- Email: [email protected]
Semidefinite and Conic Programming for Robust wireless OFDMA networks
Electronic Notes in Discrete Mathematics 36 (2010) 1225–1232 www.elsevier.com/locate/endm
Semidefinite and Conic Programming for Robust wireless OFDMA networks Pablo Adasme a,1 , Abdel Lisser a,2 a
Laboratoire de Recherche en Informatique, Universit´ e Paris-Sud XI, Bˆ atiment 490, 91405, Orsay Cedex France
Abstract In this paper, we study three robust optimization approaches [1]. The first one is based on the worst case scenario approach from Kouvelis and Yu [8]. The second, corresponds to a scaled simplex polyhedral approach due to Bertsimas and Sim [3] whilst the third, correspond to an ellipsoidal uncertainty approach proposed by Ben Tal and Nemirovski [2]. The study of the different approaches is made on the basis of a binary quadratic constrained program (BQCP). We derive two semidefinite programming (SDP) relaxations for the first two approaches whilst we use a second order conic program for the last one. Numerical results are given for a resource allocation of OFDMA wireless networks. Keywords: Robust Optimization, Semidefinite Programming, Quadratic Constrained Programs, Resource Allocation in OFDMA.
1
Introduction
Robust Optimization is an important field in mathematical programming. It deals with the uncertainty which arises in the input data of a mathematical 1 2
Email: [email protected] Email: [email protected]
1571-0653/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.endm.2010.05.155
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model. There are mainly two research directions on robust optimization. The first one deals with uncertainty by means of scenarios while the second one is based on convex uncertainty sets. Kouvelis and Yu pioneered the scenarios approach for the special case of uncertainty on the objective function [8]. In [2], on the other hand, Ben Tal and Nemirovski consider a convex ellipsoidal uncertainty set within linear programs which derives in solving conic robust problems. In [3], Bertsimas and Sim add a convex max term in each constraint of a linear program leading to a linear robust counterpart. In this paper, we propose three robust optimization models for resource allocation in OFDMA networks [9]. The first one is based on the worst case scenario approach [8]. The second and third ones are based on convex uncertainty sets [3,2]. The study is made on the basis of an OFDMA binary quadratic constrained program (BQCP) [1]. Two robust OFDMA quadratic models are proposed using Kouvelis and Yu and Bertsimas and Sim approaches and two SDP relaxations. An OFDMA conic program is also derived using Nemirovski robust approach. We use Fortet linearization method [6] as an intermediate step to obtain the second robust quadratic model and the conic one. The paper is organized as follows: Section 2 presents two robust OFDMA quadratic models and a conic one. Section 3 presents two SDP relaxations. Section 4 gives numerical results for the different robust approaches. Finally, section 5 concludes this paper.
2
Robust Models for Resource Allocation in OFDMA
An OFDMA network is composed by a single cell with one base station (BS) and several mobile users. The BS handles a set of N sub-carriers (or subchannels) that must be assigned to a set of K users using a modulation size of c ∈ {1, . . . , M } bits in each sub-carrier. The goal is to distribute efficiently sub-carriers to users while using adaptive modulation and minimizing the total power consumption in the network. To this purpose, we consider the following OFDMA quadratic model from [1]: K N M c (1) QIP0 : min{xk,n ,yn,c } k=1 n=1 c=1 Pk,n xk,n yn,c N M st: (2) x ∀k k,n n=1 c=1 c · yn,c = Rk K (3) k=1 xk,n ≤ 1 ∀n M (4) c=1 yn,c ≤ 1 ∀n (5) xk,n , yn,c ∈ {0, 1}
