Introduction to Network Utility Maximization (NUM)

Richard T. B. Ma School of Computing

National University of Singapore

CS 4226: Internet Architecture

Outline

Utility maximization and convex optimization Assumptions of utility functions Intuitive solutions of utility maximization Convex set, function and optimization framework

Utility maximization and fairness Max-min, proportional fairness and 𝛼-fairness

Karush–Kuhn–Tucker (KKT) conditions Lagrange multiplier and shadow price

How Does Utility Look Like?

Assumptions on utility function

We assume that each 𝑈𝑖 ⋅ is defined over 0, +∞ continuously differentiable non-decreasing (not critical) concave (most intriguing)

Interpretations and reasons non-negative resource allocation the more you get, the happier you are marginal happiness decreases (diminishing return) guarantee existence of optimal solution

Utility maximization

Multiple flows share a resource:

Maximize 𝑈1 𝑥1 + 𝑈2 𝑥2 subject to: 𝑥1 + 𝑥2 ≤ 𝐶, 𝑥1 ≥ 0 and 𝑥2 ≥ 0

Maximize 𝑈 𝒙 = � 𝑈𝑖 𝑥𝑖𝑛

𝑖=1

subject to: � 𝑥𝑖𝑛

𝑖=1≤ 𝐶 and 𝒙 ≥ 𝟎

𝒙𝟏 𝒙𝟐

𝑪

How does 𝒙∗ look like? 𝒙𝟏 𝒙𝟐

𝑪

𝟏 𝟐

𝟑 𝟒

𝟓 𝟔

𝟕 𝟖

𝟗

𝑼𝟏

𝟏 𝟐

𝟑 𝟒

𝟓 𝟔

𝟕 𝟖

𝟗

𝟎 𝟏 𝟐 𝟑 𝟒 𝟓 𝟔 𝟕 𝟖

𝑼𝟐

𝟏 𝟐

𝟑 𝟒

𝟎 𝟏 𝟐 𝟑 𝟒 𝟓 𝟔 𝟕 𝟖

𝑼𝟏′

𝟏 𝟐

𝟑 𝟒

𝟎 𝟏 𝟐 𝟑 𝟒 𝟓 𝟔 𝟕 𝟖

𝑼𝟐′

Sort marginal utilities by a descending order

Assign resource unit by unit

𝟏 𝟐

𝟑 𝟒

𝟎 𝟏 𝟐 𝟑 𝟒 𝟓 𝟔 𝟕 𝟖 𝟗

𝑼′

Utility maximization

Maximize 𝑈 𝒙 = � 𝑈𝑖 𝑥𝑖𝑛

𝑖=1

subject to: � 𝑥𝑖𝑛

𝑖=1≤ 𝐶 and 𝒙 ≥ 𝟎

A necessary condition: 𝒙∗ = 𝑥1∗,⋯ , 𝑥𝑛∗ is optimal only if for any 𝑖

𝑈𝑖′ 𝑥𝑖∗ ≥ 𝑈𝑗′ 𝑥𝑗∗ ∀𝑗 unless 𝑥𝑖∗ = 0

𝒙𝟏 𝒙𝟐

𝑪

Convex set and convex function

A set 𝑺 is convex iff ∀𝒙,𝒚 ∈ 𝑺 and 𝛼 ∈ 0,1 , 1 − 𝛼 𝒙 + 𝛼𝒚 ∈ 𝑺

A function 𝑓 is convex iff for any 𝛼 ∈ 0,1

𝑓 1 − 𝛼 𝒙 + 𝛼𝒚 ≤ 1 − 𝛼 𝑓 𝒙 + 𝛼𝑓 𝒚

if and only if its epigraph is a convex set

convex set non-convex set convex function

Concave function and property

A function 𝑓 is concave if for any 𝛼 ∈ 0,1

𝑓 1 − 𝛼 𝒙 + 𝛼𝒚 ≥ 1 − 𝛼 𝑓 𝒙 + 𝛼𝑓 𝒚

Properties of concave functions: A function 𝑓 𝒙 is concave over a convex set iff − 𝑓 𝒙 is a convex function over the set.

