No Arabic abstract
We consider a linear relaxation of a generalized minimum-cost network flow problem with binary input dependencies. In this model the flows through certain arcs are bounded by linear (or more generally, piecewise linear concave) functions of the flows through other arcs. This formulation can be used to model interrelated systems in which the components of one system require the delivery of material from another system in order to function (for example, components of a subway system may require delivery of electrical power from a separate system). We propose and study randomized rounding schemes for how this model can be used to approximate solutions to a related mixed integer linear program for modeling binary input dependencies. The introduction of side constraints prevents this problem from being solved using the well-known network simplex algorithm, however by characterizing its basis structure we develop a generalization of network simplex algorithm that can be used for its efficient solution.
For each integer $n$ we present an explicit formulation of a compact linear program, with $O(n^3)$ variables and constraints, which determines the satisfiability of any 2SAT formula with $n$ boolean variables by a single linear optimization. This contrasts with the fact that the natural polytope for this problem, formed from the convex hull of all satisfiable formulas and their satisfying assignments, has superpolynomial extension complexity. Our formulation is based on multicommodity flows. We also discuss connections of these results to the stable matching problem.
The increasing penetration of renewables in distribution networks calls for faster and more advanced voltage regulation strategies. A promising approach is to formulate the problem as an optimization problem, where the optimal reactive power injection from inverters are calculated to maintain the voltages while satisfying power network constraints. However, existing optimization algorithms require the exact topology and line parameters of underlying distribution system, which are not known for most cases and are difficult to infer. In this paper, we propose to use specifically designed neural network to tackle the learning and optimization problem together. In the training stage, the proposed input convex neural network learns the mapping between the power injections and the voltages. In the voltage regulation stage, such trained network can find the optimal reactive power injections by design. We also provide a practical distributed algorithm by using the trained neural network. Theoretical bounds on the representation performance and learning efficiency of proposed model are also discussed. Numerical simulations on multiple test systems are conducted to illustrate the operation of the algorithm.
Cascading failures in complex systems have been studied extensively using two different models: $k$-core percolation and interdependent networks. We combine the two models into a general model, solve it analytically and validate our theoretical results through extensive simulations. We also study the complete phase diagram of the percolation transition as we tune the average local $k$-core threshold and the coupling between networks. We find that the phase diagram of the combined processes is very rich and includes novel features that do not appear in the models studying each of the processes separately. For example, the phase diagram consists of first and second-order transition regions separated by two tricritical lines that merge together and enclose a novel two-stage transition region. In the two-stage transition, the size of the giant component undergoes a first-order jump at a certain occupation probability followed by a continuous second-order transition at a lower occupation probability. Furthermore, at certain fixed interdependencies, the percolation transition changes from first-order $rightarrow$ second-order $rightarrow$ two-stage $rightarrow$ first-order as the $k$-core threshold is increased. The analytic equations describing the phase boundaries of the two-stage transition region are set up and the critical exponents for each type of transition are derived analytically.
In this paper, we consider the systems with trajectories originating in the nonnegative orthant becoming nonnegative after some finite time transient. First we consider dynamical systems (i.e., fully observable systems with no inputs), which we call eventually positive. We compute forward-invariant cones and Lyapunov functions for these systems. We then extend the notion of eventually positive systems to the input-output system case. Our extension is performed in such a manner, that some valuable properties of classical internally positive input-output systems are preserved. For example, their induced norms can be computed using linear programming and the energy functions have nonnegative derivatives.
We present a flexible public transit network design model which optimizes a social access objective while guaranteeing that the systems costs and transit times remain within a preset margin of their current levels. The purpose of the model is to find a set of minor, immediate modifications to an existing bus network that can give more communities access to the chosen services while having a minimal impact on the current networks operator costs and user costs. Design decisions consist of reallocation of existing resources in order to adjust line frequencies and capacities. We present a hybrid tabu search/simulated annealing algorithm for the solution of this optimization-based model. As a case study we apply the model to the problem of improving equity of access to primary health care facilities in the Chicago metropolitan area. The results of the model suggest that it is possible to achieve better primary care access equity through reassignment of existing buses and implementation of express runs, while leaving overall service levels relatively unaffected.