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Here, the decision variable xk,n is equal to 1 if user k is assigned sub-carrier n and 0 otherwise. Similarly, yn,c is equal to 1 if sub-carrier n uses a modulation size of c bits. The objective function represents the power consumption of the network. The first constraint represents the amount of bits needed by user k. The second constraint ensures that each sub-carrier should be assigned to only one user at a time while the third, represents a modulation linear constraint imposing that each sub-carrier must use only one integer modulation size. To formulate a first OFDMA robust model under the worst case scenario approach [8], we suppose we have a set S of power scenario matrices. Thus, we write the following quadratic model we call hereby RoQIP1: (6) RoQIP1 : (7)
st:
(8) (9)
min{xk,n ,yn,c ,t} t s K N M c P x y n,c ≤ t ∀s ∈ S k,n k=1 n=1 k,n c=1 to constraints (2), (3) and (4) xk,n , yn,c ∈ {0, 1}, t ∈ R
In the case of Bertsimas and Sim approach, we introduce a variable ϕck,n = xk,n yn,c to linearize the quadratic terms of QIP0 using Fortet method. Then, we consider each power entry pck,n to be in an interval [pck,n − pck,n , pck,n + pck,n ], c where pck,n = εpck,n for some ε > 0. We also define the set J = {Pk,n : (k, n, c) ∈ K × N × M } as the set of all power entries and the set Υ = {Ω ∪ (k , n , c ), Ω ⊆ J, |Ω| = Γ, (k , n , c ) ∈ J\Ω}. The latter represents all possible combinations of a subset Ω ⊆ J with cardinality |Ω| = Γ when using a parameter Γ ∈ [0, KN M ] to control the number of uncertain power entries in J. Since Γ is a continuous real value, then a fractional element (k , n , c ) ∈ J\Ω appears in Υ. Hence, we use the Fortet linearized model of QIP0 to get: (10) RoBSim st: (11) (12) (13) (14) (15) (16) (17)
min{xk,n ,yn,c ,ϕck,n ,t} K N M k=1
n=1
c=1
t c Pk,n ϕck,n +
c c c c + maxΥ (k,n,c)∈Ω Pk,n ϕk,n + (Γ − Γ)Pk ,n ϕk ,n N M c ∀k n=1 c=1 c · ϕk,n = Rk , to constraints (3) and (4) xk,n ≥ ϕck,n ∀k, n, c yn,c ≥ ϕck,n ∀k, n, c ϕck,n ≥ xk,n + yn,c − 1 ∀k, n, c xk,n , yn,c , ϕck,n ∈ [0, 1], t ∈ R
≤t
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Constraints (14),(15) and (16) are due to Fortet linearization method. We evaluate the max term of the first constraint in the optimal solution |ϕck,n |∗ . Then, we use Proposition 1 from [3] to deduce the following linear program: K N M c c ,ρ β(ϕ∗ , Γ) = minwk,n (18) k=1 n=1 c=1 wk,n + Γρ st:
c c ρ + wk,n ≥ Pk,n |ϕck,n |∗ c wk,n ≥ 0 ∀k, n, c ρ≥0
∀k, n, c
Using theorem 1 from [3] and strong duality theory from linear programming, we can remove linearization constraints in RoBSim to state the following robust quadratic model we call hereafter RoQIP2: (19)
(20) (21) (22) (23) (24) (25)
c ,ρ} min{xk,n ,yn,c ,t,wk,n t M N c x P y + st: K k,n n,c k=1 n=1 c=1 k,n K N M c + k=1 n=1 c=1 wk,n + Γρ ≤ t ρ + wc ≥ Pc xk,n yn,c ∀k, n, c
RoQIP2 :
k,n
k,n
to constraints in (8) c wk,n ≥ 0 ∀k, n, c ρ≥0 xk,n , yn,c ∈ {0, 1}, t ∈ R
Another model can be obtained using Ben Tal and Nemirovski robust approach [2]. Using the linearized Fortet model, we state the following conic model: c ,t} (26) RoNem : min{ϕck,n ,xk,n ,yn,c ,φck,n ,ψk,n t K N M c c K N M c c st: k=1 n=1 c=1 Pk,n ϕk,n + k=1 n=1 c=1 |Pk,n |φk,n +
K N M c c 2 +ϑ (27) k=1 n=1 c=1 (Pk,n ψk,n ) ≤ t
(28) (29) (30)
c ≤ φck,n ∀k, n, c −φck,n ≤ ϕck,n − ψk,n to constraints (3), (4), (12), (14), (15) and (16) c ϕck,n , xk,n , yn,c ∈ [0, 1], φck,n , ψk,n ≥ 0, t ∈ R
Here, a positive root square term multiplied by a parameter ϑ is added to control the uncertainty level. Although it has more constraints and variables than Bertsimas and Sim robust approach, we study both approaches in order to see how the optimal values behave when varying parameters ϑ and Γ respectively.
