Optimized Randomness! Why and How? Shuzhong Zhang Department of - - PowerPoint PPT Presentation

LNMB, The Netherlands, January 16 – 18, 2007 1

Optimized Randomness! Why and How?

Shuzhong Zhang

Department of Systems Engineering and Engineering Management The Chinese University of Hong Kong Based on joint works with my collaborators:

• S. He, Z. Luo, J. Nie, N. Sidiropoulos, P. Tseng, J. Xie

32nd Conference on the Mathematics of Operations Research ‘De Werelt’, Lunteren, The Netherlands January 17, 2007

Shuzhong Zhang, The Chinese University of Hong Kong

LNMB, The Netherlands, January 16 – 18, 2007 2

An Example for Randomization

Zhi-Quan Luo, An Isotropic Universal Decentralized Estimation Scheme for a Bandwidth Constrained Ad Hoc Sensor Network. IEEE Journal on Selected Areas in Communications, 23 (4), 735 – 744, 2005.

Shuzhong Zhang, The Chinese University of Hong Kong

LNMB, The Netherlands, January 16 – 18, 2007 3

Data Transmission in Communication

• Ad hoc sensor network with K sensors.
• Each sensor observes a real data in [−U, U] independently.
• Each sensor sends back the data to the base-station.
• The base-station operates a least square estimation.

Shuzhong Zhang, The Chinese University of Hong Kong

LNMB, The Netherlands, January 16 – 18, 2007 4

Matters of Facts

• Sensors have weak batteries.
• The above scheme is an unbiased estimation.
• The statistical error is

U 2 K

Shuzhong Zhang, The Chinese University of Hong Kong

LNMB, The Netherlands, January 16 – 18, 2007 5

A Randomized Transmission Scheme!

• Each sensor observes the data in binary digits:

a1a2a3 · · · .

• Each sensor, say sensor k, independently tosses a coin to decide

which single binary digit to transmit: ξ = j with probability 1/2j, j = 1, 2, ...

• Then, sends this one bit data aξ back to the station.
• The base-station simply adds up all the received digits.
• This is an unbiased estimation, with statistical error

4U 2 K

Shuzhong Zhang, The Chinese University of Hong Kong

LNMB, The Netherlands, January 16 – 18, 2007 6

Another Example: Transmit Beamforming

A transmitter utilizes an array of n transmitting antennas to broadcast information within its service area to m radio receivers. The constraints model the requirement that the total received signal power at receiver i must be above a given threshold (normalized to 1);

• r, equivalently, a signal-to-noise ratio (SNR) condition for receiver i,

as commonly used in data communication. The objective is to minimize the total transmit power subject to individual SNR requirements (one at each receiver).

Shuzhong Zhang, The Chinese University of Hong Kong

LNMB, The Netherlands, January 16 – 18, 2007 7

Measuring Quality of Decisions

In a minimization problem, the quality measure of a solution x is a guaranteed bound θ such that v(x) ≤ θ × v∗ In this context, θ ≥ 1, e.g., θ = 150%. In a maximization problem, the quality measure of a solution x is a guaranteed bound θ such that v(x) ≥ θ × v∗ In this context, θ ≤ 1, e.g., θ = 85%. The value θ is called approximation ratio of a method.

Shuzhong Zhang, The Chinese University of Hong Kong

LNMB, The Netherlands, January 16 – 18, 2007 8

Transmit Beaforming: A Quadratic Model

The problem of transmit beamforming as stated before can be precisely modelled by homogeneous complex quadratic minimization: (QPc)min min zHCz s.t. zHQiz ≥ 1, i = 1, ..., m, z ∈ Cn.

Shuzhong Zhang, The Chinese University of Hong Kong

LNMB, The Netherlands, January 16 – 18, 2007 9

Homogeneous Quadratic Minimization

In general, let us consider: (QPr)min min xT Cx s.t. xT Qix ≥ 1, i = 1, ..., m, x ∈ ℜn. All data matrices are assumed to be positive semidefinite. This problem is clearly NP-hard. Also, (QPc)min is NP-hard.

