# Gérard CornuéjolsCarnegie Mellon University | CMU · Tepper School of Business

Gérard Cornuéjols

Cornell University - IEOR

## About

240

Publications

31,706

Reads

**How we measure 'reads'**

A 'read' is counted each time someone views a publication summary (such as the title, abstract, and list of authors), clicks on a figure, or views or downloads the full-text. Learn more

9,886

Citations

Citations since 2017

Introduction

Additional affiliations

September 1978 - present

## Publications

Publications (240)

A filter oracle for a clutter consists of a finite set V and an oracle which, given any set X⊆V, decides in unit time whether X contains a member of the clutter. Let A2n be an algorithm that, given any clutter C over 2n elements via a filter oracle, decides whether C is ideal. We prove that in the worst case, A2n makes at least 2n−1 calls to the fi...

A clutter is clean if it has no delta or the blocker of an extended odd hole minor, and it is tangled if its covering number is two and every element appears in a minimum cover. Clean tangled clutters have been instrumental in progress towards several open problems on ideal clutters, including the τ=2 Conjecture.
Let C be a clean tangled clutter. I...

A vector is dyadic if each of its entries is a dyadic rational number, i.e. of the form a2k for some integers a, k with k≥0. A linear system Ax≤b with integral data is totally dual dyadic if whenever min{b⊤y:A⊤y=w,y≥0} for w integral, has an optimal solution, it has a dyadic optimal solution. In this paper, we study total dual dyadicness, and give...

A filter oracle for a clutter consists of a finite set $V$ along with an oracle which, given any set $X\subseteq V$, decides in unit time whether or not $X$ contains a member of the clutter. Let $\mathfrak{A}_{2n}$ be an algorithm that, given any clutter $\mathcal{C}$ over $2n$ elements via a filter oracle, decides whether or not $\mathcal{C}$ is i...

In this paper, we make some progress in addressing Woodall's Conjecture, and the refuted Edmonds-Giles Conjecture on packing dijoins in unweighted and weighted digraphs. Let $D=(V,A)$ be a digraph, and let $w\in \mathbb{Z}^A_{\geq 0}$. Suppose every dicut has weight at least $\tau$, for some integer $\tau\geq 2$. Let $\rho(\tau,D,w):=\frac{1}{\tau}...

A Correction to this paper has been published: 10.1007/s10107-020-01587-x

A vector is \emph{dyadic} if each of its entries is a dyadic rational number, i.e. of the form $\frac{a}{2^k}$ for some integers $a,k$ with $k\geq 0$. A linear system $Ax\leq b$ with integral data is \emph{totally dual dyadic} if whenever $\min\{b^\top y:A^\top y=w,y\geq {\bf 0}\}$ for $w$ integral, has an optimal solution, it has a dyadic optimal...

A vector is dyadic if each of its entries is a dyadic rational number, i.e., an integer multiple of 1 2k for some nonnegative integer k. We prove that every clean clutter with a covering number of at least two has a dyadic fractional packing of value two. This result is best possible for there exist clean clutters with a covering number of three an...

The τ=2 Conjecture predicts that every ideal minimally non-packing clutter has covering number two. In the original paper where the conjecture was proposed, in addition to an infinite class of such clutters, thirteen small instances were provided. The construction of the small instances followed an ad-hoc procedure and why it worked has remained a...

A subset of the unit hypercube {0,1}n is cube-ideal if its convex hull is described by hypercube and generalized set covering inequalities. In this paper, we provide a structure theorem for cube-ideal sets S⊆{0,1}n such that, for any point x∈{0,1}n, S−{x} and S∪{x} are cube-ideal. As a consequence of the structure theorem, we see that cuboids of su...

A clutter is k - wise intersecting if every k members have a common element, yet no element belongs to all members. We conjecture that, for some integer $$k\ge 4$$ k ≥ 4 , every k -wise intersecting clutter is non-ideal. As evidence for our conjecture, we prove it for $$k=4$$ k = 4 for the class of binary clutters. Two key ingredients for our proof...

