lecture5 - IE 495 Lecture 5 Stochastic Programming Math...

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IE 495 – Lecture 5 Stochastic Programming – Math Review and MultiPeriod Models Prof. Jeff Linderoth January 27, 2003 January 27, 2003 Stochastic Programming – Lecture 5 Slide 1
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Outline Homework – questions? ƒ I would start on it fairly soon if I were you... A fairly lengthy math (review?) session ƒ Differentiability ƒ KKT Conditions Modeling Example s ƒ Jacob and MIT ƒ “Multi-period” production planning January 27, 2003 Stochastic Programming – Lecture 5 Slide 2
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Yucky Math Review – Derivative Let f be a function from < n 7→ < . The directional derivative f 0 of f with respect to the direction d is f 0 ( x, d ) = lim λ 0 f ( x + λd ) - f ( x ) λ If this direction derivative exists and has the same value for all d ∈ < n , then f is differentiable . The unique value of the derivative is called the gradient of f at x ƒ We denote its value as f ( x ). January 27, 2003 Stochastic Programming – Lecture 5 Slide 3
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Not Everything is Differentiable Probably, everything you have ever tried to optimize has been differentiable. ? This will not be the case in this class! Even nice, simple, convex functions may not be differentiable at all points in their domain. ƒ Examples? A vector η ∈ < n is a subgradient of a convex function f at a point x iff (if and only if) ƒ f ( z ) f ( x ) + η T ( z - x ) z ∈ < n ƒ The graph of the (linear) function h ( z ) = f ( x ) + η T ( z - x ) is a supporting hyperplane to the convex set epi ( f ) at the point ( x, f ( x )). January 27, 2003 Stochastic Programming – Lecture 5 Slide 4
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More Definitions The set of all subgradients of f at x is called the subdifferential of f at x . ƒ Denoted by ∂f ( x ) ? Is ∂f ( x ) a convex set? Thm: η ∂f ( x ) iff ƒ f 0 ( x, d ) η T d d ∈ < n January 27, 2003 Stochastic Programming – Lecture 5 Slide 5
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Optimality Conditions We are interested in determining conditions under which we can verify that a solution is optimal. To KISS, we will (for now) focus on minimizing functions that are ƒ One-dimensional ƒ Continuous ( | f ( a ) - f ( b ) | ≤ L | a - b | ) ƒ Differentiable Recall: a function f ( x ) is convex on a set S if for all a S and b S, f ( λa + (1 - λ ) b ) λf ( a ) + (1 - λ ) b . January 27, 2003 Stochastic Programming – Lecture 5 Slide 6
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Why do we care? Because they are important Because Prof. Linderoth says so! Many optimization algorithms work to find points that satisfy these conditions When faced with a problem that you don’t know how to handle, write down the optimality conditions Often you can learn a lot about a problem, by examining the properties of its optimal solutions. January 27, 2003 Stochastic Programming – Lecture 5 Slide 7
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Preliminaries Call the following problem P: z * = min f ( x ) : x S Def: Any point x * S that gives a value of f ( x * ) = z * is the global minimum of P.
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  • Spring '08
  • Linderoth
  • Optimization, stochastic programming, Programming – Lecture

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