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: a 1 a 2 a 3 · · · . • Each sensor, say sensor k , independently tosses a coin to decide which single binary digit to transmit: ξ = j with probability 1 / 2 j , j = 1 , 2 , ... • Then, sends this one bit data a ξ back to the station. • The base-station simply adds up all the received digits. • This is an unbiased estimation, with statistical error 4 U 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); or, 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: z H Cz ( QPc ) min min z H Q i z ≥ 1 , s . t . i = 1 , ..., m, z ∈ C n . Shuzhong Zhang, The Chinese University of Hong Kong
LNMB, The Netherlands, January 16 – 18, 2007 9 Homogeneous Quadratic Minimization In general, let us consider: x T Cx ( QPr ) min min x T Q i x ≥ 1 , s . t . i = 1 , ..., m, x ∈ ℜ n . 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 C • X ( SDPr ) min min s . t . Q i • X ≥ 1 , i = 1 , ..., m, X � 0 , and similarly for ( QPc ) min : C • Z ( SDPc ) min min Q i • Z ≥ 1 , s . t . 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 ∈ ℜ n from the real-valued normal distribution N (0 , X ∗ ). 2. Let ξ x ∗ ( ξ ) = . ξ T Q i ξ � min 1 ≤ i ≤ m 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 ( QPr min ) ≤ 27 m 2 v ( SDPr min ) . π Moreover, there is an instance such that v ( QPr min ) ≥ 2 m 2 π 2 v ( SDPr min ) . 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 ∈ C n from the complex-valued normal distribution N c (0 , Z ∗ ). 2. Let ξ x ∗ ( ξ ) = . � min 1 ≤ i ≤ m ξ H Q i ξ 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 ( QPc min ) ≤ 8 m · v ( SDPc min ) . Moreover, there is an instance such that m v ( QPc min ) ≥ π 2 (2 + π/ 2) 2 v ( SDPc min ) . 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): x T Cx ( QPr ) max max x T Q i x ≤ 1 , s . t . i = 1 , ..., m, x ∈ ℜ n , where Q i � 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 . Q i • X ≤ 1 , i = 1 , ..., m, X � 0 . Theorem. (Nemirovski, Roos, Terlaky; 1999) It holds that 1 v (( QPr ) max ) ≥ 2 ln(2 mµ ) v (( SDPr ) max ) , where µ = min { m, max i Rank( Q i ) } . Shuzhong Zhang, The Chinese University of Hong Kong
LNMB, The Netherlands, January 16 – 18, 2007 17 Complex Quadratic Maximization Problem Consider z H Cz ( QPc ) max max z H Q i z ≤ 1 , s . t . i = 1 , ..., m, z ∈ C n . The SDP relaxation is ( SDPc ) max max C • Z Q i • Z ≤ 1 , s . t . 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 ( QPc max ) Similar as before, we propose to solve the problem as follows 1. Generate a random vector ξ ∈ C n from the complex-valued normal distribution N c (0 , Z ∗ ). 2. Let ξ x ∗ ( ξ ) = . � max 1 ≤ i ≤ m ξ H Q i ξ 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 1 v ( QPc max ) ≥ 4 ln(100 µ ) v ( SDPc max ) , i =1 min { rank( Q i ) , √ m } . where µ = � 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 Q i ’s are indefinite! Theorem. (He, Luo, Nie, and Z.; 2007) If exactly one of Q i ’s is indefinite, then v ( QPr min ) ≤ 10 6 m 2 v ( SDPr min ) . π Theorem. (He, Luo, Nie, and Z.; 2007) If exactly one of Q i ’s is indefinite, then v ( QPc min ) ≤ 2400 m · v ( SDPc min ) . 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 Q i ’s are indefinite. Theorem. (Ben-Tal, Nemirovski, Roos; 2002) If one of the Q i ’s is indefinite and C indefinite, then 1 v ( QPr max ) ≥ 2 log(16 n 2 mµ ) v ( SDPr max ) , i =1 min { rank( Q i ) , √ m } . where µ = � 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 Q i ’s is indefinite and C indefinite, then 1 v ( QPr max ) ≥ 2 log(174 mµ ) v ( SDPr max ) , i =1 min { rank( Q i ) , √ m } . where µ = � 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 ≥ 1 ξ T Aξ ≤ E ξ T Aξ � � �� ∀ A symmetric matrix , Prob 4 , for i.i.d. ξ i ’s, with Prob { ξ i = +1 } = Prob { ξ i = − 1 } = 1 2 . 1 But they only managed to show a lower bound of 8 n 2 . 1 We have established a lower bound of 87 . Shuzhong Zhang, The Chinese University of Hong Kong
LNMB, The Netherlands, January 16 – 18, 2007 24 Put Theory to Work: Simulation Results full rank with one indefinite 1.9 maximum mean minimum 1.8 1.7 1.6 1.5 min min / v sdp v qp 1.4 1.3 1.2 1.1 1 0 10 20 30 40 50 60 70 80 90 100 m Minimization model: one indefinite constraint Shuzhong Zhang, The Chinese University of Hong Kong
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