Sonia Cafieri

Reformulations in Mathematical Programming: A Computational Approach

Mathematical programming is a language for describing optimization problems; it is based on parameters, decision variables, objective function(s) subject to various types of constraints. The present treatment is concerned with the case when objective(s) and constraints are algebraic mathematical expressions of the parameters and decision variables, and therefore excludes optimization of black-box functions. A reformulation of a mathematical program P is a mathematical program Q obtained from P via symbolic transformations applied to the sets of variables, objectives and constraints. We present a survey of existing reformulations interpreted along these lines, some example applications, and describe the implementation of a software framework for reformulation and optimization.

On convex relaxations of quadrilinear terms

The best known method to find exact or at least ε-approximate solutions to polynomial-programming problems is the spatial Branch-and-Bound algorithm, which rests on computing lower bounds to the value of the objective function to be minimized on each region that it explores. These lower bounds are often computed by solving convex relaxations of the original program. Although convex envelopes are explicitly known (via linear inequalities) for bilinear and trilinear terms on arbitrary boxes, such a description is unknown, in general, for multilinear terms of higher order. In this paper, we study convex relaxations of quadrilinear terms. We exploit associativity to rewrite such terms as products of bilinear and trilinear terms. Using a general technique, we formally establish the intuitive fact that any relaxation for k-linear terms that employs a successive use of relaxing bilinear terms (via the bilinear convex envelope) can be improved by employing instead a relaxation of a trilinear term (via the trilinear convex envelope). We present a computational analysis which helps establish which relaxations are strictly tighter, and we apply our findings to two well-studied applications: the Molecular Distance Geometry Problem and the Hartree–Fock Problem.

Aircraft deconfliction with speed regulation: new models from mixed-integer optimization

Detecting and solving aircraft conflicts, which occur when aircraft sharing the same airspace are too close to each other according to their predicted trajectories, is a crucial problem in Air Traffic Management. We focus on mixed-integer optimization models based on speed regulation. We first solve the problem to global optimality by means of an exact solver. Since the problem is very difficult to solve, we also propose a heuristic procedure where the problem is decomposed and it is locally exactly solved. Computational results show that the proposed approach provides satisfactory results.

Improving heuristics for network modularity maximization using an exact algorithm

Heuristics are widely applied to modularity maximization models for the identification of communities in complex networks. We present an approach to be applied as a post-processing to heuristic methods in order to improve their performances. Starting from a given partition, we test with an exact algorithm for bipartitioning if it is worthwhile to split some communities or to merge two of them. A combination of merge and split actions is also performed. Computational experiments show that the proposed approach is effective in improving heuristic results.

Edge ratio and community structure in networks

A hierarchical divisive algorithm is proposed for identifying communities in complex networks. To that effect, the definition of community in the weak sense of Radicchi [Proc. Natl. Acad. Sci. U.S.A. 101, 2658 (2004)] is extended into a criterion for a bipartition to be optimal: one seeks to maximize the minimum for both classes of the bipartition of the ratio of inner edges to cut edges. A mathematical program is used within a dichotomous search to do this in an optimal way for each bipartition. This includes an exact solution of the problem of detecting indivisible communities. The resulting hierarchical divisive algorithm is compared with exact modularity maximization on both artificial and real world data sets. For two problems of the former kind optimal solutions are found; for five problems of the latter kind the edge ratio algorithm always appears to be competitive. Moreover, it provides additional information in several cases, notably through the use of the dendrogram summarizing the resolution. Finally, both algorithms are compared on reduced versions of the data sets of Girvan and Newman [Proc. Natl. Acad. Sci. U.S.A. 99, 7821 (2002)] and of Lancichinetti [Phys. Rev. E 78, 046110 (2008)]. Results for these instances appear to be comparable.

