Nonlinear Opt.: Basic concepts 3

Theory

---

“Nonlinear Optimization problems”(NLP) described as a special type of optimization problems, where some of the constraints or the objective function is nonlinear. In contrast to the Linear Optimization, which solution method is the simplex algorithm, there is no comparable solution method for NLP. Instead of the Simplex algorithm there are different solution methods which are specific for a given problem. In the field of nonlinear optimization we divide between restricted and non-restricted optimization problems. While the domain of restricted optimization problems is a subset of R^n and usually described by a system of equalities and inequalities, the domain of non-restricted optimization problems is the entire range of R^n. In the following we will point out some special algorithms, developed for restricted optimization problems with linear or nonlinear constraints. Convex optimization problems as a special case of restricted optimization problems have a convex objective function and a convex range. Therefore local and global minima coincide.

In addition to the nonlinear objective function and at least one nonlinear constraint the result of our nonlinear optimization problem does not have to be the optimal solution.

The gereral form of a non linear optimization can be stated as:

$max~z=f(x_{1},...,x_{n})$

under the constraints that:

Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): h_{i}(x_{1},...,x_{n})\leq b_{i}\qquad i=1,...,m

and:

Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): x_{j}\geq 0\qquad j=1,...,n

By transforming the restriction

Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): h_{i}(x_{1},...,x_{n})\leq b_{i}

in

Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): h_{i}(x_{1},...,x_{n})-b_{i}\leq 0

and nominating this inequality with $g_{i}$

$max~z=f(x_{1},...,x_{n})$

in consideration of the restrictions:

Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): g_{i}(x_{1},...,x_{n})\leq 0\qquad i=1,...,m

Example

---

Nowadays “Nonlinear Optimization” is an important technology in applied mathematics, science and engineering. “Nonlinear Optimization problems” appear in many applications, e.g., shape optimization in engineering, robust portfolio optimization in finance, parameter identification, optimal control, etc. “Nonlinear Optimization” has emerged as a key technology in modern scientific and industrial applications.

Example 1 (Maximization)

Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): f(x)=5x-2x^2

Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): \frac{\partial }{\partial x} f(x)=5-4x

Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): \frac{\partial^2 }{\partial x^2}f(x)=4>0 \rightarrow min

Example 2 (Minimization)

$f(x)=5x+2x^{2}$

Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): \frac{\partial }{\partial x}f(x)=5+4x

Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): \frac{\partial^2 }{\partial x^2}f(x)=4>0 \rightarrow max

Example 3: Hessian Matrix

Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): F(x_1,x_2)= 88x_1-x_1^2+88x_2-x_2^2

Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): \begin{pmatrix} 88-2x_1\\ 88-2x_2 \end{pmatrix}