Optimization Theory

Optimization Theory is a branch of applied mathematics and computational science dedicated to finding the best possible solution from a set of available alternatives under a given set of constraints. It provides a formal framework for modeling decision-making problems by defining an objective function—a quantity to be maximized (e.g., profit, efficiency) or minimized (e.g., cost, error)—and a set of variables and constraints that define the feasible solution space. As a cornerstone of Systems Science, optimization offers the essential tools to analyze, design, and control complex systems, enabling the determination of the most effective resource allocation, operational strategy, or system configuration to achieve a desired outcome.

1. Foundations of Optimization
   1. Defining the Optimization Problem
      1. Objective Function
         1. Definition and Role
         2. Types of Objective Functions
            1. Linear Objective Functions
            2. Quadratic Objective Functions
            3. Polynomial Objective Functions
            4. Exponential and Logarithmic Functions
            5. Composite Functions
         3. Single-Objective Functions
         4. Multi-Objective Functions
         5. Minimization vs Maximization Formulations
      2. Decision Variables
         1. Definition and Mathematical Notation
         2. Types of Decision Variables
            1. Continuous Variables
            2. Discrete Variables
            3. Integer Variables
            4. Binary Variables
            5. Mixed Variables
         3. Variable Bounds and Domains
         4. Vector Notation for Multiple Variables
      3. Constraints
         1. Definition and Purpose
         2. Types of Constraints
            1. Equality Constraints
            2. Inequality Constraints
            3. Box Constraints
            4. Nonlinear Constraints
         3. Feasible Region
            1. Definition and Properties
            2. Bounded vs Unbounded Regions
            3. Empty Feasible Regions
            4. Geometric Interpretation
   2. Classification of Optimization Problems
      1. By Problem Structure
         1. Linear Programming Problems
         2. Quadratic Programming Problems
         3. Nonlinear Programming Problems
         4. Convex Optimization Problems
         5. Non-Convex Optimization Problems
      2. By Variable Types
         1. Continuous Optimization
         2. Discrete Optimization
         3. Integer Programming
         4. Mixed-Integer Programming
      3. By Constraint Presence
         1. Unconstrained Optimization
         2. Constrained Optimization
      4. By Uncertainty
         1. Deterministic Optimization
         2. Stochastic Optimization
         3. Robust Optimization
      5. By Number of Objectives
         1. Single-Objective Optimization
         2. Multi-Objective Optimization
   3. Mathematical Formulation Standards
      1. General Optimization Problem Form
      2. Standard Form Conversions
      3. Canonical Forms for Specific Problem Types
      4. Notation Conventions
   4. Optimality Concepts
      1. Global Optimum
         1. Definition and Uniqueness
         2. Existence Conditions
      2. Local Optimum
         1. Definition and Characterization
         2. Strict vs Non-Strict Local Optima
      3. Saddle Points
      4. Relationship Between Global and Local Optima