial equivalence between tree ensemble optimization and optimization of multilinear functions over the Cartesian product of simplices. Using this, we derive new formulations for tree ensemble optimization problems and obtain new convex hull results for multilinear polytopes. A computational experiment on multicommodity transportation problems with costs modeled using tree ensembles shows the practical advantage of our formulation relative to existing formulations of tree ensembles and other piecewiselinear modeling techniques. We then consider piecewise polyhedral relaxation of multilinear optimization problems. We provide the first ideal formulation over nonregular partitions. We also improve the relaxations over regular partitions by adding linking constraints. These relaxations significantly improve performance of ALPINE and are included in the software.
This is joint work with Jongeun Kim (Google) and J.P. P. Richard (University of Minnesota).
Friday, March 3, 2023

9:00  9:45 am ESTMarkov Chainbased Policies for Multistage Stochastic Integer Linear Programming11th Floor Lecture Hall
 Speaker
 Merve Bodur, University of Toronto
 Session Chair
 Pietro Belotti, Politecnico di Milano
Abstract
We introduce a novel aggregation framework to address multistage stochastic programs with mixedinteger state variables and continuous local variables (MSILPs). Our aggregation framework imposes additional structure to the integer state variables by leveraging the information of the underlying stochastic process, which is modeled as a Markov chain (MC). We present a novel branchandcut algorithm integrated with stochastic dual dynamic programming as an exact solution method to the aggregated MSILP, which can also be used in an approximation form to obtain dual bounds and implementable feasible solutions. Moreover, we apply twostage linear decision rule (2SLDR) approximations and propose MCbased variants to obtain highquality decision policies with significantly reduced computational effort. We test the proposed methodologies in a novel MSILP model for hurricane disaster relief logistics planning.

10:00  10:30 am ESTCoffee Break11th Floor Collaborative Space

10:30  11:15 am ESTOn practical first order methods for LP11th Floor Lecture Hall
 Speaker
 Daniel Espinoza, Google Research
 Session Chair
 Pietro Belotti, Politecnico di Milano
Abstract
Solving linear programs is nowadays an everyday task. Used even in embedded systems, but also run on very large hardware. However, solving very large models has remained a major challenge. Either because most successful algorithms require more than linear space to solve such models, or because they become extremely slow in practice. Although the concept of first ordermethods, and potential function methods have been around for a long time, they have failed to be broadly applicable, or no competitive implementations are widely available. In this talk we will be motivating the need for such a class of algorithms, share some (known) evidence that such schemes have worked in special situations, and present results of experiments running one such algorithm (PDLP) both in standard benchmark models, and also on very large models arising from network planning. And also on a first order method for general LPs using an exponential potential function to deal with unstructured constraints.

11:30 am  1:30 pm ESTLunch/Free Time

1:30  2:15 pm ESTIdeal polyhedral relaxations of nonpolyhedral sets11th Floor Lecture Hall
 Speaker
 Andres Gomez, University of Southern California
 Session Chair
 Marcia Fampa, Federal University of Rio de Janeiro
Abstract
Algorithms for mixedinteger optimization problems are based on the sequential construction of tractable relaxations of the discrete problem, until the relaxations are sufficiently tight to guarantee optimality of the resulting solution. For classical operational and logistics problems, which can be formulated as mixedinteger linear optimization problems, it is wellknown that such relaxations should be polyhedral. Thus, there has been a sustained stream of research spanning several decades on constructing and exploiting linear relaxations. As a consequence, mixedinteger linear optimization problems deemed to be intractable 30 years old can be solved to optimality in seconds or minutes nowadays. Modern statistical and decisionmaking problems call for mixedinteger nonlinear optimization (MINLO) formulations, which inherently lead to nonpolyhedral relaxations. There has been a substantial progress in extending and adapting techniques for both the mixedinteger linear optimization and continuous nonlinear literatures, but there may a fundamental limit on the effectiveness of such approaches as they fail to exploit the specific characteristics of MINLO problems. In this talk, we discuss recent progress in studying the fundamental structure of MINLO problems. In particular, we show that such problems have a hidden polyhedral substructure that captures the nonconvexities associated with discrete variables. Thus, by exploiting this substructure, convexification theory and methods based on polyhedral theory can naturally be used study nonpolyhedral sets. We also provide insights into how to design algorithms that tackle the ensuing relaxations.

2:30  3:15 pm ESTOn DantzigWolfe Relaxation of Rank Constrained Optimization: Exactness, Rank Bounds, and Algorithms11th Floor Lecture Hall
 Speaker
 Weijun Xie, Georgia Institute of Technology
 Session Chair
 Marcia Fampa, Federal University of Rio de Janeiro
Abstract
This paper studies the rank constrained optimization problem (RCOP) that aims to minimize a linear objective function over intersecting a prespecified closed rank constrained domain set with twosided linear matrix inequalities. The generic RCOP framework exists in many nonconvex optimization and machine learning problems. Although RCOP is, in general, NPhard, recent studies reveal that its DantzigWolfe Relaxation (DWR), which refers to replacing the domain set by its closed convex hull, can lead to a promising relaxation scheme. This motivates us to study the strength of DWR. Specifically, we develop the firstknown necessary and sufficient conditions under which the DWR and RCOP are equivalent. Beyond the exactness, we prove the rank bound of optimal DWR extreme points. We design a column generation algorithm with an effective separation procedure. The numerical study confirms the promise of the proposed theoretical and algorithmic results.

3:30  4:00 pm ESTCoffee Break11th Floor Collaborative Space
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