As Large Language Models (LLMs) are widely used, understanding them systematically is key to improving their safety and realizing their full potential. Although many models are aligned using techniques such as reinforcement learning from human feedback (RLHF), they are still vulnerable to jailbreaking attacks. Some of the existing adversarial attack methods search for discrete tokens that may jailbreak a target model while others try to optimize the continuous space represented by the tokens of the model's vocabulary. While techniques based on the discrete space may prove to be inefficient, optimization of continuous token embeddings requires projections to produce discrete tokens, which might render them ineffective. To fully utilize the constraints and the structures of the space, we develop an intrinsic optimization technique using exponentiated gradient descent with the Bregman projection method to ensure that the optimized one-hot encoding always stays within the probability simplex. We prove the convergence of the technique and implement an efficient algorithm that is effective in jailbreaking several widely used LLMs. We demonstrate the efficacy of the proposed technique using five open-source LLMs on four openly available datasets. The results show that the technique achieves a higher success rate with great efficiency compared to three other state-of-the-art jailbreaking techniques. The source code for our implementation is available at: https://github.com/sbamit/Exponentiated-Gradient-Descent-LLM-Attack