Ideal matrices and clutters are prevalent in Combinatorial Optimization, ranging from balanced matrices, clutters of T-joins, to clutters of rooted arborescences. Most of the known examples of ideal clutters are combinatorial in nature. In this paper, rendered by the recently developed theory of cuboids, we provide a different class of ideal clutte...

We study a class of stochastic programs in which some of the elements in the objective function are random and their probability distribution has unknown parameters. The goal is to find a good estimate for the optimal solution of the stochastic program using data sampled from the distribution of the random elements. We investigate two common optimi...

A clutter is intersecting if the members do not have a common element yet every two members intersect. It has been conjectured that for clutters without an intersecting minor, total primal integrality and total dual integrality of the corresponding set covering linear system must be equivalent. In this paper, we provide a polynomial characterizatio...

We study a class of stochastic programs where some of the elements in the objective function are random, and their probability distribution has unknown parameters. The goal is to find a good estimate for the optimal solution of the stochastic program using data sampled from the distribution of the random elements. We investigate two common optimiza...

A clutter is k-wise intersecting if every k members have a common element, yet no element belongs to all members. We conjecture that every 4-wise intersecting clutter is non-ideal. As evidence for our conjecture, we prove it in the binary case. Two key ingredients for our proof are Jaeger’s 8-flow theorem for graphs, and Seymour’s characterization...

A clutter is k-wise intersecting if every k members have a common element, yet no element belongs to all members. We conjecture that every 4-wise intersecting clutter is non-ideal. As evidence for our conjecture, we prove it in the binary case. Two key ingredients for our proof are Jaeger’s 8-flow theorem for graphs, and Seymour’s characterization...

A clutter is $k$-wise intersecting if every $k$ members have a common element, yet no element belongs to all members. We conjecture that every $4$-wise intersecting clutter is non-ideal. As evidence for our conjecture, we prove it in the binary case. Two key ingredients for our proof are Jaeger's $8$-flow theorem for graphs, and Seymour's character...

The τ=2 Conjecture, the Replication Conjecture and the f-Flowing Conjecture, and the classification of binary matroids with the sums of circuits property are foundational to Clutter Theory and have far-reaching consequences in Combinatorial Optimization, Matroid Theory and Graph Theory. We prove that these conjectures and result can equivalently be...

A clutter is {\it clean} if it has no delta or the blocker of an extended odd hole minor. There are combinatorial and geometric classes of clean clutters, namely ideal clutters, clutters without an intersecting minor, and binary clutters. Tight connections between clean clutters and the first two classes have recently been established. In this pape...

A subset of the unit hypercube {0,1}n is cube-ideal if its convex hull is described by hypercube and generalized set covering inequalities. In this note, we study sets S⊆{0,1}n such that, for any subset X⊆{0,1}n of cardinality at most 2, S∪X is cube-ideal.

A clutter is identically self-blocking if it is equal to its blocker. We prove that every identically self-blocking clutter different from \(\{\{a\}\}\) is nonideal. Our proofs borrow tools from Gauge Duality and Quadratic Programming. Along the way we provide a new lower bound for the packing number of an arbitrary clutter.

A clutter is identically self-blocking if it is equal to its blocker. We prove that every identically self-blocking clutter different from is nonideal. Our proofs borrow tools from Gauge Duality and Quadratic Programming. Along the way we provide a new lower bound for the packing number of an arbitrary clutter.

In this paper, we consider the basic problem of portfolio construction in financial engineering, and analyze how market-based and analytical approaches can be combined to obtain efficient portfolios. As a first step in our analysis, we model the asset returns as a random variable distributed according to a mixture of normal random variables. We the...

We study the following problem: given a rational polytope with Chvátal rank 1, does it contain an integer point? Boyd and Pulleyblank observed that this problem is in the complexity class NP \(\cap \) co-NP, an indication that it is probably not NP-complete. It is open whether there is a polynomial time algorithm to solve the problem, and we give s...

Cambridge Core - Mathematical Finance - Optimization Methods in Finance - by Gérard Cornuéjols

Gomory–Chvátal cuts are prominent in integer programming. The Gomory–Chvátal closure of a polyhedron is the intersection of the half spaces defined by all its Gomory–Chvátal cuts. We prove that it is NP-hard to decide whether the Gomory–Chvátal closure of a rational polyhedron P is identical to the integer hull of P. An earlier version of this pape...

