Recent research indicates that large language models (LLMs) are susceptible to jailbreaking attacks that can generate harmful content. This paper introduces a novel token-level attack method, Adaptive Dense-to-Sparse Constrained Optimization (ADC), which effectively jailbreaks several open-source LLMs. Our approach relaxes the discrete jailbreak optimization into a continuous optimization and progressively increases the sparsity of the optimizing vectors. Consequently, our method effectively bridges the gap between discrete and continuous space optimization. Experimental results demonstrate that our method is more effective and efficient than existing token-level methods. On Harmbench, our method achieves state of the art attack success rate on seven out of eight LLMs. Code will be made available. Trigger Warning: This paper contains model behavior that can be offensive in nature.