Shuzhong Zhang, The Chinese University of Hong Kong

LNMB, The Netherlands, January 16 – 18, 2007 10

The SDP Relaxation

Consider the Semidefinite Programming relaxation for (QPr)min (SDPr)min min C • X s.t. Qi • X ≥ 1, i = 1, ..., m, X 0, and similarly for (QPc)min: (SDPc)min min C • Z s.t. Qi • Z ≥ 1, i = 1, ..., m, Z 0.

Shuzhong Zhang, The Chinese University of Hong Kong

LNMB, The Netherlands, January 16 – 18, 2007 11

A Randomized Approach to (QPr)min

But what to do with the solution of a relaxed problem? Let X∗ be the optimal solution of the SDP relaxation.

• 1. Generate a random vector ξ ∈ ℜn from the real-valued normal

distribution N(0, X∗).

• 2. Let

x∗(ξ) = ξ min1≤i≤m

• ξT Qiξ

.

Shuzhong Zhang, The Chinese University of Hong Kong

LNMB, The Netherlands, January 16 – 18, 2007 12

Approximation Ratio

• Theorem. (Luo, Sidiropoulos, Tseng, and Z.; 2005)

For m ≥ 2, we have v(QPrmin) ≤ 27m2 π v(SDPrmin). Moreover, there is an instance such that v(QPrmin) ≥ 2m2 π2 v(SDPrmin).

Shuzhong Zhang, The Chinese University of Hong Kong

LNMB, The Netherlands, January 16 – 18, 2007 13

The Complex Case: (QPc)min

• 1. Generate a random vector ξ ∈ Cn from the complex-valued

normal distribution Nc(0, Z∗).

• 2. Let

x∗(ξ) = ξ min1≤i≤m

• ξHQiξ

.

Shuzhong Zhang, The Chinese University of Hong Kong

LNMB, The Netherlands, January 16 – 18, 2007 14

Approximation Ratio

• Theorem. (Luo, Sidiropoulos, Tseng, and Z.; 2005)

For m ≥ 2, we have v(QPcmin) ≤ 8m · v(SDPcmin). Moreover, there is an instance such that v(QPcmin) ≥ m π2(2 + π/2)2 v(SDPcmin).

Shuzhong Zhang, The Chinese University of Hong Kong

LNMB, The Netherlands, January 16 – 18, 2007 15

A Homogeneous Quadratic Maximization Model

The following model is considered by Nemirvoski, Roos, and Terlaky (1999): (QPr)max max xT Cx s.t. xT Qix ≤ 1, i = 1, ..., m, x ∈ ℜn, where Qi 0, i = 1, ..., m.

Shuzhong Zhang, The Chinese University of Hong Kong

LNMB, The Netherlands, January 16 – 18, 2007 16

A Homogeneous Quadratic Maximization Model

The corresponding SDP relaxation is (SDPr)max max C • X s.t. Qi • X ≤ 1, i = 1, ..., m, X 0.

• Theorem. (Nemirovski, Roos, Terlaky; 1999)

It holds that v((QPr)max) ≥ 1 2 ln(2mµ)v((SDPr)max), where µ = min{m, maxi Rank(Qi)}.

Shuzhong Zhang, The Chinese University of Hong Kong

LNMB, The Netherlands, January 16 – 18, 2007 17

Complex Quadratic Maximization Problem

Consider (QPc)max max zHCz s.t. zHQiz ≤ 1, i = 1, ..., m, z ∈ Cn. The SDP relaxation is (SDPc)max max C • Z s.t. Qi • Z ≤ 1, i = 1, ..., m, Z 0.

Shuzhong Zhang, The Chinese University of Hong Kong

LNMB, The Netherlands, January 16 – 18, 2007 18

A Randomization Method for (QPcmax)

Similar as before, we propose to solve the problem as follows

• 1. Generate a random vector ξ ∈ Cn from the complex-valued

normal distribution Nc(0, Z∗).

• 2. Let

x∗(ξ) = ξ max1≤i≤m

• ξHQiξ

.

Shuzhong Zhang, The Chinese University of Hong Kong

LNMB, The Netherlands, January 16 – 18, 2007 19

Approximation Ratio

• Theorem. (Luo, Sidiropoulos, Tseng, and Z.; 2005)

For m ≥ 2, we have v(QPcmax) ≥ 1 4 ln(100µ)v(SDPcmax), where µ = m

i=1 min{rank(Qi), √m}.

