Optimization Theory for Computer Science

Optimization is the mathematical discipline of finding the best solution from a set of available alternatives. In computer science, optimization problems arise in machine learning, algorithm design, resource allocation, and system performance tuning.

What is Optimization?

Mathematical Definition

An optimization problem seeks to minimize (or maximize) an objective function f(x) subject to constraints:

minimize f(x) subject to: g(x) ≤ 0, h(x) = 0

Where: - f(x) is the objective function - g(x) represents inequality constraints - h(x) represents equality constraints - x is the decision variable (vector)

Types of Optimization Problems

By Variable Type

  • Continuous: Variables can take any real value
  • Discrete: Variables take integer or categorical values
  • Mixed: Combination of continuous and discrete variables

By Objective Function

  • Linear: f(x) is a linear function
  • Quadratic: f(x) is quadratic
  • Convex: f(x) is convex (single global minimum)
  • Non-convex: Multiple local minima may exist

Chapter Contents

  1. Unconstrained Optimization
  2. Constrained Optimization
  3. Linear Programming
  4. Convex Optimization
  5. Gradient-Based Methods
  6. Metaheuristic Algorithms
  7. Multi-Objective Optimization

Applications in Computer Science

Machine Learning

  • Training neural networks: Minimize loss functions
  • Support Vector Machines: Quadratic programming
  • Regularization: Balance fit and complexity
  • Hyperparameter tuning: Find optimal model parameters

Algorithm Design

  • Shortest path problems: Graph optimization
  • Network flow: Resource allocation
  • Scheduling: Task assignment and timing
  • Approximation algorithms: Near-optimal solutions

System Optimization

  • Compiler optimization: Code generation and scheduling
  • Database query optimization: Execution plan selection
  • Resource allocation: CPU, memory, bandwidth
  • Load balancing: Distribute workload efficiently

Operations Research

  • Supply chain optimization: Minimize costs
  • Portfolio optimization: Risk-return tradeoffs
  • Facility location: Optimal placement problems
  • Routing problems: Vehicle routing, TSP

Mathematical Prerequisites

  • Multivariable calculus
  • Linear algebra
  • Basic probability and statistics
  • Understanding of algorithms and complexity

Tools and Libraries

Python

  • SciPy: General optimization routines
  • CVXPY: Convex optimization
  • PuLP: Linear programming
  • Optuna: Hyperparameter optimization
  • DEAP: Evolutionary algorithms

Specialized Solvers

  • Gurobi: Commercial optimization solver
  • CPLEX: IBM optimization solver
  • MOSEK: Conic optimization
  • OR-Tools: Google optimization tools