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1 OR II GSLM 52800
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2 Outline separable programming quadratic programming
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3 Separable Programs a separable NLP if f and all g j are separable functions 0 x i i, a finite number
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4 Idea of Separable Program min f(x), s.t.s.t. g j (x) 0 for j = 1, …, m. hard NLP but simple LP problems approximating a separable NL program by a LP a non-linear function by a piecewise linear one
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5 A Fact About Convex Functions f: a convex function for any > 0, possible to find a sequence of piecewise linear convex functions f n such that |f f n |
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6 Example 6.1 a separable program
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7 Example 6.1 approximating by a piecewise linear function two representations, -form and -form pointsOABC x1x1 0244.5 y041620.25 1 C B A O 16 9 4 43 2 1
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8 Form a piecewise linear function with (segment) break points any point = the convex combination of the two break points of the linear segment i ( 0) = the weight of break point i 1 C B A O 20. 25 16 9 4 43 2 1
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9 Example 6.1 the program becomes the last but one type of constraints is non-linear
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10 Fact nonlinear constraint: at most two adjacent i taking non-zero values possible to have only one i = 1 for convex f and g j : no need to have the non- linear constraint non-optimal to have more than two non-zero i, or two i not adjacent
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11 Fact non-optimal to have more than two non-zero i, or two i not adjacent e.g., f being an objective function any convex combination between two non-adjacent break points being above the piecewise non-linear function similarly, the point for three or more non-zero I ’s lying above the piecewise non-linear function think about A = 0.3, B = 0.4, and C = 0.3 1 C B A O 20.25 16 9 4 43 2 1
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12 Fact non-optimal to have more than two non-zero i, or two i not adjacent e.g., g j being a constraint g j (0.3A+0.7B) g j (0.3A+0.7C) b j the feasible set of {0.3A+0.7B} is larger than that by {0.3A+0.7C} the solution from {0.3A+0.7C} cannot be minimum similar argument for three or more non-zero i ’s lying above the piecewise non-linear function C B A O
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13 Example 6.1 the program becomes a linear program
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14 Example 6.2: Non-Convex Problem min f(x), s.t.1 x 3. approximating f(x) by a piecewise linear function y = 0 + 10 A + 6 B x = 0 + 2 A + 3 B
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15 Example 6.2: Non-Convex Problem adding slack variable s, surplus variable u, and artificial variable a 1 and a 2 :
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16 Example 6.2: Non-Convex Problem
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17 Example 6.2: Non-Convex Problem
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18 Example 6.2: Non-Convex Problem most negative 0 B in basis only A qualified to enter, not O
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19 Example 6.2: Non-Convex Problem
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20 Example 6.2: Non-Convex Problem
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21 Example 6.2: Non-Convex Problem
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22 Form again, the last constraint is unnecessary for a convex program 1 C B A O 20.25 16 9 4 43 2 1
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23 Quadratic Programming
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24 Quadratic Objective Function & Linear Constraints Langrangian function
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25 KKT Conditions positive definite Q a convex program a unique global minimum the KKT sufficient otherwise, KKT necessary
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26 KKT Conditions c T + x T Q + T A 0 Qx + A y = c Ax b 0 Ax + v = b x T (c + Qx + A ) = 0 x T y = 0 T (Ax b) = 0 T v = 0 x 0, 0, y 0, v 0 solving the set of equations phase-1 of a linear program
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27 Example 7.1 (Example 10.14 of JB)
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28 Example 7.1 (Example 10.14 of JB) KKT conditions 2x 1 + 1 + 2 y 1 = 8, 8x 2 + 1 y 2 = 16, x 1 + x 2 + v 1 = 5, x 1 + v 2 = 3. x 1 y 1 = x 2 y 2 = 1 v 1 = 2 v 2 = 0 x 1, y 1, x 2, y 2, 1, v 1, 2, v 2 0
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29 Example 7.1 (Example 10.14 of JB)
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30 Example 7.1 (Example 10.14 of JB) Example 7.1 (Example 10.14 of JB)
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