Compact Formulation Research Articles

An exponential expanding meshes sequence and finite difference method adopted for two-dimensional elliptic equations

We demonstrate a new nonuniform mesh finite difference method to obtain accurate solutions for the elliptic partial differential equations in two dimensions with nonlinear first-order partial derivative terms. The method will be based on a geometric grid network area and included among the most stable differencing scheme in which the nine-point spatial finite differences are implemented, thus arriving at a compact formulation. In general, a third order of accuracy has been achieved and a fourth-order truncation error in the solution values will follow as a particular case. The efficiency of using geometric mesh ratio parameter has been shown with the help of illustrations. The convergence of the scheme has been established using the matrix analysis, and irreducibility is proved with the help of strongly connected characteristics of the iteration matrix. The difference scheme has been applied to test convection diffusion equation, steady state Burger’s equation, ocean model and a semi-linear elliptic equation. The computational results confirm the theoretical order and accuracy of the method.

Improved compact formulations for a wide class of graph partitioning problems in sparse graphs

Given an undirected connected graph G=(V,E) where |V|=n and |E|=m, we consider a wide class of graph partitioning problems, which includes as special cases several versions classically considered in the literature. These problems are to find a partition of the nodes in V into clusters satisfying several generic constraints (of set function type) on the clusters, in order to minimize the number (or the total weight) of the edges whose end-nodes do not belong to the same cluster. Partitions of V are often modeled by using compact integer programming formulations containing O(n3) triangle inequalities. The latter is the same whatever the sparsity of graph G could be, i.e. its size does not depend on m. In this paper, we show that one can reduce the size of the integer programming formulation to O(nm) triangle inequalities. Moreover, it is shown that, when the additional constraints on the clusters satisfy some monotonicity property, the strength of the linear programming relaxation is preserved by this reduction. We present numerical experiments on two important special cases arising from applications to show the benefit in terms of computational efficiency of using the reduced formulation.

Improved compact formulations for metric and cut polyhedra

Given a graph G=(V,E) with |V|=n and |E|=m, we consider the metric cone MET(G) and the metric polytope METP(G) defined on RE. These polyhedra are relaxations of several important problems in combinatorial optimization such as the max-cut problem and the multicommodity flow problem. They are known to have non-compact formulations via the cycle inequalities in the original space RE and compact (i.e. polynomial size) extended formulations via the triangle inequalities defined on the complete graph Kn. In this paper, we show that one can reduce the number of triangle inequalities to O(nm) and still have extended formulations for MET(G) and METP(G). This is particularly interesting for sparse graphs when m=O(n).

Shape interior modeling and mass property optimization using ray-reps

We present a novel method for the modeling and optimization of the material distribution inside 3D shapes, such that their 3D printed replicas satisfy prescribed constraints regarding mass properties. In particular, we introduce an extension of ray-representation to shape interior modeling, and prove this parametrization covers the optimal interior regarding static and rotational stability criteria. This compact formulation thoroughly reduces the number of design variables compared to the general volumetric element-wise formulation. We demonstrate the effectiveness of our reduced formulation for optimizing shapes that stably float in liquids or spin around a prescribed axis.

New formulations for the elementary shortest-path problem visiting a given set of nodes

Consider a directed graph G=(V,A) with a set of nodes V and a set of arcs A, and let cuv denote the length of an arc uv ∈ A. Given two nodes s and t of V, we are interested in the problem of determining a shortest-path from s to t in G that must visit only once all nodes of a given set P⊆V−{s,t}. This problem is NP-hard for P=V−{s,t}. In this paper, we develop three new compact formulations for this problem. The first one is based on the spanning tree polytope. The second model is a primal-dual mixed integer model presenting a small number of variables and constraints; and the last one is obtained from a flow-based compact model for the Steiner traveling salesman problem (TSP). Numerical experiments indicate that the second compact model allows the efficient solution of randomly generated and benchmark (from the TSPLIB) instances of the problem with hundreds of nodes.

