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s02lec19 - 15.053 Thursday April 25 Difficulties of NLP...

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1 15.053 Thursday, April 25 z Nonlinear Programming Theory z Separable programming Handouts: Lecture Notes 2 Difficulties of NLP Models Nonlinear Programs: Linear Program: 3 Graphical Analysis of Non-linear programs in two dimensions: An example z Minimize z subject to (x - 8) 2 + (y - 9) 2 49 x 2 x 13 x + y 24 2 2 14 15 ( ) ( ) x y + 4 Where is the optimal solution? 0 2 4 6 8 10 12 14 16 18 0 2 4 6 8 10 12 14 16 18 y x Note: the optimal solution is not at a corner point. It is where the isocontour first hits the feasible region. 5 Another example: 0 2 4 6 8 10 12 14 16 18 0 2 4 6 8 10 12 14 16 18 y x Minimize (x-8) 2 + (y-8) 2 Then the global unconstrained minimum is also feasible. The optimal solution is not on the boundary of the feasible region. 6 Local vs. Global Optima There may be several locally optimal solutions. x z 1 0 z = f(x) max f(x) s.t. 0 x 1 A B C Def’n: Let x be a feasible solution, then x is a global max if f(x) f(y) for every feasible y . x is a local max if f(x) f(y) for every feasible y sufficiently close to x (i.e., x j - ε ≤ y j x j + ε for all j and some small ε ).
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7 When is a locally optimal solution also globally optimal? z We are minimizing. The objective function is convex. The feasible region is convex. W P 2 4 6 8 10 12 14 2 4 6 8 10 12 14 Convexity and Extreme Points We say that a set S is convex , if for every two points x and y in S , and for every real number λ in [0,1] , λ x + (1- λ )y ε S . The feasible region of a linear program is convex. x y We say that an element w ε S is an extreme point ( vertex, corner point ), if w is not the midpoint of any line segment contained in S . 8 9 Recognizing convex feasible regions z If all constraints are linear, then the feasible region is convex z The intersection of convex regions is convex z If for all feasible x and y, the midpoint of x and y is feasible, then the region is convex (except in totally non-realistic examples. ) Which are convex? C B B C B C B C D A 10 11 Convex Functions Convex Functions: f( λ y + (1- λ )z) ≤ λ f(y) + (1- λ )f(z) for every y and z and for 0 ≤ λ ≤1 . e.g., f((y+z)/2) f(y)/2 + f(z)/2 We say “strict” convexity if sign is “<” for 0 < λ <1 . Line joining any points is above the curve f(x) x x x y z (y+z)/2 x 12 Concave Functions Concave Functions: f( λ y + (1- λ )z) ≥ λ f(y) + (1- λ )f(z) for every y and z and for 0 ≤ λ ≤1 . e.g., f((y+z)/2) f(y)/2 + f(z)/2 We say “strict” convexity if sign is “>” for 0 < λ <1 . Line joining any points is below the curve f(x) x x x z y (y+z)/2 x
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13 Classify as convex or concave or both or neither. 14 What functions are convex? z f(x) = 4x + 7 all linear functions z f(x) = 4x 2 – 13 some quadratic functions z f(x) = e x z f(x) = 1/x for x > 0 z f(x) = |x| z f(x) = - ln (x) for x > 0 Sufficient condition: f”(x) > 0 for all x.
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