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Mixed-integer optimization problem

As explained in Optimization Algorithms: AI techniques for design, planning, and control problems

An optimization problem containing both continuous-valued and integer-valued decision variables.

A mixed-integer optimization problem contains both continuous-valued and integer-valued decision variables. Continuous variables can take real-number values, while integer variables represent discrete choices such as selections, sequences, or assignments. section 2.1.1

This classification describes the decision variables, not the number of objectives, constraints, or the linearity of the problem. When discrete variables form sets, combinations, or permutations, the problem may also be combinatorial. section 2.1.1 section 2.5

Elevator operation

Optimizing elevator speed and acceleration uses continuous variables, while choosing the sequence in which passengers are picked up uses a discrete variable. Optimizing all of them together creates a mixed-integer optimization problem. section 2.1.1

Classify decision variables as follows:

  • Use continuous optimization when variables take real-number values. section 2.1.1
  • Use integer or discrete optimization when variables take integer values. section 2.1.1
  • Use mixed-integer optimization when both variable types occur in the same problem. section 2.1.1

How Optimization Algorithms: AI techniques for design, planning, and control problems covers it

How deep it goes
Defines the variable-type distinction and gives an elevator example, but does not develop solution algorithms or computational procedures.
Examples
Uses elevator speed, acceleration, and passenger pickup sequence to demonstrate mixed variable types.
What it assumes
Assumes familiarity with decision variables and basic optimization problem classification.
What you can do after
Evaluate whether an optimization model contains continuous variables, integer variables, or both.
How it connects
Connects continuous optimization, integer optimization, combinatorial optimization, decision variables, and optimization problem classifications.
Cover of Optimization Algorithms: AI techniques for design, planning, and control problems
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Related concepts

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