Optimization problems

Optimization Problem

The task of finding the minimum or maximum of an objective function by adjusting its variables.

Objective Function

The function f(x) whose value we want to minimize or maximize.

Variable (Decision Variable)

The value(s) we can change to optimize the objective function.

Unconstrained Optimization

Optimization with no restrictions on the variable x.

Constrained Optimization

Optimization with constraints on x, such as equality (ci(x)=0) or inequality (gk(x)≥0) conditions.

Global Minimizer

A point x* where f(x*) ≤ f(x) for all x.

Local Minimizer

A point x* where f(x) ≤ f(x) for all x in a small neighborhood around x.

Feasible Set

The set of all x that satisfy the constraints in a constrained optimization problem.

Interior Minimizer

A minimum inside the feasible set, where the function cannot decrease in any direction.

Boundary Minimizer

A minimum located on the edge of the feasible set.

Lagrangian Function

Combines the objective and constraints into one function

Lagrange Multipliers

Parameters (λi, μk) that weight the influence of the constraints in the Lagrangian.

Primal Problem

The original optimization problem, minimizing f(x) subject to its constraints.

Dual Problem

The derived problem that maximizes the minimum of the Lagrangian over x. max{λ, μ} infx L(x, λ, μ)

Strong Duality

When the primal and dual problems have the same optimal value.

Weak Duality

When the dual optimum gives a lower bound on the primal optimum (always true).