Optimization is all about finding the best solution to a problem within given constraints. It's used in business, engineering, and finance to maximize profits, design efficient systems, and make smart investment choices.

Every optimization problem has three key parts: decision variables you can adjust, an objective function to maximize or minimize, and constraints that limit your choices. By modeling these mathematically, we can solve complex real-world problems.

Optimization Fundamentals

Optimization concept and applications

• Optimization involves finding the best solution to a problem considering certain constraints or limitations
  • Maximizes or minimizes an objective function to achieve the optimal outcome
• Real-world applications span various domains:
  • Business and economics: Allocating resources efficiently to maximize profits or minimize costs
  • Engineering and product design: Designing systems or products that perform optimally under given constraints (aircraft design, manufacturing processes)
  • Transportation and logistics: Scheduling and routing vehicles to minimize travel time or fuel consumption (delivery routes, airline schedules)
  • Finance: Selecting investments to maximize returns while minimizing risk in a portfolio (asset allocation, risk management)

Components of optimization problems

• Optimization problems consist of three essential components:
  • Decision variables: Adjustable quantities that influence the objective function
    • Represent the choices or decisions to be made in the optimization process
  • Objective function: Mathematical expression that quantifies the performance or goal of the system
    • Defines the criterion to be optimized, such as maximizing profit or minimizing cost
  • Constraints: Limitations or restrictions imposed on the decision variables
    • Ensure the solution is feasible and practical within the given context
    • Expressed as equalities or inequalities that the decision variables must satisfy (budget constraints, production capacities)

Mathematical Modeling and Problem Types

Mathematical models for optimization

• Mathematical modeling translates real-world optimization problems into mathematical formulations
  • Decision variables are represented using appropriate symbols ($x$, $y$, $z$)
  • Objective function is expressed as a mathematical equation in terms of the decision variables
    • Profit maximization example: $P = 3x + 2y$, where $x$ and $y$ are quantities of two products
  • Constraints are represented as mathematical inequalities or equalities
    • Resource constraint example: $x + y \leq 100$ (limited total quantity)
    • Non-negativity constraints: $x \geq 0$, $y \geq 0$ (quantities cannot be negative)

Types of optimization problems

• Linear programming (LP) problems:
  • Objective function and constraints are linear functions of the decision variables
    • Can be solved efficiently using methods like the simplex algorithm or interior point methods
• Nonlinear programming (NLP) problems:
  • Objective function and/or constraints are nonlinear functions of the decision variables
    • More complex and challenging to solve compared to LP problems
    • Require specialized solution techniques (gradient-based methods, metaheuristics)

Graphical representation of optimization

• Two-dimensional optimization problems can be visualized graphically:
  • Decision variables are plotted on the x and y axes of a coordinate plane
  • Objective function is represented by a family of lines or curves called level sets
    • Each level set corresponds to a specific value of the objective function
  • Constraints are represented as lines or regions in the plane
    • Feasible region is the area where all constraints are simultaneously satisfied
  • Optimal solution is the point within the feasible region that optimizes the objective function
    • For LP problems, the optimal solution lies at a vertex (corner point) of the feasible region