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Jakub Marecek

Publications and source records attributed to Jakub Marecek.

At least 109 records · Page 6Linked to original sources
arXiv2017

A Coordinate-Descent Algorithm for Tracking Solutions in Time-Varying Optimal Power Flows

Consider a polynomial optimisation problem, whose instances vary continuously over time. We propose to use a coordinate-descent algorithm for solving such time-varying optimisation problems. In particular, we focus on relaxations of transmission-constrained problems in power systems. On the example of the alternating-current optimal power flows (ACOPF), we bound the difference between the current approximate optimal cost generated by our algorithm and the optimal cost for a relaxation using the most recent data from above by a function of the properties of the instance and the rate of change to the instance over time. We also bound the number of floating-point operations that need to be performed between two updates in order to guarantee the error is bounded from above by a given constant.

math.OC↗
arXiv2017

Distributionally Robust Optimisation in Congestion Control

The effects of real-time provision of travel-time information on the behaviour of drivers are considered. The model of Marecek et al. [arXiv:1406.7639, Int. J. Control 88(10), 2015] is extended to consider uncertainty in the response of a driver to an interval provided per route. Specifically, it is suggested that one can optimise over all distributions of a random variable associated with the driver's response with the first two moments fixed, and for each route, over the sub-intervals within the minimum and maximum in a certain number of previous realisations of the travel time per the route.

math.OC↗
arXiv2017

On Classical Control and Smart Cities

We discuss the applicability of classical control theory to problems in smart grids and smart cities. We use tools from iterated function systems to identify controllers with desirable properties. In particular, controllers are identified that can be used to design not only stable closed-loop systems, but also to regulate large-scale populations of agents in a predictable manner. We also illustrate by means of an example and associated theory that many classical controllers are not be suitable for deployment in these applications.

math.OC↗
arXiv2016

Power Flow as an Algebraic System

Steady states of alternating-current (AC) circuits have been studied in considerable detail. In 1982, Baillieul and Byrnes derived an upper bound on the number of steady states in a loss-less AC circuit [IEEE TCAS, 29(11): 724--737] and conjectured that this bound holds for AC circuits in general. We prove this is indeed the case, among other results, by studying a certain multi-homogeneous structure in an algebraisation.

math.OC↗
arXiv2016

Hybrid Methods in Solving Alternating-Current Optimal Power Flows

Many steady-state problems in power systems, including rectangular power-voltage formulations of optimal power flows in the alternating-current model (ACOPF), can be cast as polynomial optimisation problems (POP). For a POP, one can derive strong convex relaxations, or rather hierarchies of ever stronger, but ever larger relaxations. We study means of switching from solving the convex relaxation to Newton method working on a non-convex Lagrangian of the POP.

math.OC↗
arXiv2016

A Low-Rank Coordinate-Descent Algorithm for Semidefinite Programming Relaxations of Optimal Power Flow

The alternating-current optimal power flow (ACOPF) is one of the best known non-convex non-linear optimisation problems. We present a novel re-formulation of ACOPF, which is based on lifting the rectangular power-voltage rank-constrained formulation, and makes it possible to derive alternative SDP relaxations. For those, we develop a first-order method based on the parallel coordinate descent with a novel closed-form step based on roots of cubic polynomials.

math.OC↗
arXiv2016

Signalling and obfuscation for congestion control

We aim to reduce the social cost of congestion in many smart city applications. In our model of congestion, agents interact over limited resources after receiving signals from a central agent that observes the state of congestion in real time. Under natural models of agent populations, we develop new signalling schemes and show that by introducing a non-trivial amount of uncertainty in the signals, we reduce the social cost of congestion, i.e., improve social welfare. The signalling schemes are efficient in terms of both communication and computation, and are consistent with past observations of the congestion. Moreover, the resulting population dynamics converge under reasonable assumptions.

math.OC↗
arXiv2016

Resource Allocation with Population Dynamics

Many analyses of resource-allocation problems employ simplistic models of the population. Using the example of a resource-allocation problem of Marecek et al. [arXiv:1406.7639], we introduce rather a general behavioural model, where the evolution of a heterogeneous population of agents is governed by a Markov chain. Still, we are able to show that the distribution of agents across resources converges in distribution, for suitable means of information provision, under certain assumptions. The model and proof techniques may have wider applicability.