Shuzhong Zhang, The Chinese University of Hong Kong

LNMB, The Netherlands, January 16 – 18, 2007 20

Indefinite Constraints

How about when some of the constraints are indefinite? There is no finite approximation ratio if more than one Qi’s are indefinite!

• Theorem. (He, Luo, Nie, and Z.; 2007)

If exactly one of Qi’s is indefinite, then v(QPrmin) ≤ 106m2 π v(SDPrmin).

• Theorem. (He, Luo, Nie, and Z.; 2007)

If exactly one of Qi’s is indefinite, then v(QPcmin) ≤ 2400m · v(SDPcmin).

Shuzhong Zhang, The Chinese University of Hong Kong

LNMB, The Netherlands, January 16 – 18, 2007 21

Indefinite Quadratic Maximization

The approximation ratio can be arbitrarily large, depending on the data matrices, if more than two Qi’s are indefinite.

• Theorem. (Ben-Tal, Nemirovski, Roos; 2002)

If one of the Qi’s is indefinite and C indefinite, then v(QPrmax) ≥ 1 2 log(16n2 mµ) v(SDPrmax), where µ = m

i=1 min{rank(Qi), √m}.

Shuzhong Zhang, The Chinese University of Hong Kong

LNMB, The Netherlands, January 16 – 18, 2007 22

New Bound for Indefinite Quadratic Maximization

• Theorem. (He, Luo, Nie, and Z.; 2007)

If one of the Qi’s is indefinite and C indefinite, then v(QPrmax) ≥ 1 2 log(174 mµ) v(SDPrmax), where µ = m

i=1 min{rank(Qi), √m}.

Shuzhong Zhang, The Chinese University of Hong Kong

LNMB, The Netherlands, January 16 – 18, 2007 23

A Key Ingredient

Ben-Tal, Nemirovski, Roos conjectured that Prob

• ξT Aξ ≤ E
• ξT Aξ
• ≥ 1

4, ∀ A symmetric matrix, for i.i.d. ξi’s, with Prob {ξi = +1} = Prob {ξi = −1} = 1

2.

But they only managed to show a lower bound of

1 8n2 .

We have established a lower bound of

1 87.

Shuzhong Zhang, The Chinese University of Hong Kong

LNMB, The Netherlands, January 16 – 18, 2007 24

Put Theory to Work: Simulation Results

10 20 30 40 50 60 70 80 90 100 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 m vqp

min / vsdp min

full rank with one indefinite maximum mean minimum

Minimization model: one indefinite constraint

Shuzhong Zhang, The Chinese University of Hong Kong

LNMB, The Netherlands, January 16 – 18, 2007 25 10 20 30 40 50 60 70 80 90 100 10 10

1

10

2

10

3

10

4

10

5

10

6

10

7

10

8

10

9

10

10

m vqp

min / vsdp min

full rank with 10% indefinite maximum mean minimum

Minimization model: many indefinite constraints

Shuzhong Zhang, The Chinese University of Hong Kong

LNMB, The Netherlands, January 16 – 18, 2007 26 10 20 30 40 50 60 70 80 90 100 1 1.1 1.2 1.3 1.4 1.5 1.6 m vsdp

max / vqp max

full rank with one indefinite maximum mean minimum

Maximization model: one indefinite constraint

Shuzhong Zhang, The Chinese University of Hong Kong

LNMB, The Netherlands, January 16 – 18, 2007 27 10 20 30 40 50 60 70 80 90 100 5 10 15 20 25 30 35 m vsdp

max / vqp max

full rank with 10 % indefinite maximum mean minimum

Maximization model: many indefinite constraints

Shuzhong Zhang, The Chinese University of Hong Kong

LNMB, The Netherlands, January 16 – 18, 2007 28

Randomized Investment: Theory and Practice

Portfolio Selection: Among n assets, select a set of good ones with right amounts. Classical notions (due to Markowitz, 1952):

• Return of assets

→ random variables

• Gain on investment

→ mean of the portfolio

• Risk on investment

→ variance of the portfolio

Shuzhong Zhang, The Chinese University of Hong Kong

LNMB, The Netherlands, January 16 – 18, 2007 29

The Mathematical Model

Let there be n assets, each with return rate ξi, i = 1, ..., n. Let the mean of ξi be ri = E[ξi], i = 1, ..., n. Let the covariance matrix of ξi, i = 1, ..., n, be Q. Let the initial budget be $1, and the target gain to be µ. Then the model could be minimize xT Qx subject to rT x ≥ µ eT x ≤ 1 x ∈ “a certain desirable constraint set” where e is the vector of all one’s.

