Nonlinear Opt.: Basic concepts 4

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Theory

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A nonlinear problem also so called NLP, is similar to a linear program but it is created by the objective function, general constraints and variable bounds. The significant difference from a NLP to a LP is that NLP includes minimum one nonlinear function. First we define what an optimal solution is. The function which is supposed to be minimized or maximized is ($)f(x),forx\epsilon \mathbb {R} ^{n}($) called the objective function. x* is an local minimum, if: (Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): ) \[\inline x^*\epsilon\, \mathbb{R}^n , \varepsilon > 0 , \: f(x^*)\leq f(x)\, for\, all\, x\, \epsilon A (x^*; \varepsilon)\] ( ) And the global minimum, if: (Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): ) \[\inline \: f(x^*)\leq f(x)\, for\, all\, x\, \epsilon \, \mathbb{R}^n\] ( )

The problem can be stated simply as: \[\inline x\,\epsilon\, X,\, max f(x)\,to\, maximize\, some\, variable\, such\, as\, product\, throughput\] \[\inline x\,\epsilon\, X,\, min f(x)\,to\, minimize\, a\, cost\, function\, where\,\] \[\inline f: R^n \rightarrow R\] \[\inline x\, \epsilon\, R^n\] subject to: \[\inline h_i(x) = 0, \, i\,\epsilon \, I\, = 1, ..., p\]

\[\inline g_j(x)\,\leq \,0,\,j\,\epsilon \,J\,= 1,...,m\]

Example:

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The following set of NLP are genaral subroutines:

NLPCG Conjugate Gradient Method NLPDD Double Dogleg Method NLPNMS Nelder-Mead Simplex Method NLPNRA Newton-Raphson Method NLPNRR Newton-Raphson Ridge Method NLPQN (Dual) Quasi-Newton Method NLPQUA Quadratic Optimization Method NLPTR Trust-Region Method The following subroutines are provided for solving nonlinear least-squares problems: NLPLM Levenberg-Marquardt Least-Squares Method NLPHQN Hybrid Quasi-Newton Least-Squares Methods

Example 1: Simple NLP maximization \[\inline f(x)= 3x-x^3 \Rightarrow max\] First derivative: \[\inline f'(x)= 3-3x^2 \Rightarrow 3-3x^2=0 \Rightarrow x_1 = +1, \, x_2= -1\,\] Second derivative: \[\inline f(x)= -6 \Rightarrow -6< 0 \Rightarrow max\: (local maximum)\] Example 2: NLP Gradient Method

First we need to find the gradient: \[\inline f(x,y,z)= 6x^2 +2y^2+2z^2 \:\rightarrow \triangledown_f (6x^2 +2y+2z)=\left \{ 6x,\, 2y,\, 2z \right \}\]

Gradient has to equal 0: \triangledown f(x)=0 \[\inline \left \{ 6x,\, 2y,\, 2z \right \}= 0\]

Now we get the Hesse-Matrix by the derivative once again: \triangledown^2_f = (6, 2, 2)

Solve for example with Rule of SARRUS:

\[\inline a_1_,_2=2, \, a_3=6;\: a_1_,_2_,_3> 0 \Rightarrow global\, minimum\, at\, (0,0,0)\]

Surces: http://ciser.cornell.edu/sasdoc/saspdf/iml/chap11.pdf www.wikipedia.org/nonlinear_programming www.sce.carleton.ca/faculty/chinneck/po/Chapter%2016.pdf