P. Adasme, A. Lisser / Electronic Notes in Discrete Mathematics 36 (2010) 1225–1232
3
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Semidefinite Relaxations
In this section we derive two SDP relaxations for RoQIP1 and RoQIP2. Let Sn = {Z ∈ Mn , Z = Z T } be the set of all n symmetric matrices and Sn+ = {Z ∈ Sn , a ∈ Rn , aT Za ≥ 0} be the set of all symmetric matrices satisfying the condition of positive semidefiniteness [7]. In order to formulate a SDP relaxation for RoQIP1, we define the (0-1) vector z as: (31) z T = x1,1 · · · x1,N · · · xK,1 · · · xK,N y1,1 · · · y1,M · · · yN,1 · · · yN,M Then, let Z be a symmetric positive semidefinite matrix defined by: ⎞ ⎛ T zz z 0 ⎟ ⎜ .. ⎟ ⎜ T (32) Z=⎜ z 1 .⎟ 0 ⎠ ⎝ 0 ··· t We construct symmetric matrices A, Λs for all s, Uk for all k, [exn ][exn ]T and c [eyn ][eyn ]T for all n, and ζk,n for all k, n, c to have positive terms in Z. The SDP relaxation for RoQIP1 is: (33) RoSDP1 : (34) (35) (36) (37) (38) (39) (40)
min Z
st:
T race{AZ} T race{Λs Z} − T race{AZ} ≤ 0 ∀s ∈ S T race{Uk Z} = Rk ∀k T race{[exn ][exn ]T Z} ≤ 1, ∀n T race{[eyn ][eyn ]T Z} ≤ 1, ∀n c T race{ζk,n Z} ≥ 0 ∀k, n, c diag(D) = z Z 0
Here [exn ] and [eyn ] are coefficient vectors for constraints in (3) and (4) according to vector z as defined above. Hence, the rank-1 matrices we construct with these vectors are used to strength the SDP relaxation as in [7]. In the case of RoQIP2, we use the same vector z to propose a SDP relaxation. In ad1 2 M 1 c M dition, we define vector = (w1,1 , w1,1 , . . . , w1,1 , w1,2 , . . . , wk,n , . . . , wK,N , ρ, t) c (wk,n ≥ 0) and add it to the diagonal of matrix Z to obtain a new matrix Z. The SDP relaxation for RoQIP2 can be stated as: Z} (41) RoSDP2 : min T race{A Z
(42) (43)
st:
Z} − T race{A Z} ≤0 T race{Λ c T race{Vk,n Z} ≤ 0 ∀k, n, c
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k Z} = Rk T race{U
(44)
∀k
≤ 1, ∀n T race{[exn ][exn ] Z} ≤ 1, ∀n T race{[eyn ][eyn ]T Z} c T race{ζk,n Z} ≥ 0 ∀k, n, c T
(45) (46) (47) (48)
diag(D) = z Z 0
(49)
k for all k, ζc for all k, n, c are symmetric Λ, V c for all k, n, c, U where A, k,n k,n input matrices of appropriate size according to matrix Z.
4
Numerical Results
In this section we present our numerical results concerning the robust approaches above-mentioned. We call RoIP1, RoIP2, the integer linear models obtained from RoQIP1 and RoQIP2 respectively. Similarly, we call RoLP1 and RoLP2 their LP relaxations. We generate power data using a wireless channel [9]. A Matlab program is developed using Cplex 9.1 for solving RoIP1, RoIP2, RoLP1, RoLP2. We use Csdp [4] to solve RoSDP1, RoSDP2 and Cvx [5] for solving RoNem. The numerical experiments have been carried out on a Pentium IV, 1.9GHz with 2 GoBytes of RAM under windows XP. Without loss of generality, we set M = 4 in our test instances. Numerical results are shown in table 1 for the Kouvelis and Yu approach. Table 2 shows results for the Bertsimas and Sim approach. In these tables, column RoLP1
# Users
RoSDP1
Integrality Gaps
RoIP1
Feasible
Lower bound
Cpu time (s)
Feasible
Lower bound
RoSDP1
RoLP1
K=6
0.6538
1.8558
0.4074
7.3440
1.7809
0.6377
Cpu time (s) 4131.80
0.0252
0.6048
K=8
0.6167
1.6796
0.3279
9.3130
0.9983
0.5773
9623.10
0.0683
0.8811
K=10
0.8073
1.7436
0.4858
17.6560
1.4350
0.7805
14818.00
0.0342
0.6618
K=12
1.0664
1.8629
0.5370
28.5780
1.2505
1.0423
15425.00
0.0231
0.9859
K=14
0.9428
2.0687
0.4439
76.6720
1.3376
0.9282
25724.00
0.0157
1.1239