Sum of concave functions is also concave

A differentiable function 𝑓 𝑥 is concave if 𝑓′ 𝑥 is monotonically decreasing: a concave function has a decreasing slope

Optimality for convex optimization

Minimize 𝑓 𝒙 subject to: 𝒙 ∈ 𝑺 ⊂ 𝑹𝒏

If 𝑺 is convex and 𝑓 𝒙 is convex over 𝑺 any local minimum of 𝑓(·) is a global minimum if 𝑓 ⋅ is strictly convex, then there exists at

most one global minimum

Necessary condition: if 𝒙∗ is optimal, then

𝛻𝑓 𝒙∗ 𝑻 𝒙 − 𝒙∗ ≥ 0, ∀𝒙 ∈ 𝑺. it becomes sufficient under convex optimization

Our utility maximization problem

Maximize 𝑈 𝒙 ⇔ Minimize − 𝑈 𝒙 subject to: 𝒙 ∈ 𝑺 = 𝒙:𝑔𝑖 𝒙 ≤ 0,∀𝑖 = 1,⋯ ,𝑚

Functions 𝑔𝑖 𝒙 are linear, so 𝑺 is convex 𝑈 𝒙 is concave over 𝑺

any local maximum of 𝑈(·) is a global maximum strictly concave 𝑈 ⋅ unique global maximum

Sufficient condition: 𝒙∗ is optimal if

𝛻𝑈 𝒙∗ 𝑻 𝒙 − 𝒙∗ ≤ 0, ∀𝒙 ∈ 𝑺.

Local and global maxima

Outline

Utility maximization and convex optimization Assumptions of utility functions Intuitive solutions of utility maximization Convex set, function and optimization framework

Utility maximization and fairness Max-min, proportional fairness and 𝛼-fairness

Karush–Kuhn–Tucker (KKT) conditions Lagrange multiplier and shadow price

A different point of view

utility: a measure of user satisfaction when getting a data rate from the network need to know users’ applications and utility

functions in practice

A new view: utility function is assigned to user in the network by a service provider with the goal of achieving a certain goal maximize efficiency maximize fairness

Different objectives

Maximize efficiency (aggregate data rate)

𝑈𝑖 𝑥𝑖 = 𝑥𝑖

Minimize potential delay

𝑈𝑖 𝑥𝑖 = −1𝑥𝑖

Proportional fairness

𝑈𝑖 𝑥𝑖 = log 𝑥𝑖

How about max-min fairness?

Proportional fairness

Definition: a feasible allocation solution 𝒙∗ is proportional fair if for any feasible 𝒙,

�𝑥𝑖 − 𝑥𝑖∗

𝑥𝑖∗

𝑛

𝑖=1

≤ 0.

for any allocation, the sum of proportional changes in the utilities will be non-positive.

if some rate increases, there is another user whose rate will decrease and the proportion by which it decreases is larger.

Proportional fairness

The objective function becomes

Maximize 𝑈 𝒙 = �𝑈𝑖 𝑥𝑖

𝑛

𝑖=1

= � log 𝑥𝑖

𝑛

𝑖=1

𝑈 𝒙 is strictly concave sufficient condition: if 𝒙∗ is optimal, then

𝛻𝑈 𝒙∗ 𝑻 𝒙 − 𝒙∗ = �𝑥𝑖 − 𝑥𝑖∗

𝑥𝑖∗

𝑛

𝑖=1

≤ 0, ∀𝒙 ∈ 𝑺.

A unified form of utility

Consider the form of 𝛼-fairness : 𝑈𝑖 𝑥𝑖 =

𝑥𝑖1−𝛼

1 − 𝛼 for some 𝛼 ≥ 0

maximize aggregate rate (efficiency) 𝛼 = 0 minimize potential delay 𝛼 = 2 how about 𝛼 = 1? (not defined, use L’Hospital)

𝑈𝑖 𝑥𝑖 = �𝑥𝑖1−𝛼

1 − 𝛼if 𝛼 ≠ 1

log 𝑥𝑖 if 𝛼 = 1

how about 𝛼 =→ +∞? why does it look so strange? how to remember?

