Network


Latest external collaboration on country level. Dive into details by clicking on the dots.

Hotspot


Dive into the research topics where Pietro Belotti is active.

Publication


Featured researches published by Pietro Belotti.


Optimization Methods & Software | 2009

Branching and bounds tighteningtechniques for non-convex MINLP

Pietro Belotti; Jon Lee; Leo Liberti; François Margot; Andreas Wächter

Many industrial problems can be naturally formulated using mixed integer non-linear programming (MINLP) models and can be solved by spatial Branch&Bound (sBB) techniques. We study the impact of two important parts of sBB methods: bounds tightening (BT) and branching strategies. We extend a branching technique originally developed for MILP, reliability branching, to the MINLP case. Motivated by the demand for open-source solvers for real-world MINLP problems, we have developed an sBB software package named couenne (Convex Over- and Under-ENvelopes for Non-linear Estimation) and used it for extensive tests on several combinations of BT and branching techniques on a set of publicly available and real-world MINLP instances. We also compare the performance of couenne with a state-of-the-art MINLP solver.


Acta Numerica | 2013

Mixed-integer nonlinear optimization

Pietro Belotti; Christian Kirches; Sven Leyffer; Jeff Linderoth; James R. Luedtke; Ashutosh Mahajan

Many optimal decision problems in scientific, engineering, and public sector applications involve both discrete decisions and nonlinear system dynamics that affect the quality of the final design or plan. These decision problems lead to mixed-integer nonlinear programming (MINLP) problems that combine the combinatorial difficulty of optimizing over discrete variable sets with the challenges of handling nonlinear functions. We review models and applications of MINLP, and survey the state of the art in methods for solving this challenging class of problems. Most solution methods for MINLP apply some form of tree search. We distinguish two broad classes of methods: single-tree and multitree methods. We discuss these two classes of methods first in the case where the underlying problem functions are convex. Classical single-tree methods include nonlinear branch-and-bound and branch-and-cut methods, while classical multitree methods include outer approximation and Benders decomposition. The most efficient class of methods for convex MINLP are hybrid methods that combine the strengths of both classes of classical techniques. Non-convex MINLPs pose additional challenges, because they contain non-convex functions in the objective function or the constraints; hence even when the integer variables are relaxed to be continuous, the feasible region is generally non-convex, resulting in many local minima. We discuss a range of approaches for tackling this challenging class of problems, including piecewise linear approximations, generic strategies for obtaining convex relaxations for non-convex functions, spatial branch-and-bound methods, and a small sample of techniques that exploit particular types of non-convex structures to obtain improved convex relaxations. We finish our survey with a brief discussion of three important aspects of MINLP. First, we review heuristic techniques that can obtain good feasible solution in situations where the search-tree has grown too large or we require real-time solutions. Second, we describe an emerging area of mixed-integer optimal control that adds systems of ordinary differential equations to MINLP. Third, we survey the state of the art in software for MINLP.


arXiv: Combinatorics | 2012

Linear Programming Relaxations of Quadratically Constrained Quadratic Programs

Andrea Qualizza; Pietro Belotti; François Margot

We investigate the use of linear programming tools for solving semidefinite programming relaxations of quadratically constrained quadratic problems. Classes of valid linear inequalities are presented, including sparse PSD cuts, and principal minors PSD cuts. Computational results based on instances from the literature are presented.


Archive | 2015

A Conic Representation of the Convex Hull of Disjunctive Sets and Conic Cuts for Integer Second Order Cone Optimization

Pietro Belotti; Julio C. Góez; Imre Pólik; Ted K. Ralphs; Tamás Terlaky

We study the convex hull of the intersection of a convex set E and a disjunctive set. This intersection is at the core of solution techniques for Mixed Integer Convex Optimization. We prove that if there exists a cone K (resp., a cylinder C) that has the same intersection with the boundary of the disjunction as E, then the convex hull is the intersection of E with K (resp., C).The existence of such a cone (resp., a cylinder) is difficult to prove for general conic optimization. We prove existence and unicity of a second order cone (resp., a cylinder), when E is the intersection of an affine space and a second order cone (resp., a cylinder). We also provide a method for finding that cone, and hence the convex hull, for the continuous relaxation of the feasible set of a Mixed Integer Second Order Cone Optimization (MISOCO) problem, assumed to be the intersection of an ellipsoid with a general linear disjunction. This cone provides a new conic cut for MISOCO that can be used in branch-and-cut algorithms for MISOCO problems.


integer programming and combinatorial optimization | 2005

Randomized relaxation methods for the maximum feasible subsystem problem

Edoardo Amaldi; Pietro Belotti; Raphael Hauser

In the Max FS problem, given an infeasible linear system Ax ≥ b, one wishes to find a feasible subsystem containing a maximum number of inequalities. This NP-hard problem has interesting applications in a variety of fields. In some challenging applications in telecommunications and computational biology one faces very large Max FS instances with up to millions of inequalities in thousands of variables. We propose to tackle large-scale instances of Max FS using randomized and thermal variants of the classical relaxation method for solving systems of linear inequalities. We present a theoretical analysis of one particular version of such a method in which we derive a lower bound on the probability that it identifies an optimal solution within a given number of iterations. This bound, which is expressed as a function of a condition number of the input data, implies that with probability 1 the randomized method identifies an optimal solution after finitely many iterations. We also present computational results obtained for medium- to large-scale instances arising in the planning of digital video broadcasts and in the modelling of the energy functions driving protein folding. Our experiments indicate that these methods perform very well in practice.


