Nelson Maculan

Euclidean Distance Geometry and Applications

Euclidean distance geometry is the study of Euclidean geometry based on the concept of distance. This is useful in several applications where the input data consist of an incomplete set of distances and the output is a set of points in Euclidean space realizing those given distances. We survey the theory of Euclidean distance geometry and its most important applications, with special emphasis on molecular conformation problems.

The Steiner Problem in Graphs

Publisher Summary This chapter reviews formulations and some procedures that have been suggested for the solution of the Steiner Problem in graphs. It presents the definition and results associated with the classical Steiner Problem also known as the Euclidean Steiner Problem (ESP). The chapter discusses the Steiner Problem in graphs where the Steiner Problem in an undirected graph (SPUG) and the Steiner Problem in a directed graph (SPDG) are defined. It also discusses the transformation of a SPUG in a SPDG. The chapter provides an overview on the Lawler algorithm for the SPUG. The chaptre presents four integer programming formulations of the Steiner Problem in graphs, compares the linear relaxations of three of these formulations, and describes an application of Benders method to solve one of these formulations illustrated by an example. It also presents solution methods based on these formulations and describes some computational results.

Recent advances on the Discretizable Molecular Distance Geometry Problem

The Molecular Distance Geometry Problem (MDGP) consists in finding an embedding in R3 of a nonnegatively weighted simple undirected graph with the property that the Euclidean distances between embedded adjacent vertices must be the same as the corresponding edge weights. The Discretizable Molecular Distance Geometry Problem (DMDGP) is a particular subset of the MDGP which can be solved using a discrete search occurring in continuous space; its main application is to find three-dimensional arrangements of proteins using Nuclear Magnetic Resonance (NMR) data. The model provided by the DMDGP, however, is too abstract to be directly applicable in proteomics. In the last five years our efforts have been directed towards adapting the DMDGP to be an ever closer model of the actual difficulties posed by the problem of determining protein structures from NMR data. This survey lists recent developments on DMDGP related research.

Strong bounds with cut and column generation for class-teacher timetabling

This work presents an integer programming formulation for a variant of the Class-Teacher Timetabling problem, which considers the satisfaction of teacher preferences and also the proper distribution of lessons throughout the week. The formulation contains a very large number of variables and is enhanced by cuts. Therefore, a cut and column generation algorithm to solve its linear relaxation is provided. The lower bounds obtained are very good, allowing us to prove the optimality of previously known solutions in three formerly open instances.

The volume algorithm revisited: relation with bundle methods

Abstract. We revise the Volume Algorithm (VA) for linear programming and relate it to bundle methods. When first introduced, VA was presented as a subgradient-like method for solving the original problem in its dual form. In a way similar to the serious/null steps philosophy of bundle methods, VA produces green, yellow or red steps. In order to give convergence results, we introduce in VA a precise measure for the improvement needed to declare a green or serious step. This addition yields a revised formulation (RVA) that is halfway between VA and a specific bundle method, that we call BVA. We analyze the convergence properties of both RVA and BVA. Finally, we compare the performance of the modified algorithms versus VA on a set of Rectilinear Steiner problems of various sizes and increasing complexity, derived from real world VLSI design instances.

Lagrangian decomposition for integer nonlinear programming with linear constraints

We present a Lagrangean decomposition to study integer nonlinear programming problems. Solving the dual Lagrangean relaxation we have to obtain at each iteration the solution of a nonlinear programming with continuous variables and an integer linear programming. Decreasing iteratively the primal—dual gap we propose two algorithms to treat the integer nonlinear programming.

A Function to Test Methods Applied to Global Minimization of Potential Energy of Molecules

In this paper we develop a function with a functional form similar to general potential energy functions and whose global minimum is known. We prove that the number of local minimizers of this function increases exponentially with the size of the problem, which characterizes the principal difficulty in minimizing molecular potential energy functions. In order to guarantee the global optimality and to show the difficulty in obtaining the global minimum of this function, we propose the utilization of a deterministic algorithm. The algorithm is based on a branch and bound scheme that uses interval analysis techniques to calculate the lower bounds. Computational results for problems with up to 25 degrees of freedom are presented.

Using Lagrangian dual information to generate degree constrained spanning trees

In this paper, a Lagrangian-based heuristic is proposed for the degree constrained minimum spanning tree problem. The heuristic uses Lagrangian relaxation information to guide the construction of feasible solutions to the problem. The scheme operates, within a Lagrangian relaxation framework, with calls to a greedy construction heuristic, followed by a heuristic improvement procedure. A look ahead infeasibility prevention mechanism, introduced into the greedy heuristic, allowed us to solve instances of the problem where some of the vertices are restricted to having degrees 1 or 2. Furthermore, in order to cut down on CPU time, a restricted version of the original problem is formulated and used to generate feasible solutions. Extensive computational experiments were conducted and indicate that the proposed heuristic is competitive with the best heuristics and metaheuristics in the literature.

A linear-time median-finding algorithm for projecting a vector on the simplex of Rn

An O(n) time algorithm for the projection of a vector is presented on the simplex of R^n. We use a linear-time median-finding algorithm to determine the median of the components of a vector. The problem of finding such a projection arises as a subproblem of some optimization methods.

A relax-and-cut algorithm for the prize-collecting Steiner problem in graphs

Given an undirected graph G with penalties associated with its vertices and costs associated with its edges, a Prize Collecting Steiner (PCS) tree is either an isolated vertex of G or else any tree of G, be it spanning or not. The weight of a PCS tree equals the sum of the costs for its edges plus the sum of the penalties for the vertices of G not spanned by the PCS tree. Accordingly, the Prize Collecting Steiner Problem in Graphs (PCSPG) is to find a PCS tree with the lowest weight. In this paper, after reformulating and re-interpreting a given PCSPG formulation, we use a Lagrangian Non Delayed Relax and Cut (NDRC) algorithm to generate primal and dual bounds to the problem. The algorithm is capable of adequately dealing with the exponentially many candidate inequalities to dualize. It incorporates ingredients such as a new PCSPG reduction test, an effective Lagrangian heuristic and a modification in the NDRC framework that allows duality gaps to be further reduced. The Lagrangian heuristic suggested here dominates their PCSPG counterparts in the literature. The NDRC PCSPG lower bounds, most of the time, nearly matched the corresponding Linear Programming relaxation bounds.