Maximize 𝑈 𝒙 = �𝑈𝑖 𝑥𝑖

𝑛

𝑖=1

= �𝑥𝑖1−𝛼

1 − 𝛼

𝑛

𝑖=1

𝜵𝑼 𝒙∗ =𝒙𝟏−𝜶⋮

𝒙𝒏−𝜶; 𝜵𝑼 𝒙∗ 𝑻 𝒙 − 𝒙∗ = �

𝒙𝒊 − 𝒙𝒊∗

𝒙𝒊∗ 𝜶

𝒏

𝒊=𝟏

≤ 𝟎

Consider a flow 𝑠

�𝑥𝑖 − 𝑥𝑖∗

𝑥𝑖∗𝛼

𝑖:𝑥𝑖∗≤𝑥𝑠∗

+𝑥𝑠 − 𝑥𝑠∗

𝑥𝑠∗ 𝛼 + �𝑥𝑗 − 𝑥𝑗∗

𝑥𝑗∗𝛼

𝑗:𝑥𝑗∗>𝑥𝑠∗

≤ 0

⇒ � 𝒙𝒊 − 𝒙𝒊∗𝒙𝒔∗

𝒙𝒊∗𝜶

𝒊:𝒙𝒊∗≤𝒙𝒔∗

+ 𝒙𝒔 − 𝒙𝒔∗ + � 𝒙𝒋 − 𝒙𝒋∗𝒙𝒔∗

𝒙𝒋∗𝜶

𝒋:𝒙𝒋∗>𝒙𝒔∗

≤ 𝟎

need to decrease users who got less resource

Outline

Utility maximization and convex optimization Assumptions of utility functions Intuitive solutions of utility maximization Convex set, function and optimization framework

Utility maximization and fairness Max-min, proportional fairness and 𝛼-fairness

Karush–Kuhn–Tucker (KKT) conditions Lagrange multiplier and shadow price

Karush–Kuhn–Tucker (KKT) conditions

we know a necessary/sufficient condition:

𝛻𝑈 𝒙∗ 𝑻 𝒙 − 𝒙∗ ≤ 0, ∀𝒙 ∈ 𝑺. if the constraint set can be written as

subject to: 𝑔𝑖 𝒙 ≤ 0,∀𝑖 = 1,⋯ ,𝑚

KKT: if 𝒙∗ is optimal, for each 𝑔𝑖 ⋅ , there is a Lagrange multiplier 𝜇𝑖 ≥ 0 such that

𝛻𝑈 𝒙∗ = � 𝜇𝑖𝛻𝑔𝑖 𝒙∗𝑚

𝑖=1𝑔𝑖 𝒙∗ ≤ 0, ∀𝑖 = 1,⋯ ,𝑚𝜇𝑖𝑔𝑖 𝒙∗ = 0, ∀𝑖 = 1,⋯ ,𝑚

Unconstrained problems

Maximize 𝑈 𝒙 subject to: 𝑔𝑖 𝒙 ≤ 0,∀𝑖 = 1,⋯ ,𝑚

the KKT necessary conditions 𝛻𝑈 𝒙∗ = � 𝜇𝑖𝛻𝑔𝑖 𝒙∗

𝑚

𝑖=1= 𝟎

𝑔𝑖 𝒙∗ ≤ 0, ∀𝑖 = 1,⋯ ,𝑚𝜇𝑖 ≥ 0, ∀𝑖 = 1,⋯ ,𝑚

𝜇𝑖𝑔𝑖 𝒙∗ = 0, ∀𝑖 = 1,⋯ ,𝑚

⇒𝜕𝑓 𝒙∗

𝑥𝑖= 0 ∀𝑖 = 1,⋯ ,𝑛

Utility maximization

Max 𝑈 𝒙 = 𝑈1 𝑥1 + 𝑈2 𝑥2 ; s.t. 𝑥1 + 𝑥2 ≤ 𝐶

𝛻𝑈 𝒙∗ = 𝜇𝛻𝑔 𝒙∗𝑔 𝒙∗ = 𝑥1∗ + 𝑥2∗ − 𝐶 ≤ 0

𝜇 ≥ 0𝜇𝑔 𝒙∗ = 𝜇 𝑥1∗ + 𝑥2∗ − 𝐶 = 0

Either 1) 𝜇 = 0 & 𝛻𝑈 𝒙∗ = 0 or 2) 𝜇 > 0 &

𝛻𝑈 𝒙∗ = 𝜇𝛻𝑔 𝒙∗ ⇔ �𝑈1′ 𝑥1∗ = 𝜇𝑈2′ 𝑥2∗ = 𝜇

𝒙𝟏 𝒙𝟐

𝑪

Lagrange multiplier and shadow price

Max 𝑈 𝒙 = 𝑈1 𝑥1 + 𝑈2 𝑥2 ; s.t. 𝑥1 + 𝑥2 ≤ 𝐶

the Lagrange multiplier 𝜇 corresponds to the marginal utility achieved at the maximum

