On the high temperature region of the Sherrington–Kirkpatrick model

We prove the validity of the replica-symmetric formulas for the Sherrington–Kirkpatrick (SK) model in a region of parameters that (probably) coincides with the region predicted by the physicists.     

The high temperature case for the random K-sat problem

We give a completely rigorous proof that the replica-symmetric solution holds at high enough temperature for the random K-sat problem. The most notable feature of this problem is that the order parameter of the system is a function and not a number. Received: 21 April 1998 / Revised version: 24 April 2000 / Published online: 21 December 2000     

On solving discrete two-stage stochastic programs having mixed-integer first- and second-stage variables

In this paper, we propose a decomposition-based branch-and-bound (DBAB) algorithm for solving two-stage stochastic programs having mixed-integer first- and second-stage variables. A modified Benders' decomposition method is developed, where the Benders' subproblems define lower bounding second-stage value functions of the first-stage variables that are derived by constructing a certain partial convex hull representation of the two-stage solution space. This partial convex hull is sequentially generated using a convexification scheme such as the Reformulation-Linearization Technique (RLT) or lift-and-project process, which yields valid inequalities that are reusable in the subsequent subproblems by updating the values of the first-stage variables. A branch-and-bound algorithm is designed based on a hyperrectangular partitioning process, using the established property that any resulting lower bounding Benders' master problem defined over a hyperrectangle yields the same objective value as the original stochastic program over that region if the first-stage variable solution is an extreme point of the defining hyperrectangle or the second-stage solution satisfies the binary restrictions. We prove that this algorithm converges to a global optimal solution. Some numerical examples and computational results are presented to demonstrate the efficacy of this approach.     

Bounds for diluted mean-fields spin glass models

In an important recent paper, [2], S. Franz and M. Leone prove rigorous lower bounds for the free energy of the diluted p-spin model and the K-sat model at any temperature. We show that the results for these two models are consequences of a single general principle. Our calculations are significantly simpler than those of [2], even in the replica-symmetric case.Supported by NSF grantRevised version: 30 December 2003     

A Simple Cluster Model for the Liquid–Glass Transition

Using the classical distribution-function approach to simple liquids, we estimate the orientational interaction between clusters consisting of a particle and its nearest neighbors. We show that there are density and temperature ranges where the interaction changes sign as a function of the cluster radius. On this basis, the corresponding model of interacting cubic and icosahedral clusters (of the type of a spin glass model) is proposed and solved in the replica-symmetric approximation. We show that the glass order parameter grows continuously on cooling and the replica-symmetry-breaking temperature can be identified with the glass transition temperature. We also show that on cooling a system of particles with a Lennard-Jones interaction, cubic clusters freeze first. The transition temperature for icosahedral clusters is somewhat lower; therefore, the cubic structure of the short-range order is more likely in a Lennard-Jones glass near transition.     

Geometric partitioning and robust ad-hoc network design

We consider bounds for the price of a European-style call option under regime switching. Stochastic semidefinite programming models are developed that incorporate a lattice generated by a finite-state Markov chain regime-switching model as a representation of scenarios (uncertainty) to compute bounds. The optimal first-stage bound value is equivalent to a Value at Risk quantity, and the optimal solution can be obtained via simple sorting. The upper (lower) bounds from the stochastic model are bounded below (above) by the corresponding deterministic bounds and are always less conservative than their robust optimization (min-max) counterparts. In addition, penalty parameters in the model allow controllability in the degree to which the regime switching dynamics are incorporated into the bounds. We demonstrate the value of the stochastic solution (bound) and computational experiments using the S&P 500 index are performed that illustrate the advantages of the stochastic programming approach over the deterministic strategy.     

Model Dirac-Maxwell equations with pseudounitary symmetry

We introduce a new system of equations called a model system of Dirac-Maxwell equations, reproducing the main properties of the standard system. At the same time, the model system of equations differs from the standard system in several ways; in particular, it is a tensor system and has a new symmetry with respect to the pseudounitary group. We also propose a version of the model system of Dirac-Maxwell equations with local (gauge) pseudounitary symmetry. We show that any spinor solution of the standard system of Dirac-Maxwell equations can be obtained from the corresponding tensor solution of the model system. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 157, No. 3, pp. 425–435, December, 2008.     

B&B Frameworks for the Capacity Expansion of High Speed Telecommunication Networks Under Uncertainty

The purpose of this paper is to investigate branch and bound strategies and the comparison of branch and cut with pure branch and bound approaches on high speed telecommunication network design under uncertainty. We model the problem as a two-stage stochastic program with discrete first-stage (investment) variables. Two formulations of the problem are used. The first one with general integer investment variables and the second one, a variant of the first model, with 0-1 investment variables. We present computational results for three solution approaches: the integer L-shaped (Benders) decomposition, a branch and bound framework and a disjunctive cutting plane method. This work was supported by France Telecom.     

Dynamic pricing of electricity in retail markets

This paper aims at defining a dynamic and flexible tariff structure for a distribution company that protects the retail consumers against the excessive fluctuations of the wholesales market prices. We propose a two-stage pricing scheme that sets in a first-stage a time-of-use tariff that is corrected later by a dynamic component once the real-time demand has been observed. A personalized tariff scheme may be offered by a distribution company to each dynamic customer by allowing him to choose the appropriate robustness level expressed in terms of variability between the first and the second-stage decisions. The arising limited recourse model has been tested on realistic test problems, by using a slight modification of a recently proposed interior point solution framework.      