A matheuristic for the asymmetric capacitated vehicle routing problem

In this paper, we propose a novel matheuristic for the Asymmetric Capacitated Vehicle Routing Problem (ACVRP). This optimization-based approach combines some heuristic concepts with compact mixed-integer linear programming (MILP) formulations. Basically, the proposed matheuristic includes three sequential stages. First, the problem size is heuristically reduced by discarding unpromising arcs. Second, a starting feasible solution is derived. Finally, an optimization-based improvement procedure is invoked to iteratively generate near-optimal solutions. This latter procedure requires solving a sequence of two- or three-vehicle ACVRP reduced instances. A peculiar feature of the solution strategy is that all the three stages are solely based on solving compact MILP formulations using a commercial solver and it does not resort to any constructive heuristic nor metaheuristic. We describe the results of extensive computational experiments, that were carried out on a large set of benchmark instances with up to 200 nodes, and we provide empirical evidence that the proposed matheuristic often delivers high-quality solutions.

Fitting quadrics with a Bayesian prior

Quadrics are a compact mathematical formulation for a range of primitive surfaces. A problem arises when there are not enough data points to compute the model but knowledge of the shape is available. This paper presents a method for fitting a quadric with a Bayesian prior. We use a matrix normal prior in order to favour ellipsoids when fitting to ambiguous data. The results show the algorithm copes well when there are few points in the point cloud, competing with contemporary techniques in the area.

Erratum—Exact Algorithms for the Clustered Vehicle Routing Problem

The clustered vehicle routing problem (CluVRP) is a generalization of the capacitated vehicle routing problem (CVRP) were customers are partitioned into disjoint clusters. As in the CVRP, all the customers must be visited exactly once, but a vehicle visiting one customer in a cluster must visit all the remaining customers therein before leaving it. Hence, when the first and last customer to be visited in a cluster C are p and q, finding the optimal sequence to visit the remaining customers in the cluster is equal to finding the minimum cost Hamiltonian path with p and q as endpoints. Battarra et al. (2014) presented a new compact integer programming formulation for the CluVRP. This formulation exploits the special substructure of the clusters such that only inter-cluster connections have to be determined. The authors show that this formulation provides a stronger linear relaxation than the CluVRP formulation based on the tradional CVRP two-index formulation. In the sequence of propositions that show the equivalence of the new formulation with the traditional CluVRP formulation, an erroneous notation was introduced in Battarra et al. (2014). In particular, this appears in the proof of Lemma 2, which is rewritten here in correct form. We emphasize that the incorrect notation in the proof does not affect the overall validity of Lemma 2, hence has no effect on the correctness of the paper and on the results reported therein. Let „4S5 denote the set of edges connecting the vertices in a vertex set S ⊆ V to those outside the set, and the set of edges inside a vertex set S by E4S5. The decision variables x ij and x ∗∗ ij are equal to the number of times a vehicle traverses edge 4i1 j5 for a solution of the traditional and the new formulation, respectively. The values of x ij are determined through the following transformation from x ij . For each inter-cluster edge 4i1 j5 ∈ Ē, set x ij = x ∗∗ ij . Initialize x ij = 0 for all intra-cluster edges 4i1 j5 ∈ E. For all 4p1 q5 ∈ E and for all 4i1 j5 ∈ P4p1q5, increment x ij by x ∗∗ pq . For the details of the CluVRP formulations, we refer the reader to Battarra et al. (2014). Lemma 2. Given a set of customer vertices S ⊂ V \809, with C being the minimal set of clusters covering S, i.e., ∀i ∈ S, ∃C ∈ C: i ∈C, the inequality ∑

Technique for designing antenna systems with complex-shaped reflectors

The study presents a method of defining a mathematical model for dipole antennas with reflector. We employ functional matrix operators to provide a compact formulation of the radiation field computation problem for such antenna systems. We use an asymptotic correspondence between three-dimensional and two-dimensional problem solutions to design an antenna system for distance measuring equipment. Consequently, we propose a numerical technique for investigating the mathematical model; we use a non-linear objective function for optimising the excitation function of the antenna system.

Higher-Order Logic Formalization of Conformal Geometric Algebra and its Application in Verifying a Robotic Manipulation Algorithm

Conformal geometric algebra (CGA) is an advanced geometric language used in solving three-dimensional Euclidean geometric problems due to its simple, compact and coordinate-free formulations. It promises to stimulate new methods and algorithms in all areas of science dealing with geometric properties, especially for engineering applications. This paper presents a higher-order logic formalization of CGA theories in the HOL-Light theorem prover. First, we formally define the classical algebraic operations and representations of geometric entities in the new framework. Second, we use these results to reason about the correctness of operation properties and geometric features such as the distance between the geometric entities and their rigid transformations in higher-order logic. Finally, in order to demonstrate the practical effectiveness and utilization of this formalization, we use it to formally model the grasping algorithm of a robot based on the conformal geometric control technique and verify the property that whether the robot can grasp firmly or not.