In this paper, we consider polytopes P that are contained in the unit hypercube. We provide conditions on the set of 0, 1 vectors not contained in P that guarantee that P has a small Chvátal rank. Our conditions are in terms of the subgraph induced by these infeasible 0, 1 vertices in the skeleton graph of the unit hypercube. In particular, we show...

We study a class of stochastic programs where some of the elements in the objective function are random, and their probability distribution has unknown parameters. The goal is to find a good estimate for the optimal solution of the stochastic program using data sampled from the distribution of the random elements. We investigate two common optimiza...

For a clutter 𝒞 over ground set E, a pair of distinct elements e, f ∈ E are coexclusive if every minimal cover contains at most one of them. An identification of 𝒞 is another clutter obtained after identifying coexclusive elements of 𝒞. If a clutter is nonpacking, then so is any identification of it.
Inspired by this observation, and impelled by th...

We study a class of quadratic stochastic programs where the distribution of random variables has unknown parameters. A traditional approach is to estimate the parameters using a maximum likelihood estimator (MLE) and to use this as input in the optimization problem. For the unconstrained case, we show that an estimator that shrinks the MLE towards...

For an integer linear program, Gomory's corner relaxation is obtained by ignoring the nonnegativity of the basic variables in a tableau formulation. In this paper, we do not relax these nonnegativity constraints. We generalize a classical result of Gomory and Johnson characterizing minimal cut-generating functions in terms of subadditivity, symmetr...

We consider the problem of minimizing a convex function over a subset of R^n that is not necessarily convex (minimization of a convex function over the integer points in a polytope is a special case). We define a family of duals for this problem and show that, under some natural conditions, strong duality holds for a dual problem in this family tha...

In this paper, we consider polytopes P that are contained in the unit hypercube. We provide conditions on the set of infeasible 0,1 vectors that guarantee that P has a small Chvátal rank. Our conditions are in terms of the subgraph induced by these infeasible 0,1 vertices in the skeleton graph of the unit hypercube. In particular, we show that when...

Gomory-Chvátal cuts are prominent in integer programming. The Gomory-Chvátal closure of a polyhedron is the intersection of all half spaces defined by its Gomory-Chvátal cuts. In this paper, we show that it is \(\mathcal {NP}\)-complete to decide whether the Gomory-Chvátal closure of a rational polyhedron is empty, even when this polyhedron contain...

This special issue of Mathematical Programming, Series B, is related to the 22nd International Symposium on Mathematical Programming (ISMP) which will take place in Pittsburgh from July 12 to July 17, 2015. As in previous symposia, ISMP 2015 will cover a broad range of topics in mathematical optimization. Fifteen keynote speakers will present plena...

In this paper we study general two-term disjunctions on affine cross-sections of the second-order cone. Under some mild assumptions, we derive a closed-form expression for a convex inequality that is valid for such a disjunctive set, and we show that this inequality is sufficient to characterize the closed convex hull of all two-term disjunctions o...

We consider the separation problem for sets X that are pre-images of a given set S by a linear mapping. Classical examples occur in integer programming, as well as in other optimization problems such as complementarity. One would like to generate valid inequalities that cut off some point not lying in X, without reference to the linear mapping. To...

In Chaps. 5 and 6 we have introduced several classes of valid inequalities that can be used to strengthen integer programming formulations in a cutting plane scheme. All these valid inequalities are “general purpose,” in the sense that their derivation does not take into consideration the structure of the specific problem at hand. Many integer prog...

Chapter 4 dealt with perfect formulations. What can one do when one is handed a formulation that is not perfect? A possible option is to strengthen the formulation in an attempt to make it closer to being perfect. One of the most successful strengthening techniques in practice is the addition of Gomory’s mixed integer cuts. These inequalities have...

The importance of integer programming stems from the fact that it can be used to model a vast array of problems arising from the most disparate areas, ranging from practical ones (scheduling, allocation of resources, etc.) to questions in set theory, graph theory, or number theory. We present here a selection of integer programming models, several...

