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Convex Functions (II) Lijun Zhang zlj@nju.edu.cn - - PowerPoint PPT Presentation
Convex Functions (II) Lijun Zhang zlj@nju.edu.cn - - PowerPoint PPT Presentation
Convex Functions (II) Lijun Zhang zlj@nju.edu.cn http://cs.nju.edu.cn/zlj Outline The Conjugate Function Quasiconvex Functions Log-concave and Log-convex Functions Convexity with Respect to Generalized Inequalities Summary
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Outline
The Conjugate Function Quasiconvex Functions Log-concave and Log-convex Functions Convexity with Respect to Generalized Inequalities Summary
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Conjugate Function
- Its conjugate function is
∗ ∗
∗ is always convex
∗ ∈
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Conjugate Function
- Its conjugate function is
∗ ∈
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Conjugate examples
Affine function
∗ ∈𝐒
∗ ∗
Negative logarithm
∗ ∈𝐒
∗
- ∗
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Conjugate examples
Exponential
-
∗ ∈𝐒
-
∗
- ∗
Negative entropy
∗ ∈𝐒
∗ ∗
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Conjugate examples
Inverse
∗ ∈𝐒
∗
- ∗
/
Strictly convex quadratic function
-
∗ ∈𝐒
-
∗
- ∗
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Conjugate examples
Log-determinant
-
∗ ∈𝐓
-
∗
- ∗
- Indicator function
- is not
necessarily convex
- ∗
∈
-
- ∗
is the support function of the set
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Conjugate examples
Norm
with dual norm ∗
∗ ∈𝐒
-
∗ ∗ ∗
Norm squared
- with dual norm
∗
∗ ∈𝐒
-
∗
- ∗
- ∗
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Basic properties
Fenchel’s inequality
∗ ∗
-
∗ ∈𝐒
-
- Conjugate of the conjugate
is convex and closed
∗∗
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Basic properties
Differentiable functions
𝑔 is convex and differentiable, dom 𝑔 𝐒
∗ ∈𝐒
-
∗
- ∗
∗ ∗ ∗ ∗ ∗ ∗
𝑦∗ 𝛼𝑔 𝑧
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Basic properties
Scaling with affine transformation
∗ ∗
-
is nonsingular
- ∗
∗
- ∗
- ∗
Sums of independent functions
- are convex
∗
- ∗
- ∗
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Outline
The Conjugate Function Quasiconvex Functions Log-concave and Log-convex Functions Convexity with Respect to Generalized Inequalities Summary
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Quasiconvex functions
Quasiconvex
𝑔: 𝐒 → 𝐒 𝑇 𝑦 ∈ dom 𝑔 𝑔 𝑦 𝛽, ∀𝛽 ∈ 𝐒 is convex
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Quasiconvex functions
Quasiconvex
𝑔: 𝐒 → 𝐒 𝑇 𝑦 ∈ dom 𝑔 𝑔 𝑦 𝛽, ∀𝛽 ∈ 𝐒 is convex
Quasiconcave
𝑔 is quasiconvex ⇒ 𝑔 is quasiconcave
Quasilinear
𝑔 is quasiconvex and quasiconcave ⇒ 𝑔 is quasilinear
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Examples
Some example on
Logarithm:
- n
- Ceiling function：
Linear-fractional function
-
- is convex
is Quasilinear
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Basic properties
Jensen’s inequality for quasiconvex functions
is quasiconvex is convex and
𝑔 𝜄𝑦 1 𝜄 𝑧 max 𝑔 𝑦 , 𝑔 𝑧
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Basic properties
Condition
𝑔 is quasiconvex ⇔ its restriction to any line intersecting its domain is quasiconvex
Quasiconvex functions on
A continuous function 𝑔: 𝐒 → 𝐒 is quasiconvex ⇔
- ne of the following conditions holds
- 𝑔 is nondecreasing
- 𝑔 is nonincreasing
- ∃𝑑 ∈ dom 𝑔, ∀𝑢 ∈ dom 𝑔, 𝑢 𝑑, 𝑔 is
nonincreasing, and 𝑢 𝑑, 𝑔 is nondecreasing
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Differentiable quasiconvex functions
First-order conditions
𝑔 is differentiable 𝑔 is quasiconvex ⇔ dom 𝑔 is convex,∀𝑦, 𝑧 ∈
dom 𝑔, 𝑔 𝑧 𝑔 𝑦 ⇒ 𝛼𝑔 𝑦 𝑧 𝑦 0
It is possible that 𝛼𝑔 𝑦 0, but 𝑦 is not a global minimizer of 𝑔.