Table 1 Results for Worst Case Scenario Approach for S=10 and N=128
1 gives the number of users, column 2 shows the optimal solution obtained by applying Fortet method to RoQIP1 and RoQIP2. Column 3 and 6 show feasible solutions obtained from the outputs of LP and SDP relaxations when using a simple randomized rounding algorithm [1]. Columns 4 and 7 give lower bounds for LP and SDP relaxations while columns 5 and 8 show the
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cpu times in seconds for Cplex and Csdp. Finally, columns 9 and 10 show 1−RoLP 1 integrality gaps for the LP and SDP relaxations i.e. GapLP = RoIPRoLP 1 RoIP 1−RoSDP 1 RoIP 2−RoLP 2 and GapSDP = = in table 1 and Gap and LP RoSDP 1 RoLP 2 2−RoSDP 2 GapSDP = RoIPRoSDP in table 2, respectively. We observe from these 2 RoLP2
# Users
RoSDP2
RoIP2
Feasible
Lower bound
Cpu time (s)
Feasible
Lower bound
K=6
0.4561
0.7232
0.2485
1.0000
0.4561
0.4561
Integrality Gaps Cpu time (s)
RoSDP2
RoLP2
2619.50
0
0.8356
K=8
0.3227
0.5403
0.1963
1.1410
0.3749
0.3191
6878.60
0.0111
0.6441
K=10
0.5607
0.9884
0.3435
1.3290
0.6537
0.5511
20934.00
0.0174
0.6325
K=12
0.2608
0.4444
0.1593
1.4370
0.2860
0.2599
18737.00
0.0033
0.6370
K=14
0.1816
0.3218
0.1187
1.7190
0.2566
0.1776
23060.00
0.0221
0.5295
Table 2 Results for Bertsimas and Sim Approach with ε = 5%, Γ =0.3KNM and N=64
tables that SDP outperform LP bounds. The feasible solutions obtained from the SDP solutions are close to the optimal values for all the instances. The integrality gaps confirm SDP tightness. Regarding the cpu times, Csdp requires a larger cpu time than Cplex to get a solution. We observe that cpu times are higher in table 2. This can be explained by the size of matrix Z which is larger than Z. Table 3 show numerical results for the OFDMA conic model (RoNem). Here, column 1 gives the instance number. Column 2 to 4 Instance
1
2
K
10
10
N
32
64
M
4
4
Γ
RoIP2
RoLP2
Cplex cpu time (s)
ϑ
RoNem
Cvx cpu time (s)
0
0.2496
0.1508
2.4
0
0.1507
55.6
512
0.2621
0.1583
1.1
2
0.1583
45.7
1280
0.2621
0.1583
0.8
5
0.1583
49.4
0
0.5274
0.2948
8.8
0
0.2947
323.6
1024
0.5381
0.3095
2.8
2
0.3094
262.4
2560
0.5381
0.3095
2.4
5
0.3094
260.2
Table 3 Results for RoNem and RoLP2 Relaxations
give instances sizes. Column 5 gives values for Γ ∈ [0, KN M ] while column 6 shows the optimal values obtained by RoIP2. Columns 7 and 10 give the lower bounds obtained by RoLP2 and RoNem, respectively. Columns 8 and 11 show the cpu time in seconds for Cplex and Cvx [5]. Finally, column 9 gives values for the input parameter ϑ used to control the uncertainty level in RoNem. We arbitrarily set three levels for parameters Γ and ϑ. The latter is varied within the interval [0, 5]. We observe that RoLP2 and RoNem give almost the same lower bounds. These lower bounds are weak when compared
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to the optimal solution of RoIP2. Moreover, Cvx requires larger cpu time to solve RoNem than Cplex does for solving RoLP2.
5
Conclusions
In this paper, we proposed three robust models for resource allocation in OFDMA networks. The study is made on the basis of a BQCP [1] in which Fortet method is applied as an intermediate step to derive (0-1) robust linear counterparts. In particular, we transform back the second linear robust counterpart into a quadratic one. Then, the robust models are handled by building SDP and conic relaxations. Our numerical results show tight bounds for the SDP relaxations.
