chap 2 - Basic Concepts in Minimization Problems Hong K. Lo...

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1 Basic Concepts in Minimization Problems Hong K. Lo Civil Engineering Hong Kong University of Science and Technology
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2 Problem Statement Objective function: min ( ,. .., ) z xx I 1 Subject to a set of constraints (which forms the feasible region) gx x b I 11 1 (, . . . , ) x b I 21 2 . . . , ) ... g b JI J . . . , ) 1 or can be written as: min ( ) z x s . t . g b j J jj () , , . . . , x = 1
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3 The solution, x * , is a feasible value of x that minimizes z () x That is, zz ( ) * xx for any feasible x gb jJ jj , , . . . , * x = 1 ” constraints can be introduced by multiplying both sides by -1 to convert the constraints to “ ” constraints. “=“ constraints introduced by a pair of “ ” and “ ” constraints. Maximization problems introduced by multiplying -1 to the objective function
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4 Program with one variable Local Minimum Local Minimum Local Maximum Inflection Point x z(x) Unconstrained Minimization Problem Necessary Condition (or 1st order condition): dz x dx () * = 0 , x * is called a stationary point The 1st order condition is not a sufficient condition. x * can be a maximum, point of inflection.
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5 A sufficient condition for a stationary point to be a local minimum is for the function to be strictly convex in the vicinity of the stationary point. Strict convexity means that a line segment connecting any 2 points of the function lies entirely above the function zx x z x z x θ 12 1 2 11 + < + () ( ) ( ) xx , are any 2 points, and 01 < < The if and only if condition for strict convexity can also be written as: dz x dx z x ( ) 1 1 21 2 +− < or dzx dx 2 2 0 * >
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6 zx x θ 12 1 +− () z x z x ( ) ( ) ( ) 1 z(x) x x 1 x 2 xx 1 +
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7 z(x) x x 1 x 2 zx dz x dx xx () ( ) 1 1 21 +− z x 2
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8 If a function is strictly convex in the vicinity of the stationary point, this point is a local minimum To show that a point is a global minimum, one has to compare the value of all the local minima. Alternatively, one needs to show that x is unique or the only minimum by showing that z(x) is convex for all value of x (i.e., the function is globally convex.)
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9 z(x) x Local Minimum x=a’ x=a’’ Constrained minimization program
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10 It is possible to have a minimum where the first derivative does not vanish.
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chap 2 - Basic Concepts in Minimization Problems Hong K. Lo...

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