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08. NLP (OR Models)

# 08. NLP (OR Models) - Lecture 8 Nonlinear Programming...

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Lecture 8 – Nonlinear Programming Models Topics General formulations Local vs. global solutions Solution characteristics Convexity and convex programming Examples

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In LP ... the objective function & constraints are  linear  and  the problems are “ easy   to solve. Many real-world engineering and business problems  have  nonlinear  elements and are  hard  to solve.   Nonlinear Optimization
Minimize f ( x ) s.t. g i ( x ) ( , , =) b i , i = 1,…, m x = ( x 1 ,…, x n ) is the n -dimensional vector of decision variables f ( x ) is the objective function g i ( x ) are the constraint functions b i are fixed known constants General NLP

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Example 1 Max   f   ( x ) = 3 x 1  + 2 x 2 4 s.t.    x 1  +  x 2    1,   x 1    0,  x 2  unrestricted 2 Examples  2  and  3  can be reformulated as LPs Example 2 Max   f   ( x ) =  e c 1 x e c 2 x   e c n x n s.t.    Ax  =  b x     0   n Example 3 Min =1    f ( x j   ) s.t.    Ax  =  b x     0   where each  f j ( x j   )   is of the form Problems with “decreasing efficiencies” f j ( x j ) x j Examples of NLPs
Max   f ( x 1 x 2 ) =  x 1 x 2   s.t. 4 x 1  +  x 2   8 x 0,  x 2 0 2 8 f ( x ) = 2 f ( x ) = 1 x 2 Optimal solution will lie on the line g ( x ) = 4 x 1 + x 2  – 8 = 0. x 1 NLP Graphical Solution Method

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Solution is  not  a vertex of feasible region.  For this  particular problem the solution is on the  boundary  of the  feasible region.   This is  not  always the case. In a more general case,  f   ( x 1 x 2 ) =  μ g   ( x 1 x 2 ) with  μ     0 .  ( In this case,  μ  = 1.) Solution Characteristics Gradient of  f   ( x ) =  f   ( x 1 x 2 )    ( f / x 1 f / x 2 ) T This gives   f / x 1  =  x 2 ,   f / x 2  =  x 1 and           g / x 1  = 4,   g / x 2  = 1 At optimality we have  f   ( x 1 x 2 ) =  g   ( x 1 x 2 )    or   x 1 *  = 1 and  x 2 *  = 4
f (x) x local min global max stationary point local min local max Nonconvex Function Let S

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08. NLP (OR Models) - Lecture 8 Nonlinear Programming...

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