New compact and singularity free formulations for the magnetic field produced by a finite cylinder considering linearly varying current density

This paper presents new compact analytical expressions for the magnetic field calculation produced by a finite cylindrical sheet. A linearly varying surface current density between the ends of the cylindrical sheet has been assumed as current source. The expressions presented in this work are substantially more compact if compared with the ones currently available in the literature. Since the solutions are given in terms of complete elliptic integrals of the first, second and third kind, and the last two ones diverge at certain arguments, the field expressions also diverge at these critical points, even though the field solution is finite. These singular points are located at the ends of the current sheet cylinder. New analytical expressions are also presented for these critical points avoiding the singularities in an elegant way. The radial component of the magnetic field strength isseparately discussed, and it has been proven that itssolution at singular points diverges. To improve the computational performance on theevaluation ofmagnetic fieldatcriticalpoints,an alternative formulation ispresented, where complete elliptic integrals of the second and third kind are replaced with alternative functions. Numerical algorithms for the computation of these functions are also presented.

Tight and Compact MIP Formulation of Configuration-Based Combined-Cycle Units

Private investors, flexibility, efficiency and environmental requirements from deregulated markets have led the existence and building of a significant number of combined-cycle gas turbines (CCGTs) in many power systems. These plants represent a complex optimization problem for the short-term planning unit commitment (UC) carried out by independent system operators due to their multiple operating configurations. Accordingly, this paper proposes a mixed-integer linear programming (MIP) formulation of the configuration-based model of CCGTs, which is commonly utilized for bid/offering market processes. This formulation is simultaneously tighter and more compact than analogous MIP-based models; hence, it presents a lower computational burden. The computational efficiency of the proposed formulation is demonstrated by solving network-constrained UC case studies, of different size and complexity, using three of the leading commercial MIP solvers: CPLEX, GUROBI, and XPRESS.

Compact model for non-local avalanche effect in advanced bipolar transistors: An assessment of the relaxation length and its temperature dependence

We present a physics based compact model formulation for non-local avalanche effects. It is explicit and in terms of elementary functions, hence suitable for implementation in existing compact transistor models. The formulation has only two material coefficients as parameters: the energy relaxation length and its temperature coefficient. We present a detailed verification of our model against measured avalanche characteristics, as a function of both bias and temperature, for Si and SiGe industrial bipolar transistors. We demonstrate that the model is complete and accurate enough for the parameter extraction to be taken as an in situ measurement for both the electron energy relaxation length and its temperature coefficient: values obtained correspond to the values published earlier in the semiconductor literature.

Robust storage loading problems with stacking and payload constraints

We consider storage loading problems where items with uncertain weights have to be loaded into a storage area, taking into account stacking and payload constraints. Following the robust optimization paradigm, we propose strict and adjustable optimization models for finite and interval-based uncertainties. To solve these problems, exact decomposition and heuristic solution algorithms are developed. For strict robustness, we also propose a compact formulation based on a characterization of worst-case scenarios. Computational results for randomly generated data with up to 300 items are presented showing that the robustness concepts have different potential depending on the type of data being used.

Dual Inequalities for Stabilized Column Generation Revisited

Column generation (CG) models have several advantages over compact formulations: they provide better linear program bounds, may eliminate symmetry, and can hide nonlinearities in their subproblems. However, users also encounter drawbacks in the form of slow convergence, also known as the tailing-off effect, and the oscillation of the dual variables. Among different alternatives for stabilizing the CG process, Ben Amor et al. [Ben Amor H, Desrosiers J, Valério de Carvalho JM (2006) Dual-optimal inequalities for stabilized column generation. Oper. Res. 54(3):454–463] suggest the use of dual-optimal inequalities (DOIs) in the context of cutting stock and bin packing problems. We generalize their results, provide new classes of (deep) DOIs, and show the applicability to other problems (vector packing, vertex coloring, bin packing with conflicts). We also suggest the dynamic addition of violated dual inequalities in a cutting-plane fashion and the use of dual inequalities that are not necessarily (deep) DOIs. In the latter case, a recovery procedure is needed to restore primal feasibility. Computational results proving the usefulness of the methods are presented.

