Adaptive Stochastic Optimization Techniques with Applications

2  Static Optimization Techniques

2.1  Introduction

This section presents a brief review of selected classical optimization methods such as linear programming (LP), nonlinear programming (NLP), and interior point (IP) and its variants. This review entails summaries of concepts and algorithms as well as illustrative examples applied to power system applications and other large-scale systems.

2.2  Definition

The classical optimization [1,2,3] for given scalar objective functions with or without constraints—equality and/or inequality—is mathematically stated as

$\text{Minimize}f(x,u)(2.1)$

subject to

$\begin{array}{cc}g(x,u)=0m\hfill & \text{equility}\text{constraints}\hfill \end{array}(2.2)$

$\begin{array}{cc}{C}_i\le h_i(x,u)\le D_i\hfill & m+1\text{to}n\text{inequility constraints}\hfill \end{array}(2.3)$

This class can be solved using LP methods such as the simplex ...