“shadow price”: if we can add additional resource, how much can we increase utility

Either 1) 𝜇 = 0 & 𝛻𝑈 𝒙∗ = 0 or 2) 𝜇 > 0 &

𝛻𝑈 𝒙∗ = 𝜇𝛻𝑔 𝒙∗ ⇔ �𝑈1′ 𝑥1∗ = 𝜇𝑈2′ 𝑥2∗ = 𝜇

𝒙𝟏 𝒙𝟐

𝑪

Utility maximization

𝛻𝑈 𝒙∗ = 𝜇𝛻𝑔 𝒙∗ + 𝜇1−10 + 𝜇2

0−1

𝑔 𝒙∗ = 𝑥1∗ + 𝑥2∗ − 𝐶 ≤ 0;−𝑥1 ≤ 0;−𝑥2 ≤ 0𝜇 ≥ 0; 𝜇1 ≥ 0; 𝜇2 ≥ 0

𝜇𝑔 𝒙∗ = 𝜇 𝑥1∗ + 𝑥2∗ − 𝐶 = 𝜇1𝑥1∗ = 𝜇2𝑥2∗ = 0

for the non-trivial case of 𝜇 > 0

⇒ �𝑈1′ 𝑥1∗ = 𝜇 − 𝜇1𝑈2′ 𝑥2∗ = 𝜇 − 𝜇2

𝒙𝟏 𝒙𝟐

𝑪 Max 𝑈 𝒙 = 𝑈1 𝑥1 + 𝑈2 𝑥2 ; s.t. 𝑥1 + 𝑥2 ≤ 𝐶, 𝑥1 ≥ 0 and 𝑥2 ≥ 0

Network Utility

Max 𝑈 𝒙 = 𝑈1 𝑥1 + 𝑈2 𝑥2 + 𝑈3 𝑥3 ; s.t. 𝑥1 + 𝑥2 ≤ 𝐶1, 𝑥2 + 𝑥3 ≤ 𝐶2 and 𝒙 ≥ 0

𝛻𝑈 𝒙∗ = 𝜇𝐶1110

+ 𝜇𝐶2101

+ 𝜇1−100

+ 𝜇20−10

+ 𝜇300−1

𝑥1 + 𝑥2 − 𝐶1 ≤ 0, 𝑥2 + 𝑥3 − 𝐶2 ≤ 0 and − 𝑥𝑖 ≤ 0𝜇𝐶1 ≥ 0; 𝜇𝐶2 ≥ 0; 𝜇1 ≥ 0; 𝜇2 ≥ 0; 𝜇3 ≥ 0

𝜇𝐶1 𝑥1∗ + 𝑥2∗ − 𝐶1 = 𝜇𝐶2 𝑥1

∗ + 𝑥3∗ − 𝐶2 = 𝜇𝑖𝑥𝑖∗ = 0

⇒ �𝑈1′ 𝑥1∗ = 𝜇𝐶1 + 𝜇𝐶2 − 𝜇1

𝑈2′ 𝑥2∗ = 𝜇𝐶1 − 𝜇2𝑈3′ 𝑥3∗ = 𝜇𝐶2 − 𝜇3

𝒙𝟏

𝒙𝟑

𝑪𝟐 𝑪𝟏

𝒙𝟐

Further topics

We have characterized the optimization solution. How can we find it? Different optimization algorithms

In practice, global information might not

be available. How can we solve it in a distributed manner? Primal-dual decomposition … another course Convergence of the algorithms

Further Readings F. P. Kelly, “Charging and rate control for

elastic traffic”, Euro. Trans. Telecommun., vol. 8, pp. 33–37, 1997.

F. P. Kelly, A. Maulloo, and D. Tan, “Rate control for communication networks: Shadow prices, proportional fairness and stability”, J. Oper. Res. Soc., 49(3), 1998.

Steven H. Low and David E. Lapsley, “Optimization Flow Control—I: Basic Algorithm and Convergence”, IEEE/ACM Transactions on Networking, 7(6), 1999.