math.OC↗
arXiv2016

Matrix Completion under Interval Uncertainty

Matrix completion under interval uncertainty can be cast as matrix completion with element-wise box constraints. We present an efficient alternating-direction parallel coordinate-descent method for the problem. We show that the method outperforms any other known method on a benchmark in image in-painting in terms of signal-to-noise ratio, and that it provides high-quality solutions for an instance of collaborative filtering with 100,198,805 recommendations within 5 minutes.

math.OC↗
arXiv2016

r-Extreme Signalling for Congestion Control

In many "smart city" applications, congestion arises in part due to the nature of signals received by individuals from a central authority. In the model of Marecek et al. [arXiv:1406.7639, Int. J. Control 88(10), 2015], each agent uses one out of multiple resources at each time instant. The per-use cost of a resource depends on the number of concurrent users. A central authority has up-to-date knowledge of the congestion across all resources and uses randomisation to provide a scalar or an interval for each resource at each time. In this paper, the interval to broadcast per resource is obtained by taking the minima and maxima of costs observed within a time window of length r, rather than by randomisation. We show that the resulting distribution of agents across resources also converges in distribution, under plausible assumptions about the evolution of the population over time.

math.OC↗
arXiv2016

MINLP in Transmission Expansion Planning

Transmission expansion planning requires forecasts of demand for electric power and a model of the underlying physics, i.e., power flows. We present three approaches to deriving exact solutions to the transmission expansion planning problem in the alternating-current model, for a given load.

math.OC↗
arXiv2016

Pricing Vehicle Sharing with Proximity Information

For vehicle sharing schemes, where drop-off positions are not fixed, we propose a pricing scheme, where the price depends in part on the distance between where a vehicle is being dropped off and where the closest shared vehicle is parked. Under certain restrictive assumptions, we show that this pricing leads to a socially optimal spread of the vehicles within a region.

math.OC↗
arXiv2015

Distributed Block Coordinate Descent for Minimizing Partially Separable Functions

In this work we propose a distributed randomized block coordinate descent method for minimizing a convex function with a huge number of variables/coordinates. We analyze its complexity under the assumption that the smooth part of the objective function is partially block separable, and show that the degree of separability directly influences the complexity. This extends the results in [Richtarik, Takac: Parallel coordinate descent methods for big data optimization] to a distributed environment. We first show that partially block separable functions admit an expected separable overapproximation (ESO) with respect to a distributed sampling, compute the ESO parameters, and then specialize complexity results from recent literature that hold under the generic ESO assumption. We describe several approaches to distribution and synchronization of the computation across a cluster of multi-core computers and provide promising computational results.

math.OC↗
arXiv2014

Exploiting Packing Components in General-Purpose Integer Programming Solvers

The problem of packing boxes into a large box is often a part of a larger problem. For example in furniture supply chain applications, one needs to decide what trucks to use to transport furniture between production sites and distribution centers and stores, such that the furniture fits inside. Such problems are often formulated and sometimes solved using general-purpose integer programming solvers. This chapter studies the problem of identifying a compact formulation of the multi-dimensional packing component in a general instance of integer linear programming, reformulating it using the discretisation of Allen--Burke--Marecek, and and solving the extended reformulation. Results on instances of up to 10000000 boxes are reported.

math.OC↗
arXiv2009

Some Probabilistic Results on Width Measures of Graphs

Fixed parameter tractable (FPT) algorithms run in time f(p(x)) poly(|x|), where f is an arbitrary function of some parameter p of the input x and poly is some polynomial function. Treewidth, branchwidth, cliquewidth, NLC-width, rankwidth, and booleanwidth are parameters often used in the design and analysis of such algorithms for problems on graphs. We show asymptotically almost surely (aas), there are Omega(n) lower bounds on the treewidth, branchwidth, cliquewidth, NLC-width, and rankwidth of graphs drawn from a simple random model. This raises important questions about the generality of FPT algorithms using the corresponding decompositions.

math.CO↗
arXiv2009

On a Clique-Based Integer Programming Formulation of Vertex Colouring with Applications in Course Timetabling

Vertex colouring is a well-known problem in combinatorial optimisation, whose alternative integer programming formulations have recently attracted considerable attention. This paper briefly surveys seven known formulations of vertex colouring and introduces a formulation of vertex colouring using a suitable clique partition of the graph. This formulation is applicable in timetabling applications, where such a clique partition of the conflict graph is given implicitly. In contrast with some alternatives, the presented formulation can also be easily extended to accommodate complex performance indicators (``soft constraints'') imposed in a number of real-life course timetabling applications. Its performance depends on the quality of the clique partition, but encouraging empirical results for the Udine Course Timetabling problem are reported.

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