References [1] Adasme Pablo, Abdel Lisser and Ismael Soto, Robust Semidefinite Relaxations for a New Quadratic OFDMA Resource Allocation Approach, Working Paper Number 1522, LRI, University of Paris Sud, France. [2] Ben-Tal Aharon, and Arkadi Nemirovski, Robust solutions of Linear Programming problems contaminated with uncertain data, Mathematical Programming, Springer Berlin-Heidelberg, 88 (2000), 411-424. [3] Bertsimas Dimitris, and Melvyn Sim, The Price of Robustness, Operations Research, 52 (2004). [4] CSDP is a Brian Borchers C code project of COIN-OR for SDP. https://projects.coin-or.org/Csdp/ . [5] CVX was designed by Michael Grant and Stephen Boyd, with input from Yinyu Ye. http://www.stanford.edu/∼boyd/cvx/. [6] Fortet R.,Applications de l algebre de boole en recherche operationelle, Revue Francaise de Recherche Operationelle, 4 (1960). [7] Helmberg, C., F. Rendl, and R. Weismantel, A Semidefinite Programming Approach to the Quadratic Knapsack Problem, Journal of combinatorial optimization, 4 (2000), 197-215. [8] Kouvelis, and G. Yu, Robust discrete optimization and its applications, Kluwer Academic Publishers, 1997. [9] Wong Ian C., and Brian L. Evans, Optimal OFDMA Subcarrier, Rate, and Power Allocation for Ergodic Rates Maximixation with Imperfect Channel Knowledge, Acoustics, Speech and Signal Processing, ICASSP, IEEE International Conference, 2007.
Semidefinite and Conic Programming for Robust wireless OFDMA networks Pablo Adasme a,1 , Abdel Lisser a,2 a
Laboratoire de Recherche en Informatique, Universit´ e Paris-Sud XI, Bˆ atiment 490, 91405, Orsay Cedex France
Abstract In this paper, we study three robust optimization approaches [1]. The first one is based on the worst case scenario approach from Kouvelis and Yu [8]. The second, corresponds to a scaled simplex polyhedral approach due to Bertsimas and Sim [3] whilst the third, correspond to an ellipsoidal uncertainty approach proposed by Ben Tal and Nemirovski [2]. The study of the different approaches is made on the basis of a binary quadratic constrained program (BQCP). We derive two semidefinite programming (SDP) relaxations for the first two approaches whilst we use a second order conic program for the last one. Numerical results are given for a resource allocation of OFDMA wireless networks. Keywords: Robust Optimization, Semidefinite Programming, Quadratic Constrained Programs, Resource Allocation in OFDMA.
1
Introduction
Robust Optimization is an important field in mathematical programming. It deals with the uncertainty which arises in the input data of a mathematical 1 2
Email: [email protected] Email: [email protected]
1571-0653/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.endm.2010.05.155
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model. There are mainly two research directions on robust optimization. The first one deals with uncertainty by means of scenarios while the second one is based on convex uncertainty sets. Kouvelis and Yu pioneered the scenarios approach for the special case of uncertainty on the objective function [8]. In [2], on the other hand, Ben Tal and Nemirovski consider a convex ellipsoidal uncertainty set within linear programs which derives in solving conic robust problems. In [3], Bertsimas and Sim add a convex max term in each constraint of a linear program leading to a linear robust counterpart. In this paper, we propose three robust optimization models for resource allocation in OFDMA networks [9]. The first one is based on the worst case scenario approach [8]. The second and third ones are based on convex uncertainty sets [3,2]. The study is made on the basis of an OFDMA binary quadratic constrained program (BQCP) [1]. Two robust OFDMA quadratic models are proposed using Kouvelis and Yu and Bertsimas and Sim approaches and two SDP relaxations. An OFDMA conic program is also derived using Nemirovski robust approach. We use Fortet linearization method [6] as an intermediate step to obtain the second robust quadratic model and the conic one. The paper is organized as follows: Section 2 presents two robust OFDMA quadratic models and a conic one. Section 3 presents two SDP relaxations. Section 4 gives numerical results for the different robust approaches. Finally, section 5 concludes this paper.
