Traditional network management algorithms have relied on prior knowledge of system models and networking scenarios. In practice, a universal optimization framework is desirable where a sole optimization module can be readily applied to arbitrary network management tasks without any knowledge of the system. To this end, knowledge-free optimization techniques are necessary where optimization operations are independent of scenario-specific information including objective functions, system parameters, and network setups. The major challenge is the requirement of a hyper-intelligent black-box optimizer that can establish efficient decision-making policies using its internal reasoning capabilities. This paper presents a novel knowledge-free network management paradigm with the power of large language models (LLMs). Trained on vast amounts of datasets, LLMs can understand important contexts from input prompts containing minimal system information, thereby offering remarkable inference performance even for entirely new tasks. Pretrained LLMs can be potentially leveraged as foundation models for versatile network optimization. By eliminating the dependency on prior knowledge, LLMs can be seamlessly applied to various network management tasks. The viability of this approach is demonstrated using various language models. Numerical results validate that knowledge-free LLM optimizers are able to achieve near-optimal performance.