Quadratic Minimum Research Articles

Lower bounds and exact algorithms for the quadratic minimum spanning tree problem

We address the quadratic minimum spanning tree problem (QMSTP), the problem of finding a spanning tree of a connected and undirected graph such that a quadratic cost function is minimized. We first propose an integer programming formulation based on the reformulation–linearization technique (RLT). We then use the idea of partitioning spanning trees into forests of a given fixed size and obtain a QMSTP reformulation that generalizes the RLT model. The reformulation is such that the larger the size of the forests, the stronger lower bounds provided. Thus, a hierarchy of formulations is obtained. At the lowest hierarchy level, one has precisely the RLT formulation, which is already stronger than previous formulations in the literature. The highest hierarchy level provides the convex hull of integer feasible solutions for the problem. The formulations introduced here are not compact, so the direct evaluation of their linear programming relaxation bounds is not practical. To overcome that, we introduce two lower bounding procedures based on Lagrangian relaxation. These procedures are embedded into two parallel branch-and-bound algorithms. As a result of our study, several instances in the literature were solved to optimality for the first time.

Branch‐and‐cut and Branch‐and‐cut‐and‐price algorithms for the adjacent only quadratic minimum spanning tree problem

The quadratic minimum spanning tree problem (QMSTP) consists of finding a spanning tree of a graph G such that a quadratic cost function is minimized. In its adjacent only version (AQMSTP), interaction costs only apply for edges that share an endpoint. Motivated by the weak lower bounds provided by formulations in the literature, we present a new linear integer programming formulation for AQMSTP. In addition to decision variables assigned to the edges, it also makes use of variables assigned to the stars of G. In doing so, the model is naturally linear (integer), without the need of implementing usual linearization steps, and its linear programming relaxation better estimates the interaction costs between edges. We also study a reformulation derived from the new model, obtained by projecting out the decision variables associated with the stars. Two exact solution approaches are presented: a branch‐and‐cut‐and‐price algorithm, based on the first formulation, and a branch‐and‐cut algorithm, based on its projection. Our computational results indicate that the lower bounds introduced here are much stronger than previous bounds in the literature. Being designed for the adjacent only case, our duality gaps are one order of magnitude smaller than the Gilmore–Lawler lower bounds for AQMSTP. As a result, the two exact algorithms introduced here outperform the previous exact solution approaches in the literature. In particular, the branch‐and‐cut method we propose managed to solve AQMSTP instances with as many as 50 vertices to proven optimality. © 2015 Wiley Periodicals, Inc. NETWORKS, Vol. 65(4), 367–379 2015

Optimal interior Earth–Moon Lagrange point transfer trajectories using mixed impulsive and continuous thrust

Optimal interior Earth–Moon Lagrange point transfer trajectories results with fixed flight time using mixed impulsive and continuous thrust propulsion in the framework of the planar Circular Restricted Three-Body Problem are presented. A virtual spacecraft departs geosynchronous or low-Earth orbit and enters a specified Lyapunov orbit around the interior Earth–Moon Lagrange point. For these transfer trajectory designs, a direct transcription and collocation method is employed to reformulate the continuous dynamic optimization problem into a discrete optimization problem, which then is solved using nonlinear programming software. As the design parameters, flight time and relative weighting factor between impulsive and continuous thrust are adjusted in the transfer trajectory design procedure. For practically implementable transfer trajectory designs, the constraints for coast arc and bounded control thrust are applied in the problem formulation as equality and inequality constraints, respectively. In addition, both quadratic performance index and minimum fuel performance index are used to show the relative comparison and analysis in the transfer trajectory design results. According to the relative weighting factor, the transfer trajectories are classified into direct and spiral departure trajectories whose virtual spacecraft directly departs the Earth parking orbit and departs after generating spirals around the Earth, respectively. The convergence of these transfer trajectory design methods are numerically verified by generating varying numbers of nodes and comparing their design results. Especially, the primer vector theory is utilized to analyze the transfer trajectory design results using minimum fuel performance index. Finally, the progressive homotopy continuation method is applied to design low-Earth departure transfer trajectories. These transfer trajectories show that they are numerically and theoretically valid results with the benefit of mixed impulsive and continuous thrust.

