Néstor E. Aguilera

Approximating optimization problems over convex functions

Many problems of theoretical and practical interest involve finding an optimum over a family of convex functions. For instance, finding the projection on the convex functions in Hk(Ω), and optimizing functionals arising from some problems in economics. In the continuous setting and assuming smoothness, the convexity constraints may be given locally by asking the Hessian matrix to be positive semidefinite, but in making discrete approximations two difficulties arise: the continuous solutions may be not smooth, and functions with positive semidefinite discrete Hessian need not be convex in a discrete sense. Previous work has concentrated on non-local descriptions of convexity, making the number of constraints to grow super-linearly with the number of nodes even in dimension 2, and these descriptions are very difficult to extend to higher dimensions. In this paper we propose a finite difference approximation using positive semidefinite programs and discrete Hessians, and prove convergence under very general conditions, even when the continuous solution is not smooth, working on any dimension, and requiring a linear number of constraints in the number of nodes. Using semidefinite programming codes, we show concrete examples of approximations to problems in two and three dimensions.

On Convex Functions and the Finite Element Method

Many problems of theoretical and practical interest involve finding a convex or concave function. For instance, optimization problems such as finding the projection on the convex functions in

On uniform consistent estimators for convex regression

H^k(\Omega)

The disjunctive procedure and blocker duality

, or some problems in economics. In the continuous setting and assuming smoothness, the convexity constraints may be given locally by asking the Hessian matrix to be positive semidefinite, but in making discrete approximations two difficulties arise: the continuous solutions may be not smooth, and an adequate discrete version of the Hessian must be given. In this paper we propose a finite element description of the Hessian, and prove convergence under very general conditions, even when the continuous solution is not smooth, working on any dimension, and requiring a linear number of constraints in the number of nodes. Using semidefinite programming codes, we show concrete examples of approximations to optimization problems.

On packing and covering polyhedra of consecutive ones circulant clutters

A new nonparametric estimator of a convex regression function in any dimension is proposed and its uniform convergence properties are studied. We start by using any estimator of the regression function and convexify it by taking the convex envelope of a sample of the approximation obtained. We prove that the uniform rate of convergence of the estimator is maintained after the convexification is applied. The finite-sample properties of the new estimator are investigated by means of a simulation study and the application of the new method is demonstrated in examples.

The Shapley value for arbitrary families of coalitions

In this paper we relate two rather different branches of polyhedral theory in linear optimization problems: the blocking type polyhedra and the disjunctive procedure of Balas et al. For this purpose, we define a disjunctive procedure over blocking type polyhedra with vertices in [0, 1]n, study its properties, and analyze its behavior under blocker duality. We compare the indices of the procedure over a pair of blocking clutter polyhedra, obtaining that they coincide.

Vertex adjacencies in the set covering polyhedron

Building on work by G. Cornuejols and B. Novick and by L. Trotter, we give different characterizations of contractions of consecutive ones circulant clutters that give back consecutive ones circulant clutters. Based on a recent result by G. Argiroffo and S. Bianchi, we then arrive at characterizations of the vertices of the fractional set covering polyhedron of these clutters. We obtain similar characterizations for the fractional set packing polyhedron using a result by F.B. Shepherd, and relate our findings with similar ones obtained by A. Wagler for the clique relaxation of the stable set polytope of webs. Finally, we show how our results can be used to obtain some old and new results on the corresponding fractional set covering polyhedron using properties of Farey series. Our results do not depend on Lehmans work or blocker/antiblocker duality, as is traditional in the field.

Arithmetic relations in the set covering polyhedron of circulant clutters

We address the problem of finding a suitable definition of a value similar to that of Shapleys, when the games are defined on a subfamily of coalitions with no structure. We present two frameworks: one based on the familiar efficiency, linearity and null player axioms, and the other on linearity and the behavior on unanimity games. We give several properties and examples in each case, and give necessary and sufficient conditions on the family of coalitions for the approaches to coincide.

On the facets of lift-and-project relaxations under graph operations

We describe the adjacency of vertices of the (unbounded version of the) set covering polyhedron, in a similar way to the description given by Chvtal for the stable set polytope. We find a sufficient condition for adjacency, and characterize it with similar conditions in the case where the underlying matrix is row circular. We apply our findings to show a new infinite family of minimally nonideal matrices.

A polyhedral approach to the stability of a family of coalitions

Abstract We study the structure of the set covering polyhedron of circulant clutters, P ( C n k ) , especially the properties related to contractions that yield other circulant clutters. Building on work by Cornuejols and Novick, we show that if C n k / N is isomorphic to C n ′ k ′ , then certain algebraic relations must hold and N is the union of particular disjoint simple directed cycles. We also show that this property is actually a characterization. Based on a result by Argiroffo and Bianchi, who characterize the set of null coordinates of vertices of P ( C n k ) as being one of such Ns, we then arrive at other characterizations, one of them being the conditions that hold between the existence of vertices and algebraic relations of certain parameters. With these tools at hand, we show how to obtain by algebraic means some old and new results, without depending on Lehmans work as is traditional in the field.