Nonlinear Opt.: Wolfe algorithm 1

1. Introduction

In 1956 Frank and Wolfe developed an algorithm to solve nonlinear quadratic optimization problems under certain restrictions. The difference between nonlinear and linear optimization problems only lays in the objective function F(x). Nonlinear functions include beside linear terms of cixj, also quadratic experessions of cjjxj2 and/or cjjxij for some or all i≠j. The quadratic minimization problem which is a convex objectiv function (and the quadratic maximization problem which is a concave objective function) will be solved with the Wolfe algorithm in the next step. Problems with quadratic objective functions which are not convex (not concave) can not be solved with the Wolfe algorithm.[1]

2. Theory

0. Formulate the corresponding Lagrangian function

Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): L\left ( x,\lambda_{j} \right )=F\left ( x \right )+\sum_{j=1}^{n}\lambda_{j}\cdot g_{j}(x)