arXiv subjects

Abhishek Pathapati

Publications and source records attributed to Abhishek Pathapati.

3 recordsLinked to original sources

arXiv2025

On exactness of SDP relaxation for the maximum cut problem

Semidefinite programming (SDP) provides a powerful relaxation for the maximum cut problem. In this work, we characterize a few classes of graphs for which the SDP relaxation is exact. For each of these graph classes, we establish conditions for uniqueness of the SDP optimum. We complement these findings by identifying two graph operations that preserve the solution rank, and in turn exactness. These results reveal how the SDP relaxation for the maximum cut problem can remain exact in arbitrarily large graphs, owing to the presence of a small structural core that governs exactness. We further address two open problems posed by Mirka and Williamson (2024), by demonstrating that uniqueness of the maximum cut partition in exact relaxation does not imply uniqueness of the SDP optimum, and that exact relaxation with multiple optimal partitions may admit optimal SDP solutions lying outside the convex hull of rank-1 reference solutions.

math.OC↗

arXiv2024

On generators of $k$-PSD closures of the positive semidefinite cone

Positive semidefinite (PSD) cone is the cone of positive semidefinite matrices, and is the object of interest in semidefinite programming (SDP). A computational efficient approximation of the PSD cone is the $k$-PSD closure, $1 \leq k < n$, cone of $n\times n$ real symmetric matrices such that all of their $k\times k$ principal submatrices are positive semidefinite. For $k=1$, one obtains a polyhedral approximation, while $k=2$ yields a second order conic (SOC) approximation of the PSD cone. These approximations of the PSD cone have been used extensively in real-world applications such as AC Optimal Power Flow (ACOPF) to address computational inefficiencies where SDP relaxations are utilized for convexification the non-convexities. However a theoretical discussion about the geometry of these conic approximations of the PSD cone is rather sparse. In this short communication, we attempt to provide a characterization of some family of generators of the aforementioned conic approximations.

math.OC↗

arXiv2022

Exact augmented Lagrangian duality for mixed integer convex optimization

Augmented Lagrangian dual augments the classical Lagrangian dual with a non-negative non-linear penalty function of the violation of the relaxed/dualized constraints in order to reduce the duality gap. We investigate the cases in which mixed integer convex optimization problems have an exact penalty representation using sharp augmenting functions (norms as augmenting penalty functions). We present a generalizable constructive proof technique for proving existence of exact penalty representations for mixed integer convex programs under specific conditions using the associated value functions. This generalizes the recent results for MILP (Feizollahi, Ahmed and Sun, 2017) and MIQP (Gu, Ahmed and Dey 2020) whilst also providing an alternative proof for the aforementioned along with quantification of the finite penalty parameter in these cases.

math.OC↗