Convergence of Tomlin's HOTS algorithm

The HOTS algorithm uses the hyperlink structure of the web to compute a vector of scores with which one can rank web pages. The HOTS vector is the vector of the exponentials of the dual variables of an optimal flow problem (the "temperature" of each page). The flow represents an optimal distribution of web surfers on the web graph in the sense of entropy maximization. In this paper, we prove the convergence of Tomlin's HOTS algorithm. We first study a simplified version of the algorithm, which is a fixed point scaling algorithm designed to solve the matrix balancing problem for nonnegative irreducible matrices. The proof of convergence is general (nonlinear Perron-Frobenius theory) and applies to a family of deformations of HOTS. Then, we address the effective HOTS algorithm, designed by Tomlin for the ranking of web pages. The model is a network entropy maximization problem generalizing matrix balancing. We show that, under mild assumptions, the HOTS algorithm converges with a linear convergence rate. The proof relies on a uniqueness property of the fixed point and on the existence of a Lyapunov function. We also show that the coordinate descent algorithm can be used to find the ideal and effective HOTS vectors and we compare HOTS and coordinate descent on fragments of the web graph. Our numerical experiments suggest that the convergence rate of the HOTS algorithm may deteriorate when the size of the input increases. We thus give a normalized version of HOTS with an experimentally better convergence rate.Comment: 21 page

Strongly polynomial algorithm for a class of minimum-cost flow problems with separable convex objectives

A well-studied nonlinear extension of the minimum-cost flow problem is to minimize the objective $\sum_{ij\in E} C_{ij}(f_{ij})$ over feasible flows $f$, where on every arc $ij$ of the network, $C_{ij}$ is a convex function. We give a strongly polynomial algorithm for the case when all $C_{ij}$'s are convex quadratic functions, settling an open problem raised e.g. by Hochbaum [1994]. We also give strongly polynomial algorithms for computing market equilibria in Fisher markets with linear utilities and with spending constraint utilities, that can be formulated in this framework (see Shmyrev [2009], Devanur et al. [2011]). For the latter class this resolves an open question raised by Vazirani [2010]. The running time is $O(m^4\log m)$ for quadratic costs, $O(n^4+n^2(m+n\log n)\log n)$ for Fisher's markets with linear utilities and $O(mn^3 +m^2(m+n\log n)\log m)$ for spending constraint utilities. All these algorithms are presented in a common framework that addresses the general problem setting. Whereas it is impossible to give a strongly polynomial algorithm for the general problem even in an approximate sense (see Hochbaum [1994]), we show that assuming the existence of certain black-box oracles, one can give an algorithm using a strongly polynomial number of arithmetic operations and oracle calls only. The particular algorithms can be derived by implementing these oracles in the respective settings

A constrained tropical optimization problem: complete solution and application example

The paper focuses on a multidimensional optimization problem, which is formulated in terms of tropical mathematics and consists in minimizing a nonlinear objective function subject to linear inequality constraints. To solve the problem, we follow an approach based on the introduction of an additional unknown variable to reduce the problem to solving linear inequalities, where the variable plays the role of a parameter. A necessary and sufficient condition for the inequalities to hold is used to evaluate the parameter, whereas the general solution of the inequalities is taken as a solution of the original problem. Under fairly general assumptions, a complete direct solution to the problem is obtained in a compact vector form. The result is applied to solve a problem in project scheduling when an optimal schedule is given by minimizing the flow time of activities in a project under various activity precedence constraints. As an illustration, a numerical example of optimal scheduling is also presented.Comment: 20 pages, accepted for publication in Contemporary Mathematic

Optimal Control of Transient Flow in Natural Gas Networks

We outline a new control system model for the distributed dynamics of compressible gas flow through large-scale pipeline networks with time-varying injections, withdrawals, and control actions of compressors and regulators. The gas dynamics PDE equations over the pipelines, together with boundary conditions at junctions, are reduced using lumped elements to a sparse nonlinear ODE system expressed in vector-matrix form using graph theoretic notation. This system, which we call the reduced network flow (RNF) model, is a consistent discretization of the PDE equations for gas flow. The RNF forms the dynamic constraints for optimal control problems for pipeline systems with known time-varying withdrawals and injections and gas pressure limits throughout the network. The objectives include economic transient compression (ETC) and minimum load shedding (MLS), which involve minimizing compression costs or, if that is infeasible, minimizing the unfulfilled deliveries, respectively. These continuous functional optimization problems are approximated using the Legendre-Gauss-Lobatto (LGL) pseudospectral collocation scheme to yield a family of nonlinear programs, whose solutions approach the optima with finer discretization. Simulation and optimization of time-varying scenarios on an example natural gas transmission network demonstrate the gains in security and efficiency over methods that assume steady-state behavior