No quasistable scalaron lump forms afterR2inflation

In the Einstein frame picture of Starobinky's ${R}^{2}$ inflation model, cosmic inflation is driven by a slowly rolling inflaton field, called a scalaron, and followed by a coherently oscillating scalaron phase. Since the scalaron oscillates excessively many times in its potential, which has a quadratic minimum and is a little shallower than quadratic on the positive side, it may fragment into long-living localized objects, called oscillons or I balls, due to nonlinear growth of fluctuations before reheating of the Universe. We show that, while parametric self-resonances amplify scalaron fluctuations in the Minkowski background, the growth cannot overcome the decay due to expansion in the Friedmann background after ${R}^{2}$ inflation. By taking into account the backreaction from the metric of spacetime, modes that are larger than a critical scale are indeed amplified and become nondecaying. However, those nondecaying modes are not growing enough to form spatially localized lumps of the scalaron. Thus, reheating processes are unaltered by oscillons/I balls, and they proceed through perturbative decay of the scalaron as studied in the original work.

Combinatorial optimization with one quadratic term: Spanning trees and forests

The standard linearization of a binary quadratic program yields an equivalent reformulation as an integer linear program, but the resulting LP-bounds are very weak in general. We concentrate on applications where the underlying linear problem is tractable and exploit the fact that, in this case, the optimization problem is still tractable in the presence of a single quadratic term in the objective function. We propose to strengthen the standard linearization by the use of cutting planes that are derived from jointly considering each single quadratic term with the underlying combinatorial structure. We apply this idea to the quadratic minimum spanning tree and spanning forest problems and present complete polyhedral descriptions of the corresponding problems with one quadratic term, as well as efficient separation algorithms for the resulting polytopes. Computationally, we observe that the new inequalities significantly improve dual bounds with respect to the standard linearization, particularly for sparse graphs.

Tabu search with strategic oscillation for the quadratic minimum spanning tree

The quadratic minimum spanning tree problem consists of determining a spanning tree that minimizes the sum of costs of the edges and pairs of edges in the tree. Many algorithms and methods have been proposed for this hard combinatorial problem, including several highly sophisticated metaheuristics. This article presents a simple Tabu Search (TS) for this problem that incorporates Strategic Oscillation (SO) by alternating between constructive and destructive phases. The commonalties shared by this strategy and the more recently introduced methodology called iterated greedy search are shown and implications of their differences regarding the use of memory structures are identified. Extensive computational experiments reveal that the proposed SO algorithm with embedded TS is highly effective for solving complex instances of the problem as compared with the best metaheuristics in the literature. A hybrid method that proves similarly effective for problem instances that are both simple and complex is introduced.

Uncertain Quadratic Minimum Spanning Tree Problem

The quadratic minimum spanning tree problem is to find a spanning tree on a graph that minimizes a quadratic objective function of the edge weights. In this paper, the quadratic minimum spanning tree problem is concerned on the graph with edge weights being assumed as uncertain variables. The notion of the uncertain quadratic -minimum spanning tree is introduced by using the uncertain chance constraints. It is shown that the problem of finding an uncertain quadratic - minimum spanning tree can be handled in the framework of the deterministic quadratic minimum spanning tree problem requiring no particular solving methods.

Stronger Lower Bounds for the Quadratic Minimum Spanning Tree Problem with Adjacency Costs

Abstract We address a particular case of the quadratic minimum spanning tree problem in which interaction costs only apply for adjacent edges. Motivated by the fact that Gilmore-Lawler procedures in the literature underestimate the contribution of interaction costs to compute lower bounds, we introduce a reformulation that allows stronger linear programming bounds to be computed. An algorithm based on dynamic column and row generation is presented for evaluating these bounds. Our computational experiments indicate that the reformulation introduced here is indeed much stronger than those in the literature.