On a Piecewise-Linear Approximation for Network Revenue Management

The network revenue management (RM) problem arises in airline, hotel, media, and other industries where the sale products use multiple resources. It can be formulated as a stochastic dynamic program, but the dynamic program is computationally intractable because of an exponentially large state space, and a number of heuristics have been proposed to approximate its value function. In this paper we show that the piecewise-linear approximation to the network RM dynamic program is tractable; specifically we show that the separation problem of the approximation can be solved as a relatively compact linear program. Moreover, the resulting compact formulation of the approximate dynamic program turns out to be exactly equivalent to the Lagrangian relaxation of the dynamic program, an earlier heuristic method proposed for the same problem. We perform a numerical comparison of solving the problem by generating separating cuts or as our compact linear program. We discuss extensions to versions of the network RM problem with overbooking as well as the difficulties of extending it to the choice model of network revenue RM.

Numerical minimization of a second-order functional for image segmentation

In this paper we address the numerical minimization of a variational approximation of the Blake–Zisserman functional given by Ambrosio, Faina and March. Our approach exploits a compact matricial formulation of the objective functional and its decomposition into quadratic sparse convex sub-problems. This structure is well suited for using a block-coordinate descent method that cyclically determines a descent direction with respect to a block of variables by few iterations of a preconditioned conjugate gradient algorithm. We prove that the computed search directions are gradient related and, with convenient step-sizes, we obtain that any limit point of the generated sequence is a stationary point of the objective functional. An extensive experimentation on different datasets including real and synthetic images and digital surface models, enables us to conclude that: (1) the numerical method has satisfying performance in terms of accuracy and computational time; (2) a minimizer of the proposed discrete functional preserves the expected good geometrical properties of the Blake–Zisserman functional, i.e., it is able to detect first and second order edge-boundaries in images and (3) the method allows the segmentation of large images.

A matheuristic for exam timetabling

Exam timetabling is a tedious task faced in universities. We present a matheuristic to solve the exam timetabling problem related to the Toronto benchmark. We propose first a non-linear formulation used to find an initial feasible solution. We briefly describe a former linear formulation and introduce a new compact linear formulation. This compact formulation drastically reduces by 99% the numbers of variables and constraints. We propose a matheuristic based on both formulations to solve the problem. The approach achieves better results when compared to existing mathematical-programming-based heuristics.

The speed of reaction-diffusion fronts on fractals: testing the Campos-Méndez-Fort formula

Campos, Mendez, and Fort (CMF) derived an approximate formula for the speed of reaction-diffusion fronts in fractal media. By way of a continuation of their earlier studies, we perform numerical simulations of reaction- diffusion equations with au(1 u)(1 u) for 0 ¶ ¶ 1 as the reaction term on various generalized Sierpicarpets (including infinitely ramified and random ones). The CMF formula agrees well with the mean front speed as a function of a obtained from our simulations for the classic Sierpi´ nski carpet, a randomized version of the carpet, and some finitely ramified carpets containing loops. In these cases the mean front speed also shows no significant dependence on , as predicted by the CMF formula. However, the agreement is not so good in the case of the other carpets tested and this is probably a result of the mean distance of the front from the starting point against time behaving erratically in such cases. We also propose some nomenclature for generalized Sierpi´ nski carpets and introduce a compact formulation of

Fourth-Order Compact Formulation for the Resolution of Heat Transfer in Natural Convection of Water-Cu Nanofluid in a Square Cavity with a Sinusoidal Boundary Thermal Condition

In the present work, we numerically study the laminar natural convection of a nanofluid confined in a square cavity. The vertical walls are assumed to be insulated, non-conducting, and impermeable to mass transfer. The horizontal walls are differentially heated, and the low is maintained at hot condition (sinusoidal) when the high one is cold. The objective of this work is to develop a new height accurate method for solving heat transfer equations. The new method is a Fourth Order Compact (F.O.C). This work aims to show the interest of the method and understand the effect of the presence of nanofluids in closed square systems on the natural convection mechanism. The numerical simulations are performed for Prandtl number ( ), the Rayleigh numbers varying between and for different volume fractions varies between 0% and 10% for the nanofluid (water + Cu).