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OR II GSLM 52800

OR II GSLM 52800. Outline. classical optimization – unconstrained optimization dimensions of optimization feasible direction. Classical Optimization Results  Unconstrained Optimization. different dimensions of optimization conditions nature of conditions

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OR II GSLM 52800

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  1. OR IIGSLM 52800 1

  2. Outline • classical optimization – unconstrained optimization • dimensions of optimization • feasible direction 2

  3. Classical Optimization Results Unconstrained Optimization • different dimensions of optimization conditions • nature of conditions • necessaryconditions (必要條件): satisfied by any minimum (and possibly by some non-minimum points) • sufficient conditions (充分條件): if satisfied by a point, implying that the point is a minimum (though some minima may not satisfy the conditions) • order of conditions • first-order conditions: in terms of the first derivatives of f & gj • second-order conditions: in terms of the second derivatives of f & gj • general assumptions: f, g, gjC1 (i.e., once continuously differentiable) orC2 (i.e., twice continuously differentiable) as required by the conditions 3

  4. Feasible Direction • S n: the feasible region • x  S: a feasible point • a feasible direction d of x: if there exists  > 0 such that x+d  S for 0 <  <  4

  5. Two Key Concepts for Classical Results • f: the direction of steepest accent • gradient of f at x0 being orthogonal to the tangent of the contour f(x) = c at x0 5

  6. x2 x1 The Direction of Steepest Accent f • contours of f(x1, x2) = • f: direction of steepest accent • in some sense, increment of unit move depending on the angle with f • within 90 of f: increasing • closer to 0: increasing more • beyond 90 of f: decreasing • closer to 180: decreasing more • above results generally true for any f 6

  7. Gradient of f at x0 Being Orthogonal to the Tangent of the Contour f(x) = c at x0 • f(x1, x2) = • f(x10, x20) = c • d on the tangent plane at x0 • f(x0+d) c for small  • roughly speaking, for f(x0) = c, f(x0+d) = c for small  when d is on the tangent plane at x0 7

  8. First-Order Necessary Condition (FONC) • fC1 on S and x* a local minimum of f • then for any feasible direction d at x*, Tf(x*)d  0 • increasing of f at any feasible direction f(x, y) = x2 + y2 for 0 x, y 2 f(x, y) = x2 + y2 for x 3, y 3 f(x) = x2 for 2 x 5 8

  9. FONC for Unconstrained NLP • fC1 on S & x* an interior local minimum (i.e., without touching any boundary)  Tf(x*) = 0 9

  10. FONC Not Sufficient • Example 3.2.2: f(x, y) = -(x2 + y2) for 0 x, y • Tf((0, 0))d = 0 for all feasible direction d • (0, 0): a maximum point • Example 3.2.3: f(x) = x3 • f(0) = 0 • x = 0 a stationary point 10

  11. Feasible Region with Non-negativity Constraints • Example 3.2.4. (Example 10.8 of JB) Find candidates of the minimum points by the FONC. • min f(x) = • subject to x1 0, x2 0, x2 0 or, equivalently 11

  12. Second-Order Conditions • another form of Taylor’s Theorem • f(x) = f(x*)+Tf(x*)(x-x*) +0.5(x- x*)TH(x*)(x - x*)+ , • where  being small, dominated by other terms • if Tf(x*)(x-x*) = 0, • f(x) f(x*) (x- x*)TH(x*)(x - x*)  0 12

  13. Second-Order Necessary Condition • fC2 on S • if x* is a local minimum of f, then for any feasible direction d n at x*, • (i). Tf(x*)d  0, and • (ii). if Tf(x*)d = 0, then dTH(x*)d  0 13

  14. Example 3.3.1(a) • SONC satisfied f(x, y) = x2 + y2 for 0 x, y 2 f(x, y) = x2 + y2 for x 3, y 3 f(x) = x2 for 2 x 5 14

  15. Example 3.3.1(b) • SONC: more discriminative than FONC • f(x, y) = -(x2 + y2) for 0 x, y in Example 3.2.2 • (0, 0), a maximum point, failing the SONC 15

  16. SONC for Unconstrained NLP • fC2 in S • x* an interior local minimum of f, then • (i). Tf(x*) = 0, and • (ii). for all d, dTH(x*)d  0 • (ii)  H(x*) being positive semi-definite • convex f satisfying (ii) (and actually more) 16

  17. Example 3.3.2 • identity candidates of minimum points for the f(x) = • Tf(x*) = • x = (1, -1) or (-1, -1) • H(x) = • (1, -1) satisfying SONC but not (-1, -1) 17

  18. SONC Not Sufficient • f(x, y) = -(x4 + y4) • Tf((0, 0))d = 0 for all d • (0, 0) a maximum 18

  19. SOSC for Unconstrained NLP • fC2 on S n and x* an interior point • if • (i). Tf(x*) = 0, and • (ii). H(x*) is positive definite • x* a strict local minimum of f 19

  20. SOSC Not Necessary • Example 3.3.4. • x = 0 a minimum of f(x) = x4 • SOSC not satisfied 20

  21. Example 3.3.5 • In Example 3.2.4, is (1, 1, 1) a minimum? • . • 6 > 0; • positive definite, i.e., SOSC satisfied 21

  22. Effect of Convexity • If for all y in the neighborhood of x*  S, Tf(x*)(y-x*)  0 • convexity of fimplies • f(y) f(x*) + Tf(x*)(y-x*) f(x*) • x* a local min of f in the neighborhood of x* • x* a global minimum of f 22

  23. Effect of Convexity • fC2 convex  H positive semi-definite everywhere • Taylor's Theorem, when Tf(x*)(x-x*) = 0, • f(x) = f(x*) + Tf(x*)(x-x*) + (x- x*)TH(x* + (1-)x)(x - x*) = f(x*) + (x- x*)TH(x* + (1-)x)(x - x*) f(x*) •  x* a local min  a global min 23

  24. Effect of Convexity • facts of convex functions •  (i). a local min = a global min • (ii). H(x) positive semi-definite everywhere • (iii). strictly convex function, H(x) positive definite everywhere • implications • for f C2 convex function, the FONC Tf(x*) = 0 is sufficient for x* to be a global minimum • if f strictly convex, x* the unique global min 24

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