10:00 - 10:25

Sparsest cut in planar graphs, maximum concurrent flows and their connections with the max-cut problem

We study the sparsest cut problem when the "capacity-demand" graph is planar, and give a combinatorial algorithm. In this type of graphs there is an edge for each positive capacity and also an edge for each positive demand. We extend this result to graphs with no $K_5$ minor. We also show how to find a maximum concurrent flow in these two cases. We use ideas that had been developed for the max-cut problem, and show how to exploit the connections among these problems.

11:00 - 11:25

Mathematical models with a polynomial number of variables and constraints for some NP-hard combinatorial optimization problems in graphs

We present integer linear models with a polynomial number of variables and constraints for combinatorial optimization problems in graphs: optimum elementary cycles (whose traveling salesman problem), optimum elementary paths even in a graph with negative cycles, and optimum trees (whose Steiner tree problem) problems. Computational results for the the Steiner tree problem are also presented.

11:25 - 11:50

Computational Study of Some Valid Inequalities for k-Way Graph Partitioning

We consider the maximum k-cut problem that consists in partitioning the vertex set of a graph into k subsets such that the sum of the weights of edges joining vertices in different subsets is maximized. We focus on identifying effective classes of inequalities to tighten the semidefinite programming relaxation. We carry out an experimental study of four classes of inequalities from the literature: clique, general clique, wheel and bicycle wheel. We considered multiple combinations of these classes and tested them on both dense and sparse instances for different values of k. Our computational results suggest that the bicycle wheel and wheel are the strongest inequalities for k=3, and that for greater k, the wheel inequalities are the strongest by far.

11:50 - 12:15

Relay Location and Network Design

Relays are regenerators extending the reach of optical signals in telecommunications networks; they are strategic locations where exchange of drivers, trucks or mode of transportation takes place in transportation networks or refuelling/recharging stations extending the reach of alternative fuel vehicles in green transportation. With different names and characteristics, relays play a crucial role in the design of telecommunications and transportation networks. We study the network design problem with relays and present a multi-commodity flow formulation and a branch-and-price algorithm to solve it. Motivated by the practical applications we investigate the special case where each demand has a common origin. In this special case, we can show that there exists an optimal design that is a tree. Using this fact, we replace the multi-commodity flow formulation with a tree formulation enhanced with Steiner cuts. Employing a branch-and-price-and-cut algorithm on this formulation, we are able to further extend computational efficiency.

14:20 - 14:45

Maman les petits bateaux

About the power of sailing engines.

14:45 - 15:10

Reducing some graph parameter via minimal edge contractions

Let d and k be two given integers, and let G be a graph. Can we reduce a parameter of G by at least d via at most k edge contractions?

We investigate the computational complexity of these problems for fixed values of d and k and for some subclasses of graphs. The parameter we are interested are the independence number, the clique number, or the chromatic number.

15:10 - 15:35

The Stochastic Shortest Path Problem: A polyhedral combinatorics perspective

The Stochastic Shortest Path problem (SSP) is a natural extension of the deterministic shortest path problem whereby traversing an 'arc' may now lead to several destinations with different probabilities. In this setting, vertices are called states and arcs are called actions. The goal is to decide in each time step and each state which action to take so as to converge to a predefined target with probability one over an infinite time horizon. Taking an action has a cost and we wish to find a policy that minimizes the average cost over all possible realizations. The SSP was studied thoroughly by Bertsekas and Tsitsiklis (1991) and later by Bertsekas and Yu (2016) and it is well understood when there is no nonpositive cost 'transition cycles'. In this talk we give a fresh look at the problem from a polyhedral combinatorics perspective. We study the natural linear programming relaxation of the problem and we show that actually standard methods also converge when there is no negative cost transition cycles. This closes the gap with the deterministic shortest path problem.

15:35 - 16:00

Extended (Weird) Flow based Models for the (PC)ATSP

There are many ways of modelling the Asymmetric Traveling Salesman Problem (ATSP) and the related Precedence Constrained ATSP (PCATSP). In this talk we present new formulations for the two problems that can be viewed as resulting from combining precedence variable based formulations, with network flow based formulations. As suggested in [1], the former class of formulations permits to integrate linear ordering constraints. The motivating formulation for this work is a complicated and "ugly" formulation that results from the separation of generalized subtour elimination constraints presented in [2] (see also [1]). This so called "ugly" formulation exhibits, however, one interesting feature, namely the "disjoint sub- paths" property that is further explored to create more complicated formulations that combine two (or three) "disjoint path" network flow based formulations and have a stronger linear programming bound. Some of these stronger formulations are related to the ones presented for the PCATSP in [3] and can be viewed as generalizations in the space of the precedence based variables. Several sets of projected inequalities in the space of the arc and precedence variables and in the spirit of many presented in [1] are obtained by projection from these network flow based formulations. Computational results will be given for the ATSP and PCATSP to evaluate the quality of the new models and inequalities.