A perfect formulation of a set \(S \subseteq \mathbb{R}^{n}\) is a linear system of inequalities Ax ≤ b such that \(\mathrm{conv}(S) =\{ x \in \mathbb{R}^{n}\,:\, Ax \leq b\}\). For example, Proposition 1.5 gives a perfect formulation of a 2-variable mixed integer linear set. When a perfect formulation is available for a mixed integer linear set, t...

To take advantage of the special structure occurring in a formulation of an integer program, it may be desirable to use a decomposition approach. For example, when the constraints can be partitioned into a set of nice constraints and the remaining set of complicating constraints, a Lagrangian approach may be appropriate. The Lagrangian dual provide...

What are integer programs? We introduce this class of problems and present two algorithmic ideas for solving them, the branch-and-bound and cutting plane methods. Both capitalize heavily on the fact that linear programming is a well-solved class of problems. Many practical situations can be modeled as integer programs, as will be shown in Chap. 2....

The goal of this chapter is threefold. First we present a polynomial algorithm for integer programming in fixed dimension. This algorithm is based on elegant ideas such as basis reduction and the flatness theorem. Second we revisit branch-and-cut, the most successful approach in practice for a wide range of applications. In particular we address a...

Semidefinite programs are a generalization of linear programs. Under mild technical assumptions, they can also be solved in polynomial time. In certain cases, they can provide tighter bounds on integer programming problems than linear programming relaxations. The first use of semidefinite programming in combinatorial problems dates back to Lovász [...

The focus of this chapter is on the study of systems of linear inequalities Ax ≤ b. We look at this subject from two different angles. The first, more algebraic, addresses the issue of solvability of Ax ≤ b.

Preface 1. Clutters 2. T-Cuts and T-Joins 3. Perfect Graphs and Matrices 4. Ideal Matrices 5. Odd Cycles in Graphs 6. 0,+1 Matrices and Integral Polyhedra 7. Signing 0,1 Matrices to Be Totally Unimodular or Balanced 8. Decomposition by k-Sum 9. Decomposition of Balanced Matrices 10. Decomposition of Perfect Graphs Bibliography Index.

We compare the relative strength of valid inequalities for the integer hull of the feasible region of mixed integer linear programs with two equality constraints, two unrestricted integer variables and any number of nonnegative continuous variables. In particular, we prove that the closure of Type 2 triangle (resp. Type 3 triangle; quadrilateral) i...

The concept of cut-generating function has its origin in the work of Gomory and Johnson from the 1970s. It has received renewed attention in the past few years. Recently Conforti, Cornuéjols, Daniilidis, Lemaréchal, and Malick proposed a general framework for studying cut-generating functions. However, they gave an example showing that not all cuts...

Gomory mixed-integer cuts are one of the key components in Branch-and-Cut solvers for mixed-integer linear programs. The textbook formula for generating these cuts is not used directly in open-source and commercial software that work in finite precision: Additional steps are performed to avoid the generation of invalid cuts due to the limited numer...

This paper contributes to the theory of cutting planes for mixed integer linear programs (MILPs). Minimal valid inequalities are well understood for a relaxation of an MILP in tableau form where all the nonbasic variables are continuous; they are derived using the gauge function of maximal lattice-free convex sets. In this paper we study lifting fu...

We consider a polyhedron intersected by a two-term disjunction, and we characterize the polyhedron resulting from taking its closed convex hull. This generalizes an earlier result of Conforti, Wolsey, and Zambelli on split disjunctions. We also recover as a special case the valid inequalities derived by Júdice, Sherali, Ribeiro, and Faustino for li...

In optimization problems such as integer programs or their relaxations, one encounters feasible regions of the form \(\{x\in\mathbb{R}_+^n:\: Rx\in S\}\) where R is a general real matrix and S ⊂ ℝq
is a specific closed set with 0 ∉ S. For example, in a relaxation of integer programs introduced in [ALWW2007], S is of the form ℤq
− b where \(b \not\i...

Recently, Balas and Qualizza introduced a new cut for mixed 0,1 programs, called lopsided cut. Here we present a family of cuts that comprises the Gomory mixed integer cut at one extreme and the lopsided cut at the other.