Shuzhong Zhang, The Chinese University of Hong Kong

LNMB, The Netherlands, January 16 – 18, 2007 30

A Practical Issue

In real life, n can be a large number. A reasonable investor may only wish to handle a small set of assets; that is, “a small portfolio” — terminology used by Blog, Van der Hoek, Rinnooy Kan, and Timmer, 1983. How about we explicitly require to choose k out of n assets? The problem can be formulated as (MVs) minimize xT Qx subject to rT x ≥ µ, eT x ≤ 1,

n

• i=1

|sign(xi)| ≤ k.

Shuzhong Zhang, The Chinese University of Hong Kong

LNMB, The Netherlands, January 16 – 18, 2007 31

But this is an extremely difficult problem to solve! It is NP-hard: the only guaranteed method to solve the problem to

• ptimality is basically to enumerate all the possibilities.

If n = 100 and k = 50 then there are more than 1029 possible combinations. Remember that there are only about 1021 stars in the entire universe!

Shuzhong Zhang, The Chinese University of Hong Kong

LNMB, The Netherlands, January 16 – 18, 2007 32

Randomization Approach (I)

Let xi to be the quantity invested in asset i, i = 1, ..., n. Then, consider ηi :=    xi, with probability k

n

0, with probability 1 − k

n

The portfolio is now essentially ξT η, with its variance being k n 2 ×

• n − k

k

n

• i=1

(qii + r2

i )x2 i + xT Qx

• Shuzhong Zhang, The Chinese University of Hong Kong

LNMB, The Netherlands, January 16 – 18, 2007 33

The variable xi can be regarded as the seeds of randomization. Therefore, the seeds optimization problem can be cast as (RP1) minimization

n−k k n

• i=1

(qii + r2

i )x2 i + xT Qx

subject to rT x ≥ µ, eT x ≤ 1. This is a nice and solvable convex optimization problem. Strategy: Solve (RP1) and obtain x. Then, run ξ for several times. Pick up the best run as our actual investment policy!

Shuzhong Zhang, The Chinese University of Hong Kong

LNMB, The Netherlands, January 16 – 18, 2007 34

Randomization Approach (II)

The previous approach will generate a portfolio of about k assets with high probability, but may not be exactly k. Another approach is to only consider in the sample space of choosing k

• ut of n assets.

The random seeds optimization problem is now (RP2) minimize

n(n−k) k(n−1) n

• i=1

(qii + r2

i )x2 i + n(n−k) k(n−1) xT Qx

−

n−k k(n−1)(rT x)2

subject to rT x ≥ µ eT x ≤ 1.

Shuzhong Zhang, The Chinese University of Hong Kong

LNMB, The Netherlands, January 16 – 18, 2007 35

Actual Performances

10% 15% 20% 25% 30% 35% 40% 45%

2 3 4 5 6 7 8 9 10 #of stock Standard Deviation MVs RP1 RP1 rand RP2 RP2 rand

Number of assets n = 10.

Shuzhong Zhang, The Chinese University of Hong Kong

LNMB, The Netherlands, January 16 – 18, 2007 36

5% 10% 15% 20% 25% 30% 35% 40% 45% 50% 55%

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 # of stocks Standard Deviation MVs RP1 RP1 rand RP2 RP2 rand

Number of assets n = 20.

Shuzhong Zhang, The Chinese University of Hong Kong

LNMB, The Netherlands, January 16 – 18, 2007 37

0% 5% 10% 15% 20% 25% 30% 35% 40% 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50

RP1 RP1 rand RP2 RP2 rand

Number of assets n = 50.

