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  • exposé - matière potentielle : application
  • expression écrite - matière potentielle : annotations
  • exposé - matière potentielle : the print
  • expression écrite - matière potentielle : the note
  • exposé
  • expression écrite
1 When WordHoard met Pliny: Annotation, context and application. John Bradley, Senior Lecturer, Department of the Digital Humanities, King's College London One characteristic of new technology is that it takes time to understand all the new affordances the technology provides. The earliest printers tried first to produce books that looked as much like manuscripts as possible but later discovered that print had both possibilities and requirements that were not conceived of in the pre-print era.
  • kind of annotation
  • pliny
  • personal annotation
  • dh development community
  • digital humanities
  • annotation
  • objects
  • application
  • software
  • work

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Nombre de lectures 37
Signaler un problème

Outline
Optimality conditions
Algorithms
Gradient-based algo
Derivative-free algorithms
Lecture 2: Unconstrained Optimization
Kevin Carlberg
Stanford University
July 28, 2009
Kevin Carlberg Lecture 2: Unconstrained OptimizationOutline
Optimality conditions
Algorithms
Gradient-based algo
Derivative-free algorithms
1 Optimality conditions
Univariate minimization
Multivariate
2 Algorithms
3 Gradient-based algorithms
Line search methods
Descent directions
Trust region methods
Global optimization
Computation of gradients
4 Derivative-free algorithms
Categorization
Genetic Algorithm
Kevin Carlberg Lecture 2: Unconstrained OptimizationOutline
Optimality conditions
Algorithms
Gradient-based algo
Derivative-free algorithms
Unconstrained optimization
This lecture considers unconstrained optimization
minimize f (x)
nx2R
Things become signi cantly more complicated with
constraints!
Kevin Carlberg Lecture 2: Unconstrained OptimizationOutline
Optimality conditions
Univariate minimization
Algorithms
Multivariate
Gradient-based algo
Derivative-free algorithms
Univariate minimization
Consider the unconstrained minimization of a function in one
dimension
minimize f (x) (1)
x2R
In this class, we assume all functions are \su ciently smooth"
(twice-continuously di erentiable)
f(x)
x
What is a solution to (1)?
Kevin Carlberg Lecture 2: Unconstrained OptimizationOutline
Optimality conditions
Univariate minimization
Algorithms
Multivariate
Gradient-based algo
Derivative-free algorithms
What is a solution?
f(x)
x
Global minimum: A point x satisfying f (x ) f (x)8x2R
Strong local minimum: A neighborhoodN of x exists such
that f (x )< f (x)8x2N .
Weak local minima A a neighborhoodN of x exists such
that f (x ) f (x)8x2N .
Kevin Carlberg Lecture 2: Unconstrained OptimizationOutline
Optimality conditions
Univariate minimization
Algorithms
Multivariate
Gradient-based algo
Derivative-free algorithms
Convexity
For convex objective functions in one variable,
f (x +y)f (x) +f (y)
f(x) f(x)
x x
In this case, any local minimum is a global minimum!
Kevin Carlberg Lecture 2: Unconstrained OptimizationOutline
Optimality conditions
Univariate minimization
Algorithms
Multivariate
Gradient-based algo
Derivative-free algorithms
Optimality conditions for univariate minimization
Theorem (Necessary conditions for a weak local minimum)
0 A1. f (x ) = 0 (stationary point)
00 A2. f (x ) 0.
Theorem (Su cient conditions for a strong local minimum)
f (x)
0 B1. f (x ) = 0 (stationary point) and
00 B2. f (x )> 0.
x
A1 A2
f (x)
B1, B2
x
Kevin Carlberg Lecture 2: Unconstrained OptimizationOutline
Optimality conditions
Univariate minimization
Algorithms
Multivariate
Gradient-based algo
Derivative-free algorithms
Optimality conditions for univariate minimization
A1 Maxima
A2 Saddle points
Weak minima
B1, B2
Strong minima
Kevin Carlberg Lecture 2: Unconstrained Optimization