Mathematical Optimization for Engineers - Chapter 1

 

📘 1. Introduction to Optimization

  • Optimization means finding the best solution to a problem by improving a system using mathematical models.

  • It involves decision variables, an objective function (to minimize or maximize), constraints, and a model of the system.

🧱 2. Formulating Optimization Problems

  • Key components:

    • Decision variables: What you can change.

    • Objective function: What you want to optimize.

    • Constraints: Physical, economic, or resource limits.

    • Mathematical model: Captures how the system behaves.

🎯 3. Applications

Optimization is widely used in:

  • Business (portfolio design, production planning),

  • Engineering design (equipment size, process layout),

  • Operations (real-time adjustments, control),

  • Science (model fitting, experiment design).

🧪 4. Real-World Examples

  • Pipeline insulation design: Trade-off between heating cost and insulation cost.

  • Robot motion planning: Optimize movement to minimize time and avoid collisions.

  • Semi-batch reactors: Control inputs to maximize selectivity.

  • Gem cutting: Maximize volume, minimize waste.

  • Solar and wind energy: Optimal layout of heliostat fields and wind farms.

  • Sailing & kites: Design and control of advanced sailing/kite systems for energy generation.

🧩 5. Types and Classification of Optimization Problems

  • Based on variables and function types:

    • Linear vs Nonlinear

    • Continuous vs Discrete (Integer, Mixed-Integer)

    • Convex vs Nonconvex

    • Deterministic vs Stochastic

    • Static vs Dynamic (control over time)

⚙️ 6. Solution Process

  1. Define the problem and variables

  2. Choose objective and constraints

  3. Model the system

  4. Classify the problem type

  5. Choose an algorithm

  6. Solve using numerical methods

  7. Validate and interpret results

⚠️ 7. Challenges in Optimization

  • Requires expertise (not “push-button”)

  • Model error can mislead optimization ("optimizer’s curse")

  • Global vs local solutions: hard for nonconvex problems

  • Solutions often lie on constraint boundaries

🧮 8. Mathematical Background

  • Gradient: First derivative (∇f), points in steepest ascent.

  • Hessian: Second derivative matrix (∇²f), shows curvature.

  • Directional derivative: Rate of change in any direction.

✅ 9. Optimality Conditions

  • Unconstrained problems:

    • 1st-order necessary: ∇f(x*) = 0

    • 2nd-order necessary: Hessian is positive semi-definite

    • 2nd-order sufficient: Hessian is positive definite

  • Stationary points: Where ∇f = 0; may be minimum, maximum, or saddle point.

📈 10. Convexity

  • Convex set: Any line between two points stays within the set.

  • Convex function: Lies below its chords; no local minima except global.

  • Convexity ensures that:

    • A stationary point is a global optimum.

    • 1st-order conditions are sufficient.

🧠 11. Check Yourself (Reflection Questions)

Each section ends with thought-provoking questions to reinforce understanding, such as:

  • What makes a problem convex?

  • What defines a stationary point?

  • What are the limitations of modeling?

Subscribe to: Comments (Atom)