Improving quality of service for cell-edge user in D2D-relay networks

In this paper, we investigate a scenario of D2D-relay communications where one D2D user may help cell-edge user to exchange information for improving its quality of service (QoS). We formulate a resource allocation problem, which aims at maximizing the data rate of the cell-edge user. In particular, we propose an iterative power allocation algorithm and derive the optimal closed-form power allocation expressions by Lagrangian dual method. Simulation results verify the theoretical solution and show that our D2D-relay scheme achieves higher spectrum efficiency than the traditional cellular-relay communication scheme.

Scale elasticity in the presence of integer data: An application to electricity distribution companies

Returns to scale and scale elasticity are two important issues in the field of economics and operations research. Recently, estimating returns to scale and scale elasticity using tools such as data envelopment analysis (DEA) has attracted considerable attention among researchers. The existing approaches to calculate scale elasticity in DEA context, assume all inputs and outputs are real-valued and in this sense, the underlying technology is a continuous set. In many real cases, however, we face input/output measures that are restricted to be integer-valued. Scale properties of frontier points in such cases are interesting and important. In this paper, this problem in integer-valued DEA is studied. The Lagrangian dual formulation of a mixed integer linear programming problem is used to calculate the scale elasticity of a frontier point. To illustrate the real applicability of the theoretical framework, a real case on electricity distribution companies is given.

Parallelizing Subgradient Methods for the Lagrangian Dual in Stochastic Mixed-Integer Programming

The dual decomposition of stochastic mixed-integer programs can be solved by the projected subgradient algorithm. We show how to make this algorithm more amenable to parallelization in a master-worker model by describing two approaches, which can be combined in a natural way. The first approach partitions the scenarios into batches and makes separate use of subgradient information for each batch. The second approach drops the requirement that evaluation of function and subgradient information is synchronized across the scenarios. We provide convergence analysis of both methods. We also evaluate their performance on two families of problems from SIPLIB on a single server with 32 single-core worker processes, demonstrating that when the number of workers is high relative to the number of scenarios, these two approaches (and their synthesis) can significantly reduce running time.

Lagrangian dual framework for conservative neural network solutions of kinetic equations

<p style='text-indent:20px;'>In this paper, we propose a novel conservative formulation for solving kinetic equations via neural networks. More precisely, we formulate the learning problem as a constrained optimization problem with constraints that represent the physical conservation laws. The constraints are relaxed toward the residual loss function by the Lagrangian duality. By imposing physical conservation properties of the solution as constraints of the learning problem, we demonstrate far more accurate approximations of the solutions in terms of errors and the conservation laws, for the kinetic Fokker-Planck equation and the homogeneous Boltzmann equation.</p>

On approximate solutions for convex semi-infinite programming with uncertainty

In this paper, we consider approximate solutions (also called $\varepsilon$-solutions) for semi-infinite optimization problems that objective function and constraint functions with uncertainty data are all convex, and establish robust counterpart of convex semi-infinite program and then consider approximate solutions for its. Moreover, the robust necessary condition and robust sufficient theorems are obtained. Then the duality results of the Lagrangian dual approximate solution is given by the robust optimization approach under a cone constraint qualification.

SDO and LDO relaxation approaches to complex fractional quadratic optimization

This paper examines a complex fractional quadratic optimization problem subject to two quadratic constraints. The original problem is transformed into a parametric quadratic programming problem by the well-known classical Dinkelbach method. Then a semidefinite and Lagrangian dual optimization approaches are presented to solve the nonconvex parametric problem at each iteration of the bisection and generalized Newton algorithms. Finally, the numerical results demonstrate the effectiveness of the proposed approaches.

Scale elasticity in the presence of integer data: An application to electricity distribution companies

Returns to scale and scale elasticity are two important issues in the field of economics and operations research. Recently, estimating returns to scale and scale elasticity using tools such as data envelopment analysis (DEA) has attracted considerable attention among researchers. The existing approaches to calculate scale elasticity in DEA context, assume all inputs and outputs are real-valued and in this sense, the underlying technology is a continuous set. In many real cases, however, we face input/output measures that are restricted to be integer-valued. Scale properties of frontier points in such cases are interesting and important. In this paper, this problem in integer-valued DEA is studied. The Lagrangian dual formulation of a mixed integer linear programming problem is used to calculate the scale elasticity of a frontier point. To illustrate the real applicability of the theoretical framework, a real case on electricity distribution companies is given.

An Ellipsoidal Bounding Scheme for the Quasi-Clique Number of a Graph

A γ-quasi-clique in a simple undirected graph refers to a subset of vertices that induces a subgraph with edge density at least γ. When γ equals one, this definition corresponds to a classical clique. When γ is less than one, it relaxes the requirement of all possible edges by the clique definition. Quasi-clique detection has been used in graph-based data mining to find dense clusters, especially in large-scale error-prone data sets in which the clique model can be overly restrictive. The maximum γ-quasi-clique problem, seeking a γ-quasi-clique of maximum cardinality in the given graph, can be formulated as an optimization problem with a linear objective function and a single quadratic constraint in binary variables. This article investigates the Lagrangian dual of this formulation and develops an upper-bounding technique using the geometry of ellipsoids to bound the Lagrangian dual. The tightness of the upper bound is compared with those obtained from multiple mixed-integer programming formulations of the problem via experiments on benchmark instances.

The Benders Dual Decomposition Method

Many methods that have been proposed to solve large-scale MILP problems rely on the use of decomposition strategies. These methods exploit either the primal or dual structures of the problems by applying the Benders decomposition or Lagrangian dual decomposition strategy, respectively. In “The Benders Dual Decomposition Method,” Rahmaniani, Ahmed, Crainic, Gendreau, and Rei propose a new and high-performance approach that combines the complementary advantages of both strategies. The authors show that this method (i) generates stronger feasibility and optimality cuts compared with the classical Benders method, (ii) can converge to the optimal integer solution at the root node of the Benders master problem, and (iii) is capable of generating high-quality incumbent solutions at the early iterations of the algorithm. The developed algorithm obtains encouraging computational results when used to solve various benchmark MILP problems.