The functional integral in the Hubbard model

In a new functional integral approach proposed for the model, we find the regime with a deformed integration measure in which the standard integral is replaced with the Jackson integral. We indicate the relation to a p-adic functional integral. For the magnetic and electronic subsystems in the effective functional that results from the operator formulation of the Hubbard model, we find the two-parametric quantum derivative resulting in the appearance of the quantum SU_rq (2) group. We establish the relation to the one-parametric quantum derivative and to the standard derivative.     

A kinetic formulation for a model coupling free surface and pressurised flows in closed pipes

The aim of this paper is to present a kinetic formulation of a model for the coupling of transient free surface and pressurised flows. Firstly, we revisit the system of Saint-Venant equations for free surface flow: we state some properties of Saint-Venant equations, we propose a kinetic formulation and we verify that this kinetic formulation leads to a Gibbs equilibrium that minimises (in some general case) an energy and preserves the still water steady state. Secondly, we propose a model for pressurised flows in a Saint-Venant-like conservative formulation. We then propose a kinetic formulation and we verify that this kinetic formulation leads to a Gibbs equilibrium that minimises in any case an energy and preserves the still water steady state. Finally, we propose a dual model that couples these two types of flow.     

Nonextensive thermostatistical investigation of the blackbody radiation

Thermodynamical quantities of the blackbody radiation such as free energy, entropy, total radiation energy, specific heat are calculated within the Tsallis thermostatistics where factorization method is incorporated. It is shown that basic thermodynamical relation of the blackbody radiation is form invariant with respect to the nonextensivity entropic index q. Furthermore, the nonextensive thermodynamical quantities related to the blackbody radiation are separately obtained in terms of q and the standard thermodynamical quantities of the blackbody radiation. It is indicated that the formulation may give a way to determine q which determines the degree of the nonextensivity that is one of the aims of the present study.     

Multiobjective stochastic programming for feed formulation

The minimum cost linear programming model used traditionally for feed formulation does not take account of variability of nutrients in feed ingredients. Therefore, it may be that the nutrient requirements of the animal are not adequately met. In this paper, we show how a multiobjective stochastic model that permits confronting the cost of the ration with the probabilities of meeting the nutrient requirements of the animal can enhance the process of animal diet formulation. The model presented here does not require any a priori information from the decision maker, eliciting his preferences through an interactive process. This is the main advantage in relation to other models found in the literature for treating the problem of nutrient variability, which introduce stochastic constraints in the single objective minimum cost model requiring fixing the level of probability desired for each one of the nutrients in advance.     

On the asymmetric representatives formulation for the vertex coloring problem

We consider the vertex coloring problem, which can be stated as the problem of minimizing the number of labels that can be assigned to the vertices of a graph G such that each vertex receives at least one label and the endpoints of every edge are assigned different labels. In this work, the 0-1 integer programming formulation based on representative vertices is revisited to remove symmetry. The previous polyhedral study related to the original formulation is adapted and generalized. New versions of facets derived from substructures of G are presented, including cliques, odd holes and anti-holes and wheels. In addition, a new class of facets is derived from independent sets of G. Finally, a comparison with the independent sets formulation is provided.     

Two-stage stochastic lot-sizing problem under cost uncertainty

For production planning problems, cost parameters can be uncertain due to marketing activities and interest rate fluctuation. In this paper, we consider a single-item two-stage stochastic lot-sizing problem under cost parameter uncertainty. Assuming cost parameters will increase or decrease after time period p each with certain probability, we minimize the total expected cost for a finite horizon problem. We develop an extended linear programming formulation in a higher dimensional space that can provide integral solutions by showing that its constraint matrix is totally unimodular. We also project this extended formulation to a lower dimensional space and obtain a corresponding extended formulation in the lower dimensional space. Final computational experiments demonstrate that the extended formulation is more efficient and performs more stable than the two-stage stochastic mixed-integer programming formulation.     

