Quantum computing has the potential to solve large-scale engineering optimization problems more efficiently than the classical computing scheme. Yet, most of the existing quantum optimization algorithms focus on solving problems with discrete variables on qubit-based quantum computers. Qubit-based information encoding faces the scalability challenge when solving optimization problems with continuous variables. Recent research progress on qubit-oscillator architecture has created the new opportunity to perform continuous-variable quantum computation. In this research, we propose a hybrid qubit-bosonic quantum optimization framework that allows us to efficiently encode and solve continuous and mixed-integer optimization problems. We demonstrate a variational approach to perform linear and nonlinear programming with continuous and discrete variables. Problems of mixed-integer bilinear optimization and topology optimization are also solved with the proposed framework. References [1] Wang Y., Kim J. E., Suresh K., Opportunities and challenges of quantum computing for engineering optimization, Journal of Computing and Information Science in Engineering, Vol. 23, No. 6, pp. 060817, 2023.