Examine the principles of process optimization in chemical engineering, including mathematical optimization techniques and optimization algorithms. Discuss the different types of optimization problems encountered in chemical processes, such as linear programming, nonlinear programming, and mixed$-$integer programming, and explain the methodologies used to solve these problems, such as gradient$-$based methods, evolutionary algorithms, and metaheuristic approaches. Analyze the role of process optimization in improving process performance, efficiency, and profitability.

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