Second-order conditions
𝑔 is twice differentiable ∀𝑦 ∈ dom 𝑔, ∀𝑧 ∈ 𝐒, 𝑧𝛼𝑔 𝑦 0 ⇒ 𝑧𝛼𝑔 𝑦 𝑧 0 ⇒ 𝑔 is quasiconvex
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Operations that preserve quasiconvexity
Nonnegative weighted maximum
𝑔
is quasicovex, 𝑥 0 ⇒ 𝑔
max𝑥𝑔
, … , 𝑥𝑔 is quasiconvex
𝑦, 𝑧 is quasiconvex in 𝑦 for each 𝑧, 𝑥 𝑧 0 ⇒ 𝑔 𝑦 sup
∈
𝑥 𝑧 𝑔 𝑦, 𝑧 is quasiconvex
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Operations that preserve quasiconvexity
Composition
: 𝐒 → 𝐒 is quasiconvex, ℎ: 𝐒 → 𝐒 is nondecreasing ⇒ 𝑔 ℎ ◦ is quasiconvex 𝑔: 𝐒 → 𝐒 is quasiconvex ⇒ 𝑦 𝑔𝐵𝑦 𝑐 is quasiconvex 𝑔: 𝐒 → 𝐒 is quasiconvex ⇒ 𝑦 𝑔
- is
quasiconvex,dom 𝑦|𝑑𝑦 𝑒 0, 𝐵𝑦 𝑐/𝑑 𝑦 𝑒 ∈ dom 𝑔
Minimization
𝑔𝑦, 𝑧 is quasicovex in 𝑦 and 𝑧, 𝐷 is a convex set ⇒ 𝑦 inf
∈ 𝑔𝑦, 𝑧 is quasiconvex
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Outline
The Conjugate Function Quasiconvex Functions Log-concave and Log-convex Functions Convexity with Respect to Generalized Inequalities Summary
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Log-concave and log-convex functions
Definition
- is
concave (convex) is log-concave (convex)
Condition
- is log-
concave
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Examples
- is
log-concave
- is log-
convex, is log-concave
- /
- is log-concave
- is log-convex for
and
are log-concave on
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Properties
Twice differentiable log convex/concave functions
𝑔 is twice differentiable,dom 𝑔 is convex 𝛼 log 𝑔𝑦
- 𝛼𝑔 𝑦
- 𝛼𝑔 𝑦 𝛼𝑔 𝑦
𝑔 is log convex ⇔ 𝑔 𝑦 𝛼𝑔 𝑦 ≽ 𝛼𝑔 𝑦 𝛼𝑔 𝑦 𝑔 is log concave ⇔ 𝑔 𝑦 𝛼𝑔 𝑦 ≼ 𝛼𝑔 𝑦 𝛼𝑔 𝑦
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Outline
The Conjugate Function Quasiconvex Functions Log-concave and Log-convex Functions Convexity with Respect to Generalized Inequalities Summary
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Convexity with respect to a generalized inequality
- convex
is a proper cone with associated
generalized inequality
-
- is -convex if
-
- is
−convex if
- m
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Examples
Componentwise Inequality
-
- is convex with respect to
componentwise inequality Each is a convex function
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Examples
Matrix Convexity
- is convex with respect to
matrix inequality
𝑔 𝜄𝑦 1 𝜄 𝑧 ≼ 𝜄𝑔 𝑦 1 𝜄 𝑔𝑧
- is matrix convex
is matrix convex on
- for
- r
, and matrix concave for
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Convexity with respect to generalized inequalities
Dual characterization of -convexity
A function 𝑔 is (strictly) 𝐿-convex ⇔ For every 𝑥 ≽∗ 0, the real-valued function 𝑥𝑔 is (strictly) convex in the ordinary sense.
Differentiable -convex functions
A differentiable function 𝑔 is 𝐿-convex ⇔ dom 𝑔 is convex, ∀𝑦, 𝑧 ∈ dom 𝑔, 𝑔 𝑧 ≽ 𝑔 𝑦 𝐸𝑔 𝑦 𝑧 𝑦 A differentiable function 𝑔 is strictly 𝐿-convex ⇔ dom 𝑔 is convex, ∀𝑦, 𝑧 ∈ dom 𝑔, 𝑦 𝑧, 𝑔 𝑧 ≻ 𝑔 𝑦 𝐸𝑔𝑦𝑧 𝑦
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Convexity with respect to generalized inequalities
Composition theorem
: 𝐒 → 𝐒 is 𝐿-convex, ℎ: 𝐒 → 𝐒 is convex, and ℎ (the extended-value extension of ℎ) is 𝐿- nondecreasing ⇒ ℎ ◦ is convex.
Example
: 𝐒 → 𝐓, 𝑌 𝑌𝐵𝑌 𝐶𝑌 𝑌𝐶 𝐷 is convex, where 𝐵 ≽ 0, 𝐶 ∈ 𝐒 and 𝐷 ∈ 𝐓 ℎ: 𝐓 → 𝐒, ℎ 𝑍 log det𝑍 is convex and increasing on dom ℎ 𝐓
-
𝑔 𝑌 log det𝑌𝐵𝑌 𝐶𝑌 𝑌𝐶 𝐷 is convex on dom 𝑔 𝑌 ∈ 𝐒|𝑌𝐵𝑌 𝐶𝑌 𝑌𝐶 𝐷 ≺ 0
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Monotonicity with respect to a generalized inequality
is a proper cone with
associated generalized inequality
-
- is -nondecreasing if
-
- is -increasing if
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