Lecture 15' Convexity and Concavity'

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Lecture 15. Convexity and Concavity.

• Learning objectives. By the end of this lecture
  you should
• Know the definition of a convex function, a
  convex set and a concave function.
• Understand the relationship between a convex set
  and a convex function.
• Introduction Optimization.
• In the final section of the term we look at the
  second-order conditions for a maximum or minimum.
• Todays lecture is a preliminary look at some
  important concepts

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1. Introduction.

• First order conditions yield a solution that is
  unique and a maximum in the first and fourth
  cases. In the second case the conditions yield a
  minimum in the third case there is no unique
  solution to the first order conditions.
• We seek conditions that ensure that a solution to
  the first order conditions is a global maximum.

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2. Preliminaries convex combinations.

• A convex combination of two points is a point
  that lies on the line between them.
• More formally, consider two points x and x.
• A convex combination x? ?x (1-?)x for 0 ?1
• E.g. ?0 then x? x ?1 then x? x
• A strictly convex combination x? ?x (1-?)x
  for 0 lt?lt1

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3. Convex sets.

• A convex set, X, is such that for any two
  elements of the set, x and x any convex
  combination of them is also a member of the set.
• More formally, X is convex if for all x and x e
  X, and 0 ?1, x? ?x (1-?)x e X.
• Sometimes X is described as strictly convex if
  for any 0 lt ? lt1, x? is in the interior of X
  (i.e. not on the edges)
• e.g. convex but not strictly convex

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3b. Convex sets.
U
U(x) U
p1x1p2x2 m
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3c. Non-Convex sets.
U
U(x) U
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4. Exercise which of these sets is convex?

• The set of real numbers.
• (x1,x2) x1x2 2
• (x1,x2) x1x2 2

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5. Convex functions.

• Convex functions are defined as follows.
• f(x) is convex if given any x, x , x? ?x
  (1-?)x where 0 ?1, f(x?) ?f(x)
  (1-?)f(x)
• loosely the line joining x and x lies above the
  function, f.

?f(x) (1-?)f(x)
f(x)
x
x
x
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5. Convex functions.

• f(x) is convex if given any x, x , x? ?x
  (1-?)x where 0 ?1, f(x?) ?f(x)
  (1-?)f(x)

x
x
x
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5. Strictly convex functions.

• f(x) is strictly convex if given any x, x , x?
  ?x (1-?)x where 0 lt?lt1, f(x?) lt ?f(x)
  (1-?)f(x)
• Strict convexity implies convexity.

convex, but not strictly convex
strictly convex and convex
x
x
x
not convex or strictly convex
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6. Quasi-convex functions.

• f(x) is quasi-convex if given any y the set S
  x f(x) y is convex.
• Note that a convex function is also quasi-convex.
  
• The third picture shows that the opposite isnt
  true.
• Were not going to say much more about
  quasi-convex, but it is the feature which
  guarantees a unique minimum.

y
S
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7. convex and quasi-convex.

• f(x) is strictly convex if given any x, x , x?
  ?x (1-?)x where 0 lt?lt1, f(x?) lt ?f(x)
  (1-?)f(x)
• f(x) is convex if given any x, x , x? ?x
  (1-?)x where 0 ?1, f(x?) ?f(x)
  (1-?)f(x)
• f(x) is quasi-convex if given any y the set S
  x f(x) y is convex.
• a convex function is also quasi-convex.
• Proof.
• Suppose f is convex and take some y.
• Consider the set S x f(x) y .
• Take x, x e S and any ? where 0 ?1 and so
  construct x? ?x (1-?)x .
• S is convex if x e S, in other words if f(x)
  y. But by convexity f(x) ?f(x) (1-?)f(x)
  and by the definition of x and x, f(x) y and
  f(x) y,
• So f(x) ?f(x) (1-?)f(x) ?y (1-?)y y
• i.e. f(x) y
• So x e S and therefore f is quasi convex

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8. Concave functions.

• f(x) is concave if given any x, x , x? ?x
  (1-?)x where 0 ?1, f(x?) ?f(x)
  (1-?)f(x)
• loosely the line joining x and x lies below the
  function, f.

f(x)
?f(x) (1-?)f(x)
x
x
x
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5. Strictly concave functions.

• f(x) is strictly concave if given any x, x , x?
  ?x (1-?)x where 0 lt?lt1,
  f(x?) gt ?f(x) (1-?)f(x)
• Strict concavity implies concavity.

concave, but not strictly concave
strictly concave and concave
x
x
x
not concave or strictly concave
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6. Quasi-concave functions.

• f(x) is quasi-concave if given any y the set S
  x f(x) y is convex.
• Note that a concave function is also
  quasi-concave.
• The third picture shows that the opposite isnt
  true.
• Were not going to say much more about
  quasi-concave, but it is the feature which
  guarantees a unique maximum.

y
S
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Concave sets.

• Draw a concave set here.