О необходимом условии минимума квадратичного функционала с интегралом Стилтьеса и нулевым коэффициентом при старшей производной на части интервала

О необходимом условии минимума квадратичного функционала с интегралом Стилтьеса и нулевым коэффициентом при старшей производной на части интервала

Solving the Quadratic Minimum Spanning Tree Problem

Given an undirected graph with costs associated to its edges and pairs of edges, the Quadratic Minimum Spanning Tree Problem (QMSTP) requires to determine a spanning tree of minimum total cost. This is a proper model for network problems in which both routing and interference costs need to be considered. It is NP-hard in the strong sense and not approximable unless P=NP. This paper describes a Tabu Search algorithm, with two independent and adaptively tuned tabu lists, and a Variable Neighbourhood Search algorithm. Both metaheuristics are based on the same neighbourhood, but the Tabu Search proves more effective and robust than the Variable Neighbourhood Search. To assess the quality of these results, we provide a comparison with the heuristic algorithms proposed in the recent literature and we reimplement, with minor improvements, an exact algorithm drawn from the literature, which confirms the optimality of the results obtained on small instances.

Oscillons after Inflation

Oscillons are massive, long-lived, localized excitations of a scalar field. We show that in a class of well-motivated single-field models, inflation is followed by self resonance, leading to copious oscillon generation and a lengthy period of oscillon domination. These models are characterized by an inflaton potential which has a quadratic minimum and is shallower than quadratic away from the minimum. This set includes both string monodromy models and a class of supergravity inspired scenarios and is in good agreement with the current central values of the concordance cosmology parameters. We assume that the inflaton is weakly coupled to other fields so as not to quickly drain energy from the oscillons or prevent them from forming. An oscillon-dominated universe has a greatly enhanced primordial power spectrum on very small scales relative to that seen with a quadratic potential, possibly leading to novel gravitational effects in the early Universe.

Application of an idea of Voronoĭ to lattice zeta functions

A major problem in the geometry of numbers is the investigation of the local minima of the Epstein zeta function. In this article refined minimum properties of the Epstein zeta function and more general lattice zeta functions are studied. Using an idea of Voronoĭ, characterizations and sufficient conditions are given for lattices at which the Epstein zeta function is stationary or quadratic minimum. Similar problems of a duality character are investigated for the product of the Epstein zeta function of a lattice and the Epstein zeta function of the polar lattice. Besides Voronoĭ type notions such as versions of perfection and eutaxy, these results involve spherical designs and automorphism groups of lattices. Several results are extended to more general lattice zeta functions, where the Euclidean norm is replaced by a smooth norm.

Routes to Complexity Induced by Constraints in Cournot Oligopoly Games with Linear Reaction Functions

Within a classical discrete-time Cournot oligopoly model with linear demand and quadratic cost functions, minimum and maximum production constraints are imposed in order to explore their effects on the dynamic of the system. Due to the presence of such constraints, the dynamic model assumes the form of a continuous piecewise linear map of the plane. The study of Nash equilibria of the oligopoly game, together with an analytical and numerical investigation of the different kinds of attractors of the dynamical system, shows how the presence of production constraints generates so called border collision bifurcations, a kind of global bifurcations recently introduced in the literature on non-smooth dynamical systems, which gives rise to a quite rich spectrum of dynamic scenarios, characterized by drastic changes in the qualitative dynamic properties of the system.

On the Best Quadratic Minimum Bias Non-Negative Estimator of a Two-Variance Component Model

On the Best Quadratic Minimum Bias Non-Negative Estimator of a Two-Variance Component ModelVariance components (VCs) in linear adjustment models are usually successfully computed by unbiased estimators. However, for many unbiased VC techniques estimated variance components might be negative, a result that cannot be tolerated by the user. This is, for example, the case with the simple additive VC model aσ2/1 + bσ2/2 with known coefficientsaandb, where either of the unbiasedly estimated variance components σ2/1 + σ2/2 may frequently come out negative. This fact calls for so-called non-negative VC estimators. Here the Best Quadratic Minimum Bias Non-negative Estimator (BQMBNE) of a two-variance component model is derived. A special case with independent observations is explicitly presented.

