The deployment of large language models (LLMs) at the network edge has emerged as a critical paradigm for enabling low-latency, privacy-preserving AI services. However, resource-constrained user devices (e.g., smartphones, IoT endpoints) struggle to execute computationally intensive LLM inference tasks solely. Collaborative computing frameworks that offload subtasks to edge servers offer a promising solution but face challenges in balancing task completion latency, energy consumption, and dynamic network conditions. This paper addresses these challenges by proposing a two-stage optimization framework that jointly manages task offloading decisions and computational resource allocation, aiming to minimize the total task completion latency. Simulation results demonstrate that the proposed algorithm reduces total latency by 49.6% compared to random, greedy, and genetic baselines, exhibits robust adaptability across varying energy budgets, and achieves 97.8% energy utilization in resource-constrained edge environments.

We study perspective reformulations (PRs) of semicontinuous quadratically constrained quadratic programs (SQCQPs) in this paper. Based on perspective functions, we first propose a class of PRs for SQCQPs and discuss how to find the best PR in this class via strong duality and lifting techniques. We then study the properties of the PR class and relate them to alternative formulations that are used to derive lower bounds for SQCQPs. Finally, we embed the PR bounds in branch-and-bound algorithms and conduct computational experiments to illustrate the effectiveness of the proposed approach.