In this paper we consider the infinite relaxation of the corner poly-hedron with 2 rows. For the 1-row case, Gomory and Johnson proved in their seminal paper a sufficient condition for a minimal function to be extreme, the celebrated 2-Slope Theorem. Despite increased interest in understanding the multiple row setting, no generalization of this the...

In this paper, we study MINLPs featuring “on/off” constraints. An “on/off” constraint is a constraint f(x)≤0 that is activated whenever a corresponding 0–1 variable is equal to1. Our main result is an explicit characterization
of the convex hull of the feasible region when the MINLP consists of simple bounds on the variables and one “on/off” constr...

In Mathematical Programming 2003, Gomory and Johnson conjecture that the facets of the infinite group problem are always generated by piecewise linear functions. In this paper we give an example showing that the Gomory-Johnson conjecture is false.

The corner relaxation of a mixed-integer linear program is a central concept in cutting plane theory. In a recent paper Fischetti and Monaci provide an empirical assessment of the strength of the corner and other related relaxations on benchmark problems. In this paper we give a precise characterization of the bounds given by these relaxations for...

We consider mixed-integer linear programs where free integer variables are expressed in terms of nonnegative continuous variables. When this model only has two integer variables, Dey and Louveaux characterized the intersection cuts that have infinite split rank. We show that, for any number of integer variables, the split rank of an intersection cu...

We consider mixed-integer linear programs where free integer variables are expressed in terms of nonnegative continuous variables. When this model only has two integer variables, Dey and Louveaux characterized the intersection cuts that have infinite split rank. We show that, for any number of integer variables, the split rank of an intersection cu...

In this paper we propose practical strategies for generating split cuts, by considering integer linear combinations of the
rows of the optimal simplex tableau, and deriving the corresponding Gomory mixed-integer cuts; potentially, we can generate
a huge number of cuts. A key idea is to select subsets of variables, and cut deeply in the space of the...

There has been a recent interest in cutting planes generated from two or more rows of the optimal simplex tableau. One can construct examples of integer programs for which a single cutting plane generated from two rows dominates the entire split closure. Motivated by these theoretical results, we study the effect of adding a family of cutting plane...

Branch-and-Cut is the most commonly used algorithm for solving Inte-ger and Mixed-Integer Linear Programs. In order to reduce the number of nodes that have to be enumerated before optimality of a solution can be proven, branch-ing on general disjunctions (i.e. split disjunctions involving more than one variable, as opposed to branching on simple di...

In this paper we consider the relaxation of the corner polyhedron introduced by Andersen et al., which we denote by RCP. We study the relative strength of the split and triangle cuts of RCP’s. Basu et al. showed examples where the split closure can be arbitrarily worse than the triangle closure under a ‘worst-cost’ type of measure. However, despite...

This paper considers a modification of the branch-and-cut algorithm for Mixed Integer Linear Programming where branching is
performed on general disjunctions rather than on variables. We select promising branching disjunctions based on a heuristic
measure of disjunction quality. This measure exploits the relation between branching disjunctions and...

Recently, it has been shown that minimal inequalities for a continuous relaxation of mixed integer linear programs are associated with maximal lattice-free convex sets. In this paper we show how to lift these inequalities for integral nonbasic variables by considering maximal lattice-free convex sets in a higher-dimensional space. We apply this app...

We show that, given a closed convex set $K$ containing the origin in its interior, the support function of the set $\{y\in K^* \mid \mbox{ there exists } x\in K\mbox{ such that } \langle x,y \rangle =1\}$ is the pointwise smallest among all sublinear functions $\sigma$ such that $K=\{x \mid \sigma(x)\leq 1\}$.

For a minimal inequality derived from a maximal lattice-free simplicial
polytope in $\R^n$, we investigate the region where minimal liftings are
uniquely defined, and we characterize when this region covers $\R^n$. We then
use this characterization to show that a minimal inequality derived from a
maximal lattice-free simplex in $\R^n$ with exactly...

Split cuts constitute a class of cutting planes that has been successfully em-ployed by the majority of Branch-and-Cut solvers for Mixed Integer Linear Pro-grams. Given a basis of the LP relaxation and a split disjunction, the correspond-ing split cut can be computed with a closed form expression. In this paper, we use the Lift-and-Project framewor...