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NumericalOptimizerLibrary.md

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Numerical Optimizer Library

Numerical Optimizer Library contains the supporting Functionality for Numerical Optimization - including Constrained and Mixed Integer Non-Linear Optimizers.

Documentation

Document Link
Technical Specification Latest Previous
User Guide
API Javadoc

Component Projects

  • Optimization => Necessary, Sufficient, and Regularity Checks for Gradient Descent in a Constrained Optimization Setup.

Coverage

  • Convex Optimization - Introduction and Overview
    • Motivation, Background, and Setup
    • Convex Sets and Convex Hull
    • Properties of Convex Sets/Functions
    • Convex Optimization Problems
    • References
  • Newton’s Method in Optimization
    • Method
    • Higher Dimensions
    • Wolfe Condition
    • Armijo Rule and Curvature Condition
    • Rationale for the Wolfe Conditions
    • References
  • Constrained Optimization
    • Constrained Optimization – Definition and Description
    • General Form
    • Solution Methods
    • Constraint Optimization: Branch and Bound
    • Branch-and-Bound: First-Choice Bounding Conditions
    • Branch-and-Bound: Russian Doll Search
    • Branch-and-Bound: Bucket Elimination
    • References
  • Lagrange Multipliers
    • Motivation, Definition, and Problem Formulation
    • Introduction, Background, and Overview
    • Handling Multiple Constraints
    • Modern Formulation via Differentiable Manifolds
    • Interpretation of the Lagrange Multipliers
    • Lagrange Application: Maximal Information Entropy
    • Lagrange Application: Numerical Optimization Techniques
    • Lagrange Multipliers – Common Practice Applications
    • References
  • Spline Optimizer
    • Constrained Optimization using Lagrangian
    • Least Squares Optimizer
  • Karush-Kuhn-Tucker Conditions
    • Introduction, Overview, Purpose, and Motivation
    • Necessary Conditions for Optimization Problems
    • Regularity Conditions or Constraint Qualifications
    • Sufficiency Conditions
    • KKT Conditions Application - Economics
    • KKT Conditions Application - Value Function
    • Generalizations
    • References
  • Interior Point Method
    • Motivation, Background, and Literature Survey
    • Interior Point Methodology and Algorithm
    • References
  • Portfolio Selection with Cardinality and Bound Constraints
    • Synposys
    • Introduction
    • Problem Formulation
    • Analysis of the Problem
    • Bender’s Decomposition
    • A Greedy Heuristic
    • Cutting Planes Algorithm and PROXACCPM – Concept and Tool
    • PROXACCPM Performance on the Generic Problem
    • Chvatal-Gomory Cuts and Variants
    • Chvatal-Gomory Cuts
    • Deriving the Cuts for the Setup
    • Branching Rule and Node Selection
    • Computational Results
    • Conclusion
    • References
  • Simplex Algorithm
    • Introduction
    • Overview
    • Standard Form
    • Simplex Tableau
    • Pivot Operations
    • The Algorithm
    • Entering Variable Selection
    • Leaving Variable Selection
    • Example #1
    • Finding an Initial Canonical Tableau
    • Example #2
    • Advanced Topics – Implementation
    • Degeneracy and Stalling
    • Efficiency
    • Other Algorithms
    • Linear-Fractional Programming
    • References
  • Optimal Control
    • Introduction
    • General Method
    • Linear Quadratic Control
    • Numerical Methods for Optimal Control
    • Discrete-time Optimal Control
    • Examples
      • Finite Time
    • References
  • Hamilton–Jacobi–Bellman Equation
    • Introduction
    • Optimal Control Problems
    • The Partial Differential Equation
    • Deriving the Equation
    • Solving the Equation
    • Extension to Stochastic Problems
      • Application to LQG-Control
    • See Also
    • References
  • Bellman Equation
    • Overview
    • Analytical Concepts in Dynamic Programming
    • Derivation
      • A Dynamic Decision Problem
      • Bellman's Principle of Optimality
      • The Bellman Equation
      • In a Stochastic Problem
    • Solution Methods
    • Applications in Economics
    • Example
    • See Also
    • References
  • Understanding Exact Dynamic Programming through Bellman Operators
    • Overview
    • Value Functions as Vectors
    • Expected Reward and State Transition Probabilities
    • Bellman Operators B_π and B_*
    • Contraction and Monotonicity of Operators
    • Policy Evaluation
    • Policy Improvement
    • Policy Iteration
    • Value Iteration
    • Greedy Policy from an Optimal Value Function is an Optimal Policy

DROP Specifications