2
Robust Models for Resource Allocation in OFDMA
An OFDMA network is composed by a single cell with one base station (BS) and several mobile users. The BS handles a set of N sub-carriers (or subchannels) that must be assigned to a set of K users using a modulation size of c ∈ {1, . . . , M } bits in each sub-carrier. The goal is to distribute efficiently sub-carriers to users while using adaptive modulation and minimizing the total power consumption in the network. To this purpose, we consider the following OFDMA quadratic model from [1]: K N M c (1) QIP0 : min{xk,n ,yn,c } k=1 n=1 c=1 Pk,n xk,n yn,c N M st: (2) x ∀k k,n n=1 c=1 c · yn,c = Rk K (3) k=1 xk,n ≤ 1 ∀n M (4) c=1 yn,c ≤ 1 ∀n (5) xk,n , yn,c ∈ {0, 1}
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Here, the decision variable xk,n is equal to 1 if user k is assigned sub-carrier n and 0 otherwise. Similarly, yn,c is equal to 1 if sub-carrier n uses a modulation size of c bits. The objective function represents the power consumption of the network. The first constraint represents the amount of bits needed by user k. The second constraint ensures that each sub-carrier should be assigned to only one user at a time while the third, represents a modulation linear constraint imposing that each sub-carrier must use only one integer modulation size. To formulate a first OFDMA robust model under the worst case scenario approach [8], we suppose we have a set S of power scenario matrices. Thus, we write the following quadratic model we call hereby RoQIP1: (6) RoQIP1 : (7)
st:
(8) (9)
min{xk,n ,yn,c ,t} t s K N M c P x y n,c ≤ t ∀s ∈ S k,n k=1 n=1 k,n c=1 to constraints (2), (3) and (4) xk,n , yn,c ∈ {0, 1}, t ∈ R
In the case of Bertsimas and Sim approach, we introduce a variable ϕck,n = xk,n yn,c to linearize the quadratic terms of QIP0 using Fortet method. Then, we consider each power entry pck,n to be in an interval [pck,n − pck,n , pck,n + pck,n ], c where pck,n = εpck,n for some ε > 0. We also define the set J = {Pk,n : (k, n, c) ∈ K × N × M } as the set of all power entries and the set Υ = {Ω ∪ (k , n , c ), Ω ⊆ J, |Ω| = Γ, (k , n , c ) ∈ J\Ω}. The latter represents all possible combinations of a subset Ω ⊆ J with cardinality |Ω| = Γ when using a parameter Γ ∈ [0, KN M ] to control the number of uncertain power entries in J. Since Γ is a continuous real value, then a fractional element (k , n , c ) ∈ J\Ω appears in Υ. Hence, we use the Fortet linearized model of QIP0 to get: (10) RoBSim st: (11) (12) (13) (14) (15) (16) (17)
min{xk,n ,yn,c ,ϕck,n ,t} K N M k=1
n=1
c=1
t c Pk,n ϕck,n +
c c c c + maxΥ (k,n,c)∈Ω Pk,n ϕk,n + (Γ − Γ)Pk ,n ϕk ,n N M c ∀k n=1 c=1 c · ϕk,n = Rk , to constraints (3) and (4) xk,n ≥ ϕck,n ∀k, n, c yn,c ≥ ϕck,n ∀k, n, c ϕck,n ≥ xk,n + yn,c − 1 ∀k, n, c xk,n , yn,c , ϕck,n ∈ [0, 1], t ∈ R
≤t
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Constraints (14),(15) and (16) are due to Fortet linearization method. We evaluate the max term of the first constraint in the optimal solution |ϕck,n |∗ . Then, we use Proposition 1 from [3] to deduce the following linear program: K N M c c ,ρ β(ϕ∗ , Γ) = minwk,n (18) k=1 n=1 c=1 wk,n + Γρ st:
c c ρ + wk,n ≥ Pk,n |ϕck,n |∗ c wk,n ≥ 0 ∀k, n, c ρ≥0
∀k, n, c
Using theorem 1 from [3] and strong duality theory from linear programming, we can remove linearization constraints in RoBSim to state the following robust quadratic model we call hereafter RoQIP2: (19)
(20) (21) (22) (23) (24) (25)
c ,ρ} min{xk,n ,yn,c ,t,wk,n t M N c x P y + st: K k,n n,c k=1 n=1 c=1 k,n K N M c + k=1 n=1 c=1 wk,n + Γρ ≤ t ρ + wc ≥ Pc xk,n yn,c ∀k, n, