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LECTURE 2: MOTIVATION, INTUITION, SPECULATION AND THEORIZATION
1. Basic terminologies 2. Review of background knowledge
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Preliminaries • Euclidean Space An n-dimensional Euclidean space is a space of elements specified by n coordinates in real numbers with emphasis on the structure of Euclidean geometry, such as distance and angle, using the standard “inner product” operation. It is a real vector space with an inner product operation. • Notation Commonly used 2-norm of a vector x in is denoted by Sometimes by |x| for convenience.
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General Aspects
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Unconstrained optimization problem
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Basic terminologies
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Notations
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Neighborhood
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Open sets
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Closed sets
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Open and closed sets
( )Note: ( ) ( )
( ) ( )bdry S S int S
acc S S int S= −
= −
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Relatively open and close sets
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Cones
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Bounded sets
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Compact sets
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Compact sets
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Convex sets
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Convex hulls
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Extreme points
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Characterization of convex sets
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Separation and supporting hyperplanes
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Separation and supporting hyperplanes
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Feasible directions
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Continuous functions
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Characterization of continuous functions
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Optimization of continuous functions
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Differentiable functions
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Differentiable functions
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Differentiable functions
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Taylor Theorem – 1 dimensional case
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Approximation
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Taylor Theorem – n dimensional case
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Taylor Theorem
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Approximation
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Approximation
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Approximation
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Big O and small O