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OR II GSLM 52800. Outline. separable programming quadratic programming. Separable Programs. a separable NLP if f and all g j are separable functions 0 x i i , a finite number. Idea of Separable Program. min f (x), s . t . g j (x) 0 for j = 1, …, m .
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Outline • separable programming • quadratic programming 2
Separable Programs • a separable NLP if • f and all gjare separable functions • 0 xi i, a finite number 3
Idea of Separable Program • min f(x), • s.t. • gj(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 4
A Fact About Convex Functions • f: a convex function • for any > 0, possible to find a sequence of piecewise linear convex functions fnsuch that |f fn| 5
Example 6.1 • a separable program 6
20.25 C 16 B 9 4 A 1 O 3 4 1 2 Example 6.1 • approximating by a piecewise linear function • two representations, -form and -form 7
20.25 C 16 B 9 4 A 1 O 3 4 1 2 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 8
Example 6.1 • the program becomes the last but one type of constraints is non-linear 9
Fact • nonlinear constraint: at most two adjacent itaking non-zero values • possible to have only one i = 1 • for convex f and gj: no need to have the non-linear constraint • non-optimal to have more than two non-zero i, or two i not adjacent 10
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 20.25 C 16 B 9 4 A 1 O 11 3 4 1 2
Fact • non-optimal to have more than two non-zero i, or two i not adjacent • e.g., gj being a constraint • gj(0.3A+0.7B) gj(0.3A+0.7C) bj 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 12 A O
Example 6.1 • the program becomes a linear program 13
Example 6.2: Non-Convex Problem • min f(x), • s.t. 1 x 3. • approximating f(x) by a piecewise linear function • y = 0 + 10A + 6B • x = 0 + 2A + 3B 14
Example 6.2: Non-Convex Problem • adding slack variable s, surplus variable u, and artificial variable a1 and a2: 15
Example 6.2: Non-Convex Problem • most negative 0 • B in basis only A qualified to enter, not O 18
20.25 C 16 B 9 4 A 1 O 3 4 1 2 Form • again, the last constraint is unnecessary for a convex program 22
Quadratic Objective Function & Linear Constraints • Langrangian function 24
KKT Conditions • positive definite Q • a convex program • a unique global minimum • the KKT sufficient • otherwise, KKT necessary 25
KKT Conditions • cT + xTQ + TA 0 Qx + A y = c • Ax b 0 Ax + v = b • xT(c + Qx + A) = 0 xTy = 0 • T(Ax b) = 0 Tv = 0 • x 0, 0, y 0, v 0 • solving the set of equations phase-1 of a linear program 26
Example 7.1(Example 10.14 of JB) • KKT conditions • 2x1 + 1 + 2y1 = 8, • 8x2+ 1y2 = 16, • x1+ x2 + v1 = 5, • x1 + v2 = 3. • x1y1 = x2y2 = 1v1 = 2v2 = 0 • x1, y1, x2, y2, 1, v1, 2, v2 0 28