References:

[1] L. Gouveia and P. Pesneau. On extended formulations for the precedence constrainted asymmetric traveling salesman problem. Networks, 48(2):77{89, 2006.

[2] L. Gouveia and J. M. Pires. The asymmetric travelling salesman problem: On generalizations of disaggregated Miller-Tucker-Zemlin constraints. Discrete Applied Mathematics, 112:129{145, 2001.

16:30 - 16:55

High-dimensional risk-constrained dynamic asset allocation via Markov stochastic dual dynamic programming

Dynamic portfolio optimization has a vast literature exploring different simplifications by virtue of computational tractability of the problem. Previous works provide solution methods considering unrealistic assumptions, such as no transactional costs, small number of assets, specific choices of utility functions and oversimplified price dynamics. Other more realistic strategies use heuristic solution approaches to obtain suitable investment policies. In this work, we propose a high-dimensional risk-constrained dynamic asset allocation model and efficiently solve it using the stochastic dual dynamic programming algorithm. We consider multiple assets, transactional costs and a Markov factor model for asset returns. We present results for a realistic 100-asset model guaranteeing a sufficiently small optimality gap with high probability. To the best of our knowledge, this is the first systematic approach for solving realistic high-dimensional dynamic stochastic asset allocation problems.

16:55 - 17:20

Vertex k-Separator Problem

Abstract: Given an undirected graph, a Vertex k-Cut (VKS) is a subset of the vertex set such that, when the VKS is removed from the graph, the remaining vertices can be partitioned into k subsets that are pairwise edge-disconnected. The problem of finding the minimum size VKS is called Vertex k-Cut Problem (VKSP). We give a proof that VKSPis NP-Hard. We present a compact Integer Linear Programming formulation for the problems and investigate the associated polytopes. We also present extensive formulations, for which we derive a column generation and a branching scheme. Extensive computational results prove the effectiveness of the proposed methods and of the theoretical analysis.

17:20 - 17:45

The split-demand one-commodity pickup-and-delivery travelling salesman problem

This talk is related with the article published by Kerivin, Lacroix, Mahjoub and Quilliot in 2008 on the splittable pickup and delivery problem with reloads. We give a mathematical model, an exact branch-and-cut algorithm and a matheuristic approach for a new vehicle routing problem that generalizes the capacitated vehicle routing problem with split demands and some other variants recently addressed in the literature.

09:15 - 09:40

Tight MIP formulations for unit commitment and production planning problems with bounded up/down times

We study tight mixed integer programming formulations for unit commitment and production planning problems with bound constraints on the lengths of on- and off-intervals. The resulting optimization problems admit a general Dynamic Programming (shortest path) formulation which leads to tight extended formulations with a number of binary variables that is quadratic in the number of time periods. We present tight formulations with linear numbers of binary variables, derived from a network dual formulation based on new integer cumulative variables, that treats time-varying lower and upper bounds on interval lengths. This in turn leads to more general results, including simpler proofs of known tight formulations, for problems with just lower bounds. We also present some open problems.

09:40 - 10:05

Some mathematical gems in Distance Geometry

Distance geometry focuses on the concept of distance rather than points and lines. Its fundamental problem asks to draw a weighted graph in a given K-dimensional Euclidean space, so that each edge is drawn as a segment with length equal to the weight, and it has applications to many fields of science and engineering (e.g. protein folding, wireless networks, robotic control, nanostructures and more). Distance geometry results are scattered throughout the whole history of mathematics starting with the Greeks. I will present some of those I find most beautiful.

10:05 - 10:30

Distance Transformation for Network Design Problems

We propose a new way to construct extended formulations a large for class of network design problems. The approach is based on a transformation that maps any graph into a layered graph according to a given distance function. While graphs induced by binary vectors are mapped to isomorphic graphs, those induced by fractional vectors are mapped to less connected graphs, where some vertices are split. Hence, simple connectivity inequalities in the layered graph may cut off fractional vectors that were feasible for similar inequalities in the original graph. Experiments over instances of the Steiner Forest and Survivable Network Design problems show very significant gap reductions over the state-of-the art formulations.