Detecting stage-wise outliers in hierarchical Bayesian linear models of repeated measures data

We propose numerical and graphical methods for outlier detection in hierarchical Bayes modeling and analyses of repeated measures regression data from multiple subjects; data from a single subject are generically called a “curve”. The first-stage of our model has curve-specific regression coefficients with possibly autoregressive errors of a prespecified order. The first-stage regression vectors for different curves are linked in a second-stage modeling step, possibly involving additional regression variables. Detection of thestage at which the curve appears to be an outlier and themagnitude and specific component of the violation at that stage is accomplished by embedding the null model into a larger parametric model that can accommodate such unusual observations. We give two examples to illustrate the diagnostics, develop a BUGS program to compute them using MCMC techniques, and examine the sensitivity of the conclusions to the prior modeling assumptions.     

Continuity and Stability of a Quadratic Mixed-Integer Stochastic Program

For the two-stage quadratic stochastic program where the second-stage problem is a general mixed-integer quadratic program with a random linear term in the objective function and random right-hand sides in constraints, we study continuity properties of the second-stage optimal value as a function of both the first-stage policy and the random parameter vector. We also present sufficient conditions for lower or upper semicontinuity, continuity, and Lipschitz continuity of the second-stage problem's optimal value function and the upper semicontinuity of the optimal solution set mapping with respect to the first-stage variables and/or the random parameter vector. These results then enable us to establish conclusions on the stability of optimal value and optimal solutions when the underlying probability distribution is perturbed with respect to the weak convergence of probability measures.     

Symmetry,validation, and scattering matrices for time–domain scattering in waveguides

We outline a method to compute the solution in the frequency–domain for scattering in a waveguide by exploiting symmetry. The method is illustrated by considering a simple scattering example, where soft hard boundary conditions are alternated. We show how the straightforward mode matching or eigenfunction matching solution can be easily converted to scattering and transmission matrices when symmetry is exploited. We then show how the solution for two scatterers can be found explicitly, using symmetry which allows validation of our subsequent solution by scattering matrices. We also give a series of identities which the scattering matrix must satisfy for further numerical validation. Using these frequency–domain solutions we compute the time-domain scattering by incident Gaussian wave–packets.     

Symmetry Properties of Positive Solutions of Parabolic Equations on ℝ N : I. Asymptotic Symmetry for the Cauchy Problem

We consider quasilinear parabolic equations on ? ^N satisfying certain symmetry conditions. We prove that bounded positive solutions decaying to zero at spatial infinity are asymptotically radially symmetric about a center. The asymptotic center of symmetry is not fixed a priori (and depends on the solution) but it is independent of time. We also prove a similar theorem on reflectional symmetry.     

On two-stage stochastic knapsack problems

In this paper we study a particular version of the stochastic knapsack problem with normally distributed weights: the two-stage stochastic knapsack problem. Contrary to the single-stage knapsack problem, items can be added to or removed from the knapsack at the moment the actual weights become known (second stage). In addition, a chance-constraint is introduced in the first stage in order to restrict the percentage of cases where the items chosen lead to an overload in the second stage. To the best of our knowledge, there is no method known to exactly evaluate the objective function for a given first-stage solution. Therefore, we propose methods to calculate the upper and lower bounds. These bounds are used in a branch-and-bound framework in order to search the first-stage solution space. Special interest is given to the case where the items have similar weight means. Numerical results are presented and analyzed.     

Analytic construction of the near field in a transonic flow near a thin axisymmetric body

A solution of the axisymmetric problem of unsteady transonic flow around thin bodies of revolution is proposed in the form of a double series expansion in powers of the distance to the axis of symmetry and its logarithm in a neighborhood of a given point at the symmetry axis. Chains of recurrence equations are obtained for the coefficients of the series. The convergence of the constructed series is proved by the method of special majorants. The theorem of existence and uniqueness of the solution to the boundary-value problem for a nonlinear partial differential equation with a singularity at the symmetry axis is obtained in the asymptotic model of unsteady transonic flow under consideration. Thereby the application of the proposed series is justified to the problems of unsteady transonic flow around thin axisymmetric bodies with a drift of the nonpenetration condition onto the symmetry axis. Hence, these series can be used in numerical-analytical methods and model computations.     

Symmetry analysis of a model of stochastic volatility with time-dependent parameters

We provide the solutions for the Heston model of stochastic volatility when the parameters of the model are constant and when they are functions of time. In the former case, the solution follows immediately from the determination of the Lie point symmetries of the governing 1+1 evolution partial differential equation. This is not the situation in the latter case, but we are able to infer the essential structure of the required nonlocal symmetry from that of the autonomous problem and hence can present the solution to the nonautonomous problem. As in the case of the standard Black-Scholes problem the presence of time-dependent parameters is not a hindrance to the demonstration of a solution.     

一种求解混合多目标规划问题的功效函数法

逻辑生长函数相对于龚珀兹生长函数具有拐点高和对称性的特点,采用逻辑生长函数形式的功效函数法求解混合多目标规划问题具有形式简单、计算量小、符合实际的优点。证明了用这种方法求出的最优解是有效解。讨论了满意值对有效解的影响。逻辑生长函数还可以应用于多维变量评价的功效系数法中。     

A symmetry problem with applications to finance

In this paper we will consider an overdetermined problem in an unknown ring-shaped domain, and we prove that the domain has to be spherical ring. We will apply this problem to an \(n\) -dimensional toy model in the mean field game theory introduced by Lasry-Lions too. We show that if in Lasry-Lions model, the additional extra data ”the boundary is a level set” is assumed, then the region, where the solution is harmonic, has to have spherical symmetry.