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Unit 1.4: Classification

Lesson 4 of 22 in the free Engineering Optimization notes on Siksha Sarovar, written by Rohit Jangra.

Unit 1.4: Classification of Optimization Problems

Optimization problems are classified based on the nature of equations, constraints, and variables involved.

Classification Table

BasisTypeDescription
ConstraintsConstrainedHas constraints (e.g., x + y = 10). Most engineering problems are this type.
UnconstrainedNo limits on variables. Pure math problems like "Find min of ".
LinearityLinear Programming (LP)Objective f(X) AND all Constraints g(X), l(X) are Linear functions (Straight lines/planes).
Non-Linear (NLP)ANY function involves powers (), trig (sin x), or products (xy). Most real-world problems are NLP.
VariablesIntegerVariables must be whole numbers (e.g., "Number of bolts").
Real-ValuedVariables can be decimals (e.g., "Length = 5.23 mm").
Mixed-IntegerMix of both.
TimeStaticVariables do not change with time.
Dynamic (Optimal Control)Variables are functions of time (e.g., Trajectory of a missile).
SeparabilitySeparablef(X) can be written as sum of independent functions f1(x1) + f2(x2).
Non-SeparableVariables are interlinked (e.g., f(X) = x₁ * x₂).

Why classify?

Because different types require different algorithms:

  • LP -> Simplex Method.
  • Unconstrained NLP -> Gradient Descent, Newton's Method.
  • Constrained NLP -> Penalty Methods, SQP.
  • Integer -> Branch and Bound.