arXiv subjects

Antonio Bellon

Publications and source records attributed to Antonio Bellon.

3 recordsLinked to original sources

arXiv2024

Time-Varying Semidefinite Programming: Path Following a Burer-Monteiro Factorization

We present an online algorithm for time-varying semidefinite programs (TV-SDPs), based on the tracking of the solution trajectory of a low-rank matrix factorization, also known as the Burer-Monteiro factorization, in a path-following procedure. There, a predictor-corrector algorithm solves a sequence of linearized systems. This requires the introduction of a horizontal space constraint to ensure the local injectivity of the low-rank factorization. The method produces a sequence of approximate solutions for the original TV-SDP problem, for which we show that they stay close to the optimal solution path if properly initialized. Numerical experiments for a time-varying max-cut SDP relaxation demonstrate the computational advantages of the proposed method for tracking TV-SDPs in terms of runtime compared to off-the-shelf interior point methods.

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arXiv2023

Parametric Semidefinite Programming: Geometry of the Trajectory of Solutions

In many applications, solutions of convex optimization problems are updated on-line, as functions of time. In this paper, we consider parametric semidefinite programs, which are linear optimization problems in the semidefinite cone whose coefficients (input data) depend on a time parameter. We are interested in the geometry of the solution (output data) trajectory, defined as the set of solutions depending on the parameter. We propose an exhaustive description of the geometry of the solution trajectory. As our main result, we show that only six distinct behaviors can be observed at a neighborhood of a given point along the solution trajectory. Each possible behavior is then illustrated by an example.

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arXiv2023

Optimal Power Flow Pursuit in the Alternating Current Model

Transmission-constrained problems in power systems can be cast as polynomial optimization problems whose coefficients vary over time. We consider the complications therein and suggest several approaches. On the example of the alternating-current optimal power flows (ACOPFs), we illustrate one of the approaches in detail. For the time-varying ACOPF, we provide an upper bound for the difference between the optimal cost for a relaxation using the most recent data and the current approximate optimal cost generated by our algorithm. This bound is a function of the properties of the instance and the rate of change of the coefficients over time. Moreover, we also bound the number of floating-point operations to perform between two subsequent updates to ensure a bounded error.

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