Unit 1.2: Statement of an Optimization Problem

An optimization problem can be mathematically stated as finding a vector of design variables X that minimizes (or maximizes) an objective function f(X), satisfying certain constraints.

1. Standard Mathematical Form

Find X = [x₁, x₂, ... xₙ]ᵀ

To Minimize: f(X)

Subject to the constraints:

1. Inequality Constraints: g_j(X) <= 0 (for j = 1, 2, ... m)
2. Equality Constraints: l_j(X) = 0 (for j = 1, 2, ... p)
3. Variable Bounds: X_i,min <= X_i <= X_i,max (for i = 1, 2, ... n)

Note: If the problem is to Maximize F(X), it is equivalent to Minimizing -F(X).

2. Key Terminology

A. Design Vector (X)

The set of unknowns or variables that define the system. We can change these values to get a better design.

• Example: In a gear design, X = [Number of teeth, Module, Face width].
• Represented as a column vector X.

B. Objective Function f(X)

The single criterion used to judge how "good" a design is.

• Also called the Cost Function, Merit Function, or Utility Function.
• Example: f(X) = Weight of the truss. We want to make this as small as possible.

C. Constraints

Physical, Geometric, or Practical limitations imposed on the design variables.

• Behavioral Constraints: Depend on the performance (e.g., "Stress < 100 MPa").
• Side/Geometric Constraints: Depend on physical bounds (e.g., "Diameter > 10 mm").
• Inequality (g(x) <= 0): "The tank must hold at least 50 liters" (Volume >= 50 => 50 - Volume <= 0).
• Equality (l(x) = 0): "The total current must equal 10 Amps".