Operations Research and Optimization

6.

Nonlinear Programming

6.1.

Introduction to Nonlinear Programming

6.1.1.

Nonlinear Objective Functions

6.1.2.

Nonlinear Constraints

6.1.3.

Local versus Global Optima

6.1.4.

Computational Challenges

6.2.

Convexity in Optimization

6.2.1.

6.2.1.1.

Definition and Properties

6.2.1.2.

6.2.1.3.

6.2.2.

Convex Functions

6.2.2.1.

Definition and Properties

6.2.2.2.

First-Order Conditions

6.2.2.3.

Second-Order Conditions

6.2.2.4.

Operations Preserving Convexity

6.2.3.

Importance of Convexity

6.2.3.1.

Global Optimality

6.2.3.2.

Algorithmic Advantages

6.2.3.3.

6.3.

Unconstrained Optimization

6.3.1.

Optimality Conditions

6.3.1.1.

First-Order Necessary Conditions

6.3.1.2.

Second-Order Conditions

6.3.1.3.

Global Optimality

6.3.2.

Line Search Methods

6.3.2.1.

Exact Line Search

6.3.2.2.

Inexact Line Search

6.3.2.3.

6.3.2.4.

Wolfe Conditions

6.3.3.

Gradient-Based Methods

6.3.3.1.

Steepest Descent Method

6.3.3.2.

Convergence Properties

6.3.3.3.

Step Size Selection

6.3.4.

Newton's Method

6.3.4.1.

Newton Direction

6.3.4.2.

Hessian Matrix Requirements

6.3.4.3.

Convergence Rate

6.3.4.4.

Modified Newton Methods

6.3.5.

Quasi-Newton Methods

6.3.5.1.

6.3.5.2.

6.3.5.3.

Secant Conditions

6.3.5.4.

Hessian Approximations

6.3.6.

Conjugate Gradient Methods

6.3.6.1.

Conjugate Directions

6.3.6.2.

Fletcher-Reeves Method

6.3.6.3.

Polak-Ribiere Method

6.4.

Constrained Optimization

6.4.1.

Optimality Conditions

6.4.1.1.

Lagrange Multiplier Method

6.4.1.2.

Karush-Kuhn-Tucker Conditions

6.4.1.3.

Constraint Qualifications

6.4.1.4.

Second-Order Conditions

6.4.2.

Penalty Methods

6.4.2.1.

Exterior Penalty Functions

6.4.2.2.

Interior Penalty Functions

6.4.2.3.

Barrier Methods

6.4.2.4.

Exact Penalty Functions

6.4.3.

Lagrangian Methods

6.4.3.1.

Augmented Lagrangian

6.4.3.2.

Method of Multipliers

6.4.3.3.

6.4.4.

Sequential Quadratic Programming

6.4.4.1.

6.4.4.2.

Merit Functions

6.4.4.3.

Trust Region Methods

6.4.5.

Active Set Methods

6.4.5.1.

Working Set Strategy

6.4.5.2.

Constraint Addition and Deletion

6.4.5.3.

Degeneracy Handling

6.5.

Special Classes of Nonlinear Programs

6.5.1.

Quadratic Programming

6.5.1.1.

Convex Quadratic Programs

6.5.1.2.

Active Set Methods

6.5.1.3.

Interior Point Methods

6.5.2.

Geometric Programming

6.5.2.1.

Posynomial Functions

6.5.2.2.

6.5.2.3.

Signomial Programs

6.5.3.

Separable Programming

6.5.3.1.

Piecewise Linear Approximation

6.5.3.2.

Special Ordered Sets

6.5.4.

Fractional Programming

6.5.4.1.

Linear Fractional Programming

6.5.4.2.

Dinkelbach's Algorithm

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