Archive | 2009

Optimization of Catheter Position and Dwell Time in Prostate HDR Brachytherapy using HIPO and Linear Programming

A. Karabis; Pietro Belotti; D. Baltas

In this work the problem of determination of the optimal catheter position and source loading in HDR prostate brachytherapy is modeled as a Mixed Integer Linear Programming (MILP) problem and solved by ILOG CPLEX. The results are compared to those of HIPO, a state of the art inverse treatment plan optimization algorithm. Given that MILP is guaranteed to find the global optimum, it provides the baseline for the evaluation of HIPO solutions. The quality of 12 clinical HDR brachytherapy implants for prostate utilizing HIPO, for dwell time optimization only, has been compared to alternative plans with Linear Programming (LP). All common dose-volume indices for the prostate and the organs at risk have been considered. Our results demonstrate that in the case of dwell time optimization, HIPO delivers high quality plans and the differences to the LP solutions are statistically insignificant (p > 0.05) for all indices examined. In the case of combined catheter position and dwell time optimization, the results for 3 clinical cases have been compared. MILP was able to deliver the optimal solution only for one simple case and upper- lower bounds for the rest. The plans produced by HIPO were clinically acceptable, close and clinically equivalent to the global optimum or the upper bounds delivered by MILP for all 3 cases.


Discrete Applied Mathematics | 2013

On families of quadratic surfaces having fixed intersections with two hyperplanes

Pietro Belotti; Julio C. Góez; Imre Pólik; Ted K. Ralphs; Tamás Terlaky

We investigate families of quadrics all of which have the same intersection with two given hyperplanes. The cases when the two hyperplanes are parallel and when they are nonparallel are discussed. We show that these families can be described with only one parameter and describe how the quadrics are transformed as the parameter changes. This research was motivated by an application in mixed integer conic optimization. In that application, we aimed to characterize the convex hull of the union of the intersections of an ellipsoid with two half-spaces arising from the imposition of a linear disjunction.


Computer Networks | 2008

Multi-layer MPLS network design: The impact of statistical multiplexing

Pietro Belotti; Antonio Capone; Giuliana Carello; Federico Malucelli

The possibility of adding multi protocol label switching (MPLS) support to transport networks is considered an important opportunity by telecom carriers that want to add packet services and applications to their networks. However, the question arises whether it is suitable to have MPLS nodes just at the edge of the network to collect packet traffic from users, or to introduce also MPLS facilities on a subset of the core nodes in order to exploit packet switching flexibility and multiplexing, thus inducing a better bandwidth allocation. In this paper, we propose a mathematical programming model for the design of two-layer networks where MPLS is considered on top of transport networks (SDH or WDM depending on required link speed). Our models take into account the tradeoff between the cost of adding MPLS support in the core nodes and the savings in the link bandwidth allocation due to the statistical multiplexing and the traffic grooming effects induced by MPLS nodes. The traffic matrix specifies for each point-to-point request a pair of values: a mean traffic value and an additional one. Using this traffic model, the effect of statistical multiplexing on a link allows to allocate a capacity equal to the sum of all the mean values of the traffic demands routed on the link and only the highest additional one. We propose a path-based Mixed Integer Programming (MIP) model for the problem of optimizing the number and location of MPLS nodes in the network and the link capacities. We apply Lagrangian relaxation to this model and use the subgradient method to obtain a lower bound of the network cost. As the number of path variables used to model the routing grows exponentially with the graph size, we use an initially limited number of variables and a column generation approach. We also introduce a heuristic approach to get a good feasible solution. Computational results are reported for small size and real-world instances.


A Quarterly Journal of Operations Research | 2007

A branch-and-cut method for the obnoxious p-median problem

Pietro Belotti; Martine Labbé; Francesco Maffioli; Malick Ndiaye

The obnoxious p-median (OpM) problem is the repulsive counterpart of the ore known attractive p-median problem. Given a set I of cities and a set J of possible locations for obnoxious plants, a p-cardinality subset Q of J is sought, such that the sum of the distances between each city of I and the nearest obnoxious site in Q is maximised. We formulate (OpM) as a {0,1} linear programming problem and propose three families of valid inequalities whose separation problem is polynomial. We describe a branch-and-cut approach based on these inequalities and apply it to a set of instances found in the location literature. The computational results presented show the effectiveness of these inequalities for (OpM).


cologne twente workshop on graphs and combinatorial optimization | 2007

New formulations for the Kissing Number Problem

Sergei S. Kucherenko; Pietro Belotti; Leo Liberti; Nelson Maculan

Determining the maximum number of D-dimensional spheres of radius r that can be adjacent to a central sphere of radius r is known as the Kissing Number Problem (KNP). The problem has been solved for two, three and very recently for four dimensions. We present two nonlinear (nonconvex) mathematical programming models for the solution of the KNP. We solve the problem by using two stochastic global optimization methods: a Multi Level Single Linkage algorithm and a Variable Neighbourhood Search. We obtain numerical results for two, three and four dimensions.

Collaboration


Dive into the Pietro Belotti's collaboration.

Top Co-Authors

Avatar
Top Co-Authors

Avatar

Jon Lee

University of Michigan

View shared research outputs
Top Co-Authors

Avatar
Top Co-Authors

Avatar
Top Co-Authors

Avatar
Top Co-Authors

Avatar

Andrew J. Miller

University of Wisconsin-Madison

View shared research outputs
Top Co-Authors

Avatar

François Margot

Carnegie Mellon University

View shared research outputs
Top Co-Authors

Avatar
Top Co-Authors

Avatar
Top Co-Authors

Avatar

Ayşegül Altın

TOBB University of Economics and Technology

View shared research outputs
Researchain Logo
Decentralizing Knowledge