Discrete Optimization vs Continuous Optimization

- Overview

In optimization, continuous optimization deals with problems where variables can take on any real value, while discrete optimization focuses on problems where variables are restricted to a finite or countable set of values, such as integers. Essentially, continuous optimization allows for smooth adjustments, while discrete optimization involves choosing from a set of distinct options. 

In continuous optimization, the variables in the model can take on any value within a range, usually real numbers. This is often used in engineering or economics where fine-tuning parameters is crucial. Algorithms like gradient descent are commonly used in this type of optimization. 

Discrete optimization, also known as combinatorial optimization, involves finding the best solution from a finite or countable set of possible solutions. It's often used in problems like scheduling, resource allocation, and routing where decisions must be made from a limited number of options. Integer programming and mixed-integer programming are common approaches. 

Key Differences: 

• Variables: Continuous optimization uses continuous variables, while discrete optimization uses discrete variables (e.g., integers, binary values).
• Search Space: Continuous optimization searches a continuous space, while discrete optimization searches a discrete space.
• Algorithms: Continuous optimization often relies on algorithms like gradient descent, while discrete optimization may use techniques like branch and bound or dynamic programming.

Example: 

• Continuous: Optimizing the size of a product to maximize profits, where the size can be any real number within a range.
• Discrete: Determining the best route for a delivery truck, where the route must be chosen from a set of pre-defined paths.