Feasibility-based bounds tightening via fixed points

The search tree size of the spatial Branch-and-Bound algorithm for Mixed-Integer Nonlinear Programming depends on many factors, one of which is the width of the variable ranges at every tree node. A range reduction technique often employed is called Feasibility Based Bounds Tightening, which is known to be practically fast, and is thus deployed at every node of the search tree. From time to time, however, this technique fails to converge to its limit point in finite time, thereby slowing the whole Branch-and-Bound search considerably. In this paper we propose a polynomial time method, based on solving a linear program, for computing the limit point of the Feasibility Based Bounds Tightening algorithm applied to linear equality and inequality constraints.

On the composition of convex envelopes for quadrilinear terms

Within the framework of the spatial Branch-and-Bound algorithm for solving Mixed-Integer Nonlinear Programs, different convex relaxations can be obtained for multilinear terms by applying associativity in different ways. The two groupings ((x1x2)x3)x4 and (x1x2x3)x4 of a quadrilinear term, for example, give rise to two different convex relaxations. In [6] we prove that having fewer groupings of longer terms yields tighter convex relaxations. In this paper we give an alternative proof of the same fact and perform a computational study to assess the impact of the tightened convex relaxation in a spatial Branch-and-Bound setting.

Mixed-integer nonlinear programming for aircraft conflict avoidance by sequentially applying velocity and heading angle changes

We consider the problem of aircraft conflict avoidance in Air Traffic Management systems. Given an initial configuration of a number of aircraft sharing the same airspace, the main goal of conflict avoidance is to guarantee that a minimum safety distance between each pair of aircraft is always respected during their flights. We consider aircraft separation achieved by heading angle deviations, and propose a mixed 0–1 nonlinear optimization model, that is then combined with another one which is based on aircraft speed regulation. A two-step solution approach is proposed, where the two models are sequentially solved using a state-of-the-art mixed-integer nonlinear programming solver. Numerical results validate the proposed approach and clearly show the benefit of combining the two considered separation maneuvers.

Maximizing the number of conflict-free aircraft using mixed-integer nonlinear programming

We address the conflict detection and resolution problem in air traffic control, where an aircraft conflict is a loss of separation between aircraft trajectories. Conflict avoidance is crucial to ensure flight safety and remains a challenging traffic control problem. We focus on speed control to separate aircraft and consider two approaches: (i) maximize the number of conflicts resolved and (ii) identify the largest set of conflict-free aircraft. Both problems are modeled using mixed-integer nonlinear programming and a tailored greedy algorithm is proposed for the latter. Computational efficiency is improved through a pre-processing algorithm which attempts to reduce the size of the conflict resolution models by detecting the existence of pairwise potential conflicts. Numerical results are provided after implementing the proposed models and algorithms on benchmark conflict resolution instances. The results highlight the benefits of using the proposed pre-processing step as well as the versatility and the efficiency of the proposed models.

Plunge milling time optimization via mixed-integer nonlinear programming

A new approach to minimize time machining through Mixed-Integer Nonlinear Programming.Mathematical formulation of cutting parameter optimization for plunge milling.Machine-tool and cutter-related constraints taking into account control laws.Optimal cutting parameters obtained, tailored to each elementary tool trajectory.Gains as high as 55% are obtained when compared with standard industrial methods. Plunge milling is a recent and efficient production mean for machining deep workpieces, notably in aeronautics. This paper focuses on the minimization of the machining time by optimizing the values of the cutting parameters. Currently, neither Computer-Aided Manufacturing (CAM) software nor standard approaches take into account the tool path geometry and the control laws driving the tool displacements to propose optimal cutting parameter values, despite their significant impact. This paper contributes to plunge milling optimization through a Mixed-Integer NonLinear Programming (MINLP) approach, which enables us to determine optimal cutting parameter values that evolve along the tool path. It involves both continuous (cutting speed, feed per tooth) and, in contrast with standard approaches, integer (number of plunges) optimization variables, as well as nonlinear constraints. These constraints are related to the Computer Numerical Control (CNC) machine tool and to the cutting tool, taking into account the control laws. Computational results, validated on CNC machines and on representative test cases of engine housing, show that our methodology outperforms standard industrial engineering know-how approaches by up to 55% in terms of machining time.