Shuzhong Zhang, The Chinese University of Hong Kong

LNMB, The Netherlands, January 16 – 18, 2007 38

250-stocks

0 % 1 0 % 2 0 % 3 0 % 4 0 % 5 0 % 6 0 % 7 0 % 2 8 1 4 2 0 26 32 3 8 4 4 5 0 5 6 6 2 6 8 7 4 8 0 8 6 9 2 9 8 1 0 4 11 0 1 16 1 22 1 28 13 4 14 0 14 6 15 2 1 5 8 1 6 4 1 7 0 17 6 1 8 2 1 8 8 1 9 4 2 0 0 2 0 6 2 1 2 2 1 8 2 2 4 2 30 2 36 2 42 2 4 8

R P1 R P1 rand R P2 R P2 rand

Number of assets n = 250.

Shuzhong Zhang, The Chinese University of Hong Kong

LNMB, The Netherlands, January 16 – 18, 2007 39

Do It Not or Do It Well!

Consider decision problem (MVq) minimize xT Qx subject to xT Qix = 0, i = 1, ..., s xT Qix =   

• r

≥ 1 i = s + 1, ..., k xT Qix ≥ 1, i = k + 1, ..., m This is an extremely difficult combinatorial problem.

Shuzhong Zhang, The Chinese University of Hong Kong

LNMB, The Netherlands, January 16 – 18, 2007 40

A relaxed version of the problem is (DSDP) minimize Q • X subject to Qi • X = 0 for i = 1, ..., s, Qi • X =   

• r ,

≥ 1 i = s + 1, ..., k, Qi • X ≥ 1 for i = k + 1, ..., m, X 0 This is still a hard combinatorial problem. But it has a much better structure.

Shuzhong Zhang, The Chinese University of Hong Kong

LNMB, The Netherlands, January 16 – 18, 2007 41

A Quality Assurance

• Theorem. (He, Xie, and Z.; 2006)

There is an algorithm that solves (DSDP) such that its objective value is no ore than k − s times the optimal value. Let such a solution be ˆ X. We can view this as a randomization seed. What do we do next?

Shuzhong Zhang, The Chinese University of Hong Kong

LNMB, The Netherlands, January 16 – 18, 2007 42

A Randomized Rounding Method

Step 1. Generate ξ → N(0, ˆ X). Step 2. Let η := ξ/

• min

s+1≤i≤m ξT Qiξ > 0.

• Theorem. (He, Xie, and Z.; 2006)

The above algorithm yields an O(m3) approximation with probability

• f at least 7.5%.

Shuzhong Zhang, The Chinese University of Hong Kong

LNMB, The Netherlands, January 16 – 18, 2007 43

A Simulation Test

100 200 300 400 500 600 700 800 900 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6

Upper bound on v(MVq)/v∗, n = 33, m = 34, 1000 realizations.

Shuzhong Zhang, The Chinese University of Hong Kong

LNMB, The Netherlands, January 16 – 18, 2007 44 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 100 200 300 400 500 600 700 800 900

The histogram of the previous figure.

Shuzhong Zhang, The Chinese University of Hong Kong

LNMB, The Netherlands, January 16 – 18, 2007 45

Conclusions

• Randomness is a force of nature
• Beyond Natural Science, so is the case in Management Science
• Randomization can be controlled and used in making decisions
• Theory embraces practice in this endeavor

Shuzhong Zhang, The Chinese University of Hong Kong

LNMB, The Netherlands, January 16 – 18, 2007 46

URL of the reports

http://www.se.cuhk.edu.hk/~zhang/#workingpaper

• Semidefnite Relaxation Bounds for Indefinite Homogeneous

Quadratic Optimization, Technical Report SEEM2007-01, Department of Systems Engineering & Engineering Management, The Chinese University of Hong Kong, 2007 (with Simai He, Zhi-Quan Luo, and Jiawang Nie).

• Approximation Bounds for Quadratic Optimization with

Homogeneous Quadratic Constraints, Technical Report SEEM2005-07, Department of Systems Engineering & Engineering Management, The Chinese University of Hong Kong, 2005 (with Z.Q. Luo, N.D. Sidiropoulos, and P. Tseng).

Shuzhong Zhang, The Chinese University of Hong Kong