One-dimensional Cooper pairing

We study electron pairing in a one-dimensional (1D) fermion gas at zero temperature under zero- and finite-range, attractive, two-body interactions. The binding energy of Cooper pairs (CPs) with zero total or center-of-mass momentum (CMM) increases with attraction strength and decreases with interaction range for fixed strength. The excitation energy of 1D CPs with nonzero CMM display novel, unique properties. It satisfies a dispersion relation with two branches: a phonon-like linear excitation for small CP CMM; this is followed by roton-like quadratic excitation minimum for CMM greater than twice the Fermi wavenumber, but only above a minimum threshold attraction strength. The expected quadratic-in-CMM dispersion in vacuo when the Fermi wavenumber is set to zero is recovered for any coupling. This paper completes a three-part exploration initiated in 2D and continued in 3D.

METAHEURISTIC APPROACHES FOR THE QUADRATIC MINIMUM SPANNING TREE PROBLEM

Given an undirected graph with costs associated both with its edges and unordered pairs of edges, the quadratic minimum spanning tree problem asks to find a spanning tree that minimizes the sum of costs of all edges and pairs of edges in the tree. We present multistart simulated annealing, hybrid genetic and iterated tabu search algorithms for solving this problem. We report on computational experiments that compare these algorithms on random graphs of size up to 50 vertices. The results indicate that the iterated tabu search algorithm is superior to the other two approaches in terms of both solution quality and computation time.

A swarm intelligence approach to the quadratic minimum spanning tree problem

The quadratic minimum spanning tree problem (Q-MST) is an extension of the minimum spanning tree problem (MST). In Q-MST, in addition to edge costs, costs are also associated with ordered pairs of distinct edges and one has to find a spanning tree that minimizes the sumtotal of the costs of individual edges present in the spanning tree and the costs of the ordered pairs containing only edges present in the spanning tree. Though MST can be solved in polynomial time, Q-MST is NP -Hard. In this paper we present an artificial bee colony (ABC) algorithm to solve Q-MST. The ABC algorithm is a new swarm intelligence approach inspired by intelligent foraging behavior of honey bees. Computational results show the effectiveness of our approach.

Nonlinear Network Optimization—An Embedding Vector Space Approach

This paper proposes a normed-space vector representation of networks which allows defining evolutionary operators for network optimization that resemble continuous-space operators. These operators are employed here to build a genetic algorithm which becomes generic for the optimization of tree networks, without the requirement of any special encoding scheme. Such a genetic algorithm has been compared with several encoding-based genetic algorithms, on 25 and 50-node instances of the optimal communication spanning tree and of the quadratic minimum spanning tree, and has been shown to outperform all other algorithms in a stochastic dominance analysis. The proposed approach has also been applied to an electric power distribution network design (a multibranch problem), outperforming the results presented in a former reference (which have been obtained with an Ant Colony algorithm). The results of some landscape dispersion analysis suggest that the proposed normed-space network vector representation is analogous to some continuous-variable space dilation operations, which define favorable space coordinates for optimization.

The quadratic minimum spanning tree problem: A lower bounding procedure and an efficient search algorithm

In this paper we consider the quadratic minimum spanning tree problem (QMSTP) which is known to be NP-hard. Given a complete graph, the QMSTP consists of finding a minimum spanning tree (MST) where interaction costs between pairs of edges are prescribed. A Lagrangian relaxation procedure is devised and an efficient local search algorithm with tabu thresholding is developed. Computational experiments are reported on standard test instances, randomly generated test instances and quadratic assignment problem (QAP) instances from the QAPLIB by using a transformation scheme. The local search heuristic yields very good performance and the Lagrangian relaxation procedure gives the tightest lower bounds for all instances when compared to previous lower bounding approaches.

On the applicability of the equipartition theorem

Generalization of the equipartition theorem is presented for a broad range of potentials U(x) with quadratic minimum. It is shown, that the equipartition of energy in its standard form appears at the low temperatures limit. For potentials demonstrating the quadratic behavior for large displacements from the equilibrium the equipartition holds also in the high temperature limit. The temperature range of applicability of the equipartition theorem for the potential U = ax2 + bx4 was established.