11:00 - 11:25

The Polyhedral Lens

I would like to recall to and share with Ridha and the audience some simple old and new examples of the use of polyhedral combinatorics for solving purely combinatorial problems.

11:25 - 11:50

On the Convex Piecewise Linear Unsplittable Multicommodity Flow Problem

We consider the problem of finding the cheapest routing for a set of commodities over a directed graph, such that: i) each commodity flows through a single path, ii) the routing cost of each arc is given by a convex piecewise linear function of the load i.e. the total flow) traversing it.

We propose a new mixed-integer programming formulation for this problem. The linear relaxation of this formulation gives an optimal solution for the single commodity case, and produces very tight linear programming bounds for the multi-commodity case.

We also derive new valid inequalities for the compact basic model based on the projection of the extended formulation.

11:50 - 12:15

Connectivity with Ridha

I will present my work with Ridha on survivable network design along with some souvenirs and personal appreciations.

12:15 - 12:40

Reduced-size formulations for metric and cut polyhedra in sparse graphs

Given a graph G=(V,E) of n vertices and m edges, we consider the metric cone MET(G) and the metric polytope METP(G) defined on the real space indexed by E. These polyhedra are relaxations of several important problems in combinatorial optimization such as the max-cut problem and the multicommodity flow problem. They are known to have non-compact formulations via the cycle inequalities in the original space inline image and compact (i.e., polynomial size) extended formulations via the triangle inequalities defined on the complete graph of n vertices. In this talk, we show that one can reduce the number of triangle inequalities to O(nm) and still have extended formulations for MET(G) and METP(G). We also present several extensions of the result for special graphs and for graph partitioning problem.

14:30 - 14:55

Min Cuts with Mahjoub and McCormick

I have worked with Ridha Mahjoub and others for 20 years on a variety of network optimization problems, all related to min cuts of various types. I will survey these results, which include the 2-edge connected subgraph problem with bounded rings, max flow and min cut with bounded-length paths, separation algorithms for single-machine scheduling with precedence constraints, and strongly polynomial bounds for multiobjective and parametric global minimum cuts in graphs and hypergraphs.

14:55 - 15:20

A Matheuristic for the Multicommodity Fixed-Charge Network Design Problem

We propose an efficient iterative linear programming-based heuristic for solving the Multicommodity Fixed-Charge Network Design problem. The computational results show that the approach is competitive, finding more best solutions than any other heuristic of the current state-of-the-art.

15:20 - 15:45

Campaign planning optimization @ OM Partners

OM Partners is a leading provider of supply chain software, with the OMP Plus planning and scheduling platform embedded with different optimization techniques. In this talk, we first present various kinds of problems that manufacturing companies are facing (lotsizing, job sequencing, facility location,...), before going into the specifics of campaign planning. Two cases coming from the chemical industry are highlighted further: the so-called "small bucket" and "large bucket" cases. Finally, we give some hints on how these industrial-size problems can be tackled in practice by the use of mathematical programming... and more.

16:15 - 16:40

Optimization in Mechanism Design

Mechanism design is an area of micro-economic theory where optimization is very useful. We give a very simple introduction to optimization in mechanism design using the simple problem of the sale of an object between a seller and a buyer. The buyer has a valuation (drawn from a discrete set) of the object unknown to the seller. The seller wants to design a selling mechanism that will maximize his/her expected revenue. It is customary to assume that the probability distribution of the values is known to the seller. We relax this assumption and examine the problem from a robustness perspective under a worst-case optimization lens. Several extensions and research directions will be pointed out.

16:40 - 17:05

A business dinner problem

We are given suppliers and customers, and a set of tables. Every evening of the forthcoming days, there will be a dinner. Each customer must eat with each supplier exactly once, but two suppliers may meet at most once at a table. The number of customers and the number of suppliers who can sit together at a table are bounded above by fixed parameters. What is the minimum number of evenings to be scheduled in order to reach this objective? This question was submitted by a firm to the Junior company of a French engineering school some years ago. We provide lower and upper bounds, proven optimal solutions with closed-form expressions for some cases, and many conjectures.