arXiv subjects

Avinash Bhardwaj

Publications and source records attributed to Avinash Bhardwaj.

14 recordsLinked to original sources

arXiv2026

The Complexity of Recognizing SDP Exactness for the Maximum Cut Problem

The standard semidefinite programming (SDP) relaxation of Max-Cut is exact when its optimum equals the maximum cut value. Delorme and Poljak resolved NP-completeness of recognizing exactness for weighted graphs and left the unweighted case open. We show that recognition is NP-complete even for connected simple unweighted graphs, and hence strongly NP-complete for nonnegative integer edge weights. The reduction provides an explicit SDP optimum and makes the additive integrality gap equal to the minimum number of unsatisfied clauses in the source formula. Recognition remains NP-complete even when an exact rational optimal primal--dual pair is supplied. We also establish strong NP-hardness of recognizing exactness of the Frieze--Jerrum Max-$k$-Cut relaxation for every fixed $k\ge3$, even for connected graphs with nonnegative integer edge weights. An independent bounded-weight construction gives a second proof for Max-Cut. Finally, reductions preserving the additive gap up to explicit factors establish strong NP-completeness of exactness recognition for a basic Max-DiCut SDP and NP-hardness for a Max-Bisection SDP.

math.OC↗

arXiv2026

Max-$k$-Cut via Node Features

We study the Max-$k$-Cut problem from a node-feature perspective, where each vertex is associated with a feature vector and edge weights are given by pairwise inner products. We first examine the semidefinite relaxation of Max-$k$-Cut from this perspective. Using a normal-cone argument, we derive a general sufficient condition for exactness of the Frieze--Jerrum relaxation and show that it is satisfied in two feature-structural regimes: perfect feature balance, where the aggregate feature vectors of the parts are equal, and feature dominance, where a small set of large nonnegative feature vectors determines the structure of an optimal partition. We then show that the Max-$k$-Cut objective is equivalent to minimizing the sum of squared norms of the aggregate feature vectors assigned to the $k$ parts, thereby connecting the problem to vector balancing. Motivated by this observation, we show that a greedy feature-balancing algorithm retains the classical $1-1/k$ worst-case approximation guarantee and recovers an optimal partition under feature dominance. For rank-$1$ feature graphs with nonnegative features, classical bounds of Chandra and Wong for greedy load balancing yield a computable \emph{a posteriori} optimality-gap certificate that depends only on the returned partition and requires no knowledge of the optimum.

math.OC↗

arXiv2026

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↗

arXiv2025

Short note on phase retrievable weaving fusion frames

Fusion frames are extensively studied due to their effectiveness in recovering signals from large-scale data. They are applicable in distributed processing, wireless sensor networks, and packet encoding systems due to their robustness and redundancy. Motivated by the foundational work of Bemrose et al.\cite{Be16} and Balan\cite{Ba13}, this paper investigates the theoretical properties and characterizations of phase retrievable weaving fusion frames. These frames offer enhanced redundancy and stability in signal reconstruction. We present key results that deepen the understanding of their structure and behaviour. Lastly, an application involving probabilistic erasure is explored to demonstrate their practical utility.

math.FA↗

arXiv2025

Study of weaving frames in Krein spaces

Inspired by the work of Bemrose et al. \cite{Be16}, we delve into the study of weaving frames in Krein spaces. This paper presents a comprehensive exploration of various properties and characterizations of Krein space weaving frames. In support of our findings, several examples and counter examples are provided, illustrating the applicability of the theoretical results. Additionally, we extend the discussion to an important application in probabilistic erasure, highlighting how weaving frames can be used to mitigate data loss in such scenarios. This work contributes to the broader understanding of frame theory in indefinite inner product spaces.

math.FA↗

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↗

arXiv2024

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↗

arXiv2023

Online Universal Dirichlet Factor Portfolios

We revisit the online portfolio allocation problem and propose universal portfolios that use factor weighing to produce portfolios that out-perform uniform dirichlet allocation schemes. We show a few analytical results on the lower bounds of portfolio growth when the returns are known to follow a factor model. We also show analytically that factor weighted dirichlet sampled portfolios dominate the wealth generated by uniformly sampled dirichlet portfolios. We corroborate our analytical results with empirical studies on equity markets that are known to be driven by factors.

q-fin.PM↗

arXiv2023

A few more Lonely Runners

Lonely Runner Conjecture, proposed by Jörg M. Wills and so nomenclatured by Luis Goddyn, has been an object of interest since it was first conceived in 1967 : Given positive integers $k$ and $n_1,n_2,\ldots,n_k$ there exists a positive real number $t$ such that the distance of $t\cdot n_j$ to the nearest integer is at least $\frac{1}{k+1}$, $\forall~~1\leq j\leq k$. In a recent article Beck, Hosten and Schymura described the Lonely Runner polyhedron and provided a polyhedral approach to identifying families of lonely runner instances. We revisit the Lonely Runner polyhedron and highlight some new families of instances satisfying the conjecture. In addition, we relax the sufficiency of existence of an integer point in the Lonely Runner polyhedron to prove the conjecture. Specifically, we propose that it suffices to show the existence of a lattice point of certain superlattices of the integer lattice in the Lonely Runner polyhedron.

math.CO↗

arXiv2022

Almost Exact Risk Budgeting with Return Forecasts for Portfolio Allocation

In this paper, we revisit the portfolio allocation problem with designated risk-budget [Qian, 2005]. We generalize the problem of arbitrary risk budgets with unequal correlations to one that includes return forecasts and transaction costs while keeping the no-shorting (long-only positions) constraint. We offer a convex second order cone formulation that scales well with the number of assets and explore solutions to the problem in different settings. In particular, the problem is solved on a few practical cases - on equity and bond asset allocation problems as well as formulating index constituents for the NASDAQ100 index, illustrating the benefits of this approach.

cs.CE↗

arXiv2022

Unsupervised Early Exit in DNNs with Multiple Exits

Deep Neural Networks (DNNs) are generally designed as sequentially cascaded differentiable blocks/layers with a prediction module connected only to its last layer. DNNs can be attached with prediction modules at multiple points along the backbone where inference can stop at an intermediary stage without passing through all the modules. The last exit point may offer a better prediction error but also involves more computational resources and latency. An exit point that is `optimal' in terms of both prediction error and cost is desirable. The optimal exit point may depend on the latent distribution of the tasks and may change from one task type to another. During neural inference, the ground truth of instances may not be available and error rates at each exit point cannot be estimated. Hence one is faced with the problem of selecting the optimal exit in an unsupervised setting. Prior works tackled this problem in an offline supervised setting assuming that enough labeled data is available to estimate the error rate at each exit point and tune the parameters for better accuracy. However, pre-trained DNNs are often deployed in new domains for which a large amount of ground truth may not be available. We model the problem of exit selection as an unsupervised online learning problem and use bandit theory to identify the optimal exit point. Specifically, we focus on Elastic BERT, a pre-trained multi-exit DNN to demonstrate that it `nearly' satisfies the Strong Dominance (SD) property making it possible to learn the optimal exit in an online setup without knowing the ground truth labels. We develop upper confidence bound (UCB) based algorithm named UEE-UCB that provably achieves sub-linear regret under the SD property. Thus our method provides a means to adaptively learn domain-specific optimal exit points in multi-exit DNNs. We empirically validate our algorithm on IMDb and Yelp datasets.

cs.LG↗

arXiv2021

On the quality of the $k-$PSD closure approximation

Postive 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. In a recent series of articles Blekharman et al. provided bounds on the quality of these approximations. In this work, we revisit some of their results and also propose a new dominant bound on quality of the $k$-PSD closure approximation of the PSD cone. In addition, we characterize the extreme rays of the $2$-PSD closure.

math.OC↗

arXiv2017

Network Design with Probabilistic Capacities

We consider a network design problem with random arc capacities and give a formulation with a probabilistic capacity constraint on each cut of the network. To handle the exponentially-many probabilistic constraints a separation procedure that solves a nonlinear minimum cut problem is introduced. For the case with independent arc capacities, we exploit the supermodularity of the set function defining the constraints and generate cutting planes based on the supermodular covering knapsack polytope. For the general correlated case, we give a reformulation of the constraints that allows to uncover and utilize the submodularity of a related function. The computational results indicate that exploiting the underlying submodularity and supermodularity arising with the probabilistic constraints provides significant advantages over the classical approaches.

math.OC↗

arXiv2015

Deciding polyhedrality of spectrahedra

Spectrahedra are linear sections of the cone of positive semidefinite matrices that, as convex bodies, generalize the class of polyhedra. In this paper we investigate the problem of recognizing when a spectrahedron is polyhedral. We reprove a result of Ramana (1998) regarding the structure of spectrahedra and we devise a normal form of representations of spectrahedra. This normal form is effectively computable and leads to an algorithm for deciding polyhedrality.

math.OC↗