The automated real time control of an electrical network is achieved through the estimation of its state using phasor measurement units. Given an undirected graph representing the network, we study the problem of finding the minimum number of phasor measurement units to place on the edges such that the graph is fully observed. This problem is also known as the Power Edge Set problem, a variant of the Power Dominating Set problem. It is naturally modeled using an iteration-indexed binary linear program, whose size turns out to be too large for practical purposes. We use a fixed-point argument to remove the iteration indices and obtain a more compact bilevel formulation. We then reformulate the latter to a single-level mixed-integer linear program, which performs better than the natural formulation. Lastly, we provide an algorithm that solves the bilevel program directly and much faster than a commercial solver can solve the previous models. We also discuss robust variants and extensions of the problem.
Random projections are random linear maps, sampled from appropriate distributions, that approximately preserve certain geometrical invariants so that the approximation improves as the dimension of the space grows. The well-known Johnson-Lindenstrauss lemma states that there are random matrices with surprisingly few rows that approximately preserve pairwise Euclidean distances among a set of points. This is commonly used to speed up algorithms based on Euclidean distances. We prove that these matrices also preserve other quantities, such as the distance to a cone. We exploit this result to devise a probabilistic algorithm to solve linear programs approximately. We show that this algorithm can approximately solve very large randomly generated LP instances. We also showcase its application to an error correction coding problem.The paper [10] proposes a randomized dimensionality reduction based on PAC learning [3]: from a small training set, it is possible to forecast some properties of large data sets while keeping the error low. This is exploited in LPs with very few variables and huge numbers of inequality constraints: it is found that this number can be greatly reduced while keeping the optimality error bounded. In order to have PAC learning assumptions work, the authors focus on application cases which have a specific structure, i.e. there is an order on the constraints which makes their slope vary in a controlled way (an example is
We solve a linear robust problem with mixed-integer first-stage variables and continuous second stage variables. We consider column wise uncertainty. We first focus on a problem with right hand-side uncertainty which satisfies a "full recourse property" and a specific definition of the uncertainty. We propose a solution based on a generation constraint algorithm. Then we give several generalizations of the approach: for left-hand side uncertainty, for the cases where the "full recourse property" is not satisfied and for uncertainty sets defined by a polytope.
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