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# s02lec17 - 15.053 Thursday April 18 Linear Programming...

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1 15.053 Thursday, April 18 z Nonlinear Programming (NLP) Modeling Examples Convexity Local vs. Global Optima z Handouts: Lecture Notes 2 Linear Programming Model 1 1 2 2 11 1 12 2 1n n 1 21 1 22 2 2n n 2 m1 1 2 2 mn n m Maximize ..... subject to a x + a x + ... +a x b a x + a x + ... +a x b a x + a x + ... +a x b n n m c x c x c x x + + + # # # 1 2 , ,..., 0 n x x ASSUMPTIONS : z Proportionality Assumption Objective function Constraints z Additivity Assumption Objective function Constraints 3 What is a non-linear program? z maximize 3 sin x + xy + y 3 - 3z + log z Subject to x 2 + y 2 = 1 x + 4z 2 z 0 z A non-linear program is permitted to have non-linear constraints or objectives. z A linear program is a special case of non- linear programming! 4 Nonlinear Programs (NLP) Nonlinear objective function f(x) and/or Nonlinear constraints g i (x) Could include x i 0 by adding the constraints x i = y i 2 for i=1,…,n. ( ) 1 2 , , , ( ) ( ) , 1,2, , n i i Let x x x x Max f x g x b i m = ∀ = 5 Unconstrained Facility Location 0 2 4 6 8 10 12 14 16 y 0 2 4 6 8 10 12 14 16 C (2) (7) B A (19) P ? D (5) x Loc. Dem. A: (8,2) 19 B: (3,10) 7 C: (8,15) 2 D: (14,13) 5 P: ? This is the warehouse location problem with a single warehouse that can be located anywhere in the plane. Distances are “Euclidean.” 6 z Costs proportional to distance; known daily demands An NLP 2 2 8 2 ( ) ( ) x y + d(P,A) = 2 2 14 13 ( ) ( ) x y + d(P,D) = minimize 19 d(P,A) + … + 5 d(P,D) subject to: P is unconstrained

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7 Here are the objective values for 55 different locations. 0 50 100 150 200 250 300 350 values for y Objective value x = 0 x = 2 x = 4 x = 6 x = 8 x = 10 x = 12 8 Facility Location. What happens if P must be within a specified region? 0 2 4 6 8 10 12 14 16 y 0 2 4 6 8 10 12 14 16 C (2) (7) B A (19) P ? D (5) x 9 The model 2 2 19 8 2 ( ) ( ) x y + 2 2 5 14 13 ( ) ( ) x y + + …+ Minimize Subject to x 7 5 y 11 x + y 24 10 0-1 integer programs as NLPs minimize Σ j c j x j subject to Σ j a ij x j = b i for all i x j is 0 or 1 for all j is “nearly” equivalent to minimize Σ j c j x j + 10 6 Σ j x j (1- x j ). subject to Σ j a ij x j = b i for all i 0 x j 1 for all j 11 Some comments on non-linear models z The fact that non-linear models can model so much is perhaps a bad sign How can we solve non-linear programs if we have trouble with integer programs? Recall, in solving integer programs we use techniques that rely on the integrality. z Fact: some non-linear models can be solved, and some are WAY too difficult to solve. More on this later. 12 Variant of exercise from Bertsimas and Freund z Buy a machine and keep it for t years, and then sell it. (0 t 10) all values are measured in \$ million Cost of machine = 1.5 Revenue = 4(1 - .75 t ) Salvage value = 1(1 + t)
13 Machine values 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0.2 1 1.8 2.6 3.4 4.2 5 5.8 6.6 7.4 8.2 9 9.8 Time Millions of dollars revenue salvage total 14 How long should we keep the machine?

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