Lower bounds for nonlinear assignment problems using many body interactions

This paper concerns lower bounding techniques for the general α-adic assignment problem. The nonlinear objective function is linearized by the introduction of additional variables and constraints, thus yielding a mixed integer linear programming formulation of the problem. The concept of many body interactions is introduced to strengthen this formulation and incorporated in a modified formulation obtained by lifting the original representation to a higher dimensional space. This process involves two steps — (i) addition of new variables and constraints and (ii) incorporation of the new variables in the objective function. If this lifting process is repeated β times on an α-adic assignment problem along with the incorporation of higher order interactions, it results in the mixed-integer formulation of an equivalent (α + β)-adic assignment problem. The incorporation of many body interactions in the higher dimensional formulation improves its degeneracy properties and is also critical to the derivation of decomposition methods for the solution of these large scale mathematical programs in the higher dimensional space. It is shown that a lower bound to the optimal solution of the corresponding linear programming relaxation can be obtained by dualizing a subset of constraints in this formulation and solving O(N^2(α+β−1)) linear assignment problems, whose coefficients depend on the dual values. Moreover, it is proved that the optimal solution to the LP relaxation is obtained if we use the optimal duals for the solution of the linear assignment problems. This concept of many body interactions could be applied in designing algorithms for the solution of formulations obtained by lifting general MILP's. We illustrate all these concepts on the quadratic assignment problems With these decomposition bounds, we have found the provably optimal solutions of two unsolved QAP's of size 32 and have also improved upon existing lower bounds for other QAP's.     

A Shared-Constraint Approach to Multi-Leader Multi-Follower Games

Multi-leader multi-follower games are a class of hierarchical games in which a collection of leaders compete in a Nash game constrained by the equilibrium conditions of another Nash game amongst the followers. The resulting equilibrium problem with equilibrium constraints is complicated by nonconvex agent problems and therefore providing tractable conditions for existence of global or even local equilibria has proved challenging. Consequently, much of the extant research on this topic is either model specific or relies on weaker notions of equilibria. We consider a modified formulation in which every leader is cognizant of the equilibrium constraints of all leaders. Equilibria of this modified game contain the equilibria, if any, of the original game. The new formulation has a constraint structure called shared constraints, and our main result shows that if the leader objectives admit a potential function, the global minimizers of the potential function over this shared constraint are equilibria of the modified formulation. We provide another existence result using fixed point theory that does not require potentiality. Additionally, local minima, B-stationary, and strong-stationary points of this minimization problem are shown to be local Nash equilibria, Nash B-stationary, and Nash strong-stationary points of the corresponding multi-leader multi-follower game. We demonstrate the relationship between variational equilibria associated with this modified shared-constraint game and equilibria of the original game from the standpoint of the multiplier sets and show how equilibria of the original formulation may be recovered. We note through several examples that such potential multi-leader multi-follower games capture a breadth of application problems of interest and demonstrate our findings on a multi-leader multi-follower Cournot game.     

A time finite element method for structural dynamics

A time (Galerkin) finite element method (time FEM) for structural dynamics is proposed in this paper. The key lies in a variational formulation that is well-posed and equivalent to the conventional strong form of governing equations of structural dynamics. Based on the variational formulation, a time finite element formulation is naturally established and its convergence property is easily derived through an a priori error analysis. Technical details on practical implementation of the time FEM are presented. Numerical examples are studied to verify the proposed time FEM.     

Exploring the trade-off between generalization and empirical errors in a one-norm SVM

We propose a one-norm support vector machine (SVM) formulation as an alternative to the well-known formulation that uses parameter C in order to balance the two inherent objective functions of the problem. Our formulation is motivated by the ?-constraint approach that is used in bicriteria optimization and we propose expressing the objective of minimizing total empirical error as a constraint with a parametric right-hand-side. Using dual variables we show equivalence of this formulation to the one with the trade-off parameter. We propose an algorithm that enumerates the entire efficient frontier by systematically changing the right-hand-side parameter. We discuss the results of a detailed computational analysis that portrays the structure of the efficient frontier as well as the computational burden associated with finding it. Our results indicate that the computational effort for obtaining the efficient frontier grows linearly in problem size, and the benefit in terms of classifier performance is almost always substantial when compared to a single run of the corresponding SVM. In addition, both the run time and accuracy compare favorably to other methods that search part or all of the regularization path of SVM.     

A priori error estimation for the dual mixed finite element method of the elastodynamic problem in a polygonal domain, I

In this paper we analyze a new dual mixed formulation of the elastodynamic system in polygonal domains. In this formulation the symmetry of the strain tensor is relaxed by the rotation of the displacement. For the time discretization of this new dual mixed formulation, we use an explicit scheme. After the analysis of stability of the fully discrete scheme, L^∞ in time, L² in space a priori error estimates are derived for the approximation of the displacement, the strain, the pressure and the rotation. Numerical experiments confirm our theoretical predictions.     

Degeneration of Riemann surfaces and intermediate polarization of the moduli space of flat connections

We construct a family of polarizations of the moduli space of flat SU(n)-connections on a closed 2-manifold of genus g(≧2). These are generalizations of various polarizations known until now. That is, our family of polarizations includes Weitsman’s real polarizations in the case of n=2 [17], as well as the Kahler polarizations which are well known since [2] and [18]. Our construction is based on an original formulation of degeneration of Riemann surfaces. The relation between our polarizations and the complex structures of the moduli Spaces of parabolic bundles are also studied.     

The Steiner connectivity problem

The Steiner connectivity problem has the same significance for line planning in public transport as the Steiner tree problem for telecommunication network design. It consists in finding a minimum cost set of elementary paths to connect a subset of nodes in an undirected graph and is, therefore, a generalization of the Steiner tree problem. We propose an extended directed cut formulation for the problem which is, in comparison to the canonical undirected cut formulation, provably strong, implying, e.g., a class of facet defining Steiner partition inequalities. Since a direct application of this formulation is computationally intractable for large instances, we develop a partial projection method to produce a strong relaxation in the space of canonical variables that approximates the extended formulation. We also investigate the separation of Steiner partition inequalities and give computational evidence that these inequalities essentially close the gap between undirected and extended directed cut formulation. Using these techniques, large Steiner connectivity problems with up to 900 nodes can be solved within reasonable optimality gaps of typically less than five percent.     

An arc cover–path-cover formulation and strategic analysis of alternative-fuel station locations

In this study, we present a new formulation of the generalized flow-refueling location model that takes vehicle range and trips between origin–destination pairs into account. The new formulation, based on covering the arcs that comprise each path, is more computationally efficient than previous formulations or heuristics. Next, we use the new formulation to provide managerial insights for some key concerns of the industry, such as: whether infrastructure deployment should focus on locating clusters of facilities serving independent regions or connecting these regions by network of facilities; what is the impact of uncertainty in the origin–destination demand forecast; whether station locations will remain optimal as higher-range vehicles are introduced; and whether infrastructure developers should be willing to pay more for stations at higher-cost intersections. Experiments with real and random data sets are encouraging for the industry, as optimal locations tend to be robust under various conditions.     

Robust multiobjective optimization & applications in portfolio optimization

Motivated by Markowitz portfolio optimization problems under uncertainty in the problem data, we consider general convex parametric multiobjective optimization problems under data uncertainty. For the first time, this uncertainty is treated by a robust multiobjective formulation in the gist of Ben-Tal and Nemirovski. For this novel formulation, we investigate its relationship to the original multiobjective formulation as well as to its scalarizations. Further, we provide a characterization of the location of the robust Pareto frontier with respect to the corresponding original Pareto frontier and show that standard techniques from multiobjective optimization can be employed to characterize this robust efficient frontier. We illustrate our results based on a standard mean–variance problem.     

Multi‐Trace Boundary Integral Formulation for Acoustic Scattering by Composite Structures

We study the scattering of an acoustic wave by an object composed of several adjacent parts with different material properties. For this problem we derive a new integral equation formulation of the first kind. This formulation involves two Dirichlet data and two Neumann data at each point of each material interface of the diffracting object. It is immune to spurious resonances, and it enjoys a stability property that ensures quasi‐optimal convergence of conforming Galerkin boundary element discretization. In addition, the operator of this formulation satisfies a relation similar to the standard Calderón identity. © 2012 Wiley Periodicals, Inc.     

Stable hp mixed finite elements based on the Hellinger–Reissner principle

In the Hellinger–Reissner formulation for linear elasticity, both the displacement u and the stress σ are taken as unknowns, giving rise to a saddle point problem. We present new pairings of quadrilateral ‘trunk’ finite element spaces for this method and prove stability (and optimality) in terms of both h and p. The effect of mesh shape regularity on the stability constant is explicitly tracked. Our results provide a theoretical basis for recent numerical experiments (in the context of a mixed p formulation for viscoelasticity) that showed these spaces worked well computationally.