arXiv subjects

Hansi Abeynanda

Publications and source records attributed to Hansi Abeynanda.

2 recordsLinked to original sources

arXiv2024

On the Characteristics of the Conjugate Function Enabling Effective Dual Decomposition Methods

We investigate a novel characteristic of the conjugate function associated to a generic convex optimization problem, which can subsequently be leveraged for efficient dual decomposition methods. In particular, under mild assumptions, we show that there is a specific region in the domain of the conjugate function such that for any point in the region, there is always a ray originating from that point along which the gradients of the conjugate remain constant. We refer to this characteristic as a fixed gradient over rays (FGOR). We further show that this characteristic is inherited by the corresponding dual function. Then we provide a thorough exposition of the application of the FGOR characteristic to dual subgradient methods. More importantly, we leverage FGOR to devise a simple stepsize rule that can be prepended with state-of-the-art stepsize methods enabling them to be more efficient. Furthermore, we investigate how the FGOR characteristic is used when solving the global consensus problem, a prevalent formulation in diverse application domains. We show that FGOR can be exploited not only to expedite the convergence of the dual decomposition methods but also to reduce the communication overhead. FGOR is extended to nonconvex formulations, and its advantages in stochastic optimization are demonstrated. Numerical experiments using quadratic objectives and a regularized least squares regression with real datasets are conducted. The results show that FGOR can significantly improve the performance of existing stepsize methods and outperform the state-of-the-art splitting methods on average in terms of both convergence behavior and communication efficiency.

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arXiv2022

On the Primal Feasibility in Dual Decomposition Methods Under Additive and Bounded Errors

With the unprecedented growth of signal processing and machine learning application domains, there has been a tremendous expansion of interest in distributed optimization methods to cope with the underlying large-scale problems. Nonetheless, inevitable system-specific challenges such as limited computational power, limited communication, latency requirements, measurement errors, and noises in wireless channels impose restrictions on the exactness of the underlying algorithms. Such restrictions have appealed to the exploration of algorithms' convergence behaviors under inexact settings. Despite the extensive research conducted in the area, it seems that the analysis of convergences of dual decomposition methods concerning primal optimality violations, together with dual optimality violations is less investigated. Here, we provide a systematic exposition of the convergence of feasible points in dual decomposition methods under inexact settings, for an important class of global consensus optimization problems. Convergences and the rate of convergences of the algorithms are mathematically substantiated, not only from a dual-domain standpoint but also from a primal-domain standpoint. Analytical results show that the algorithms converge to a neighborhood of optimality, the size of which depends on the level of underlying distortions. Further elaboration of a generalized problem formulation is also furnished, together with the convergence properties of the underlying algorithms. Finally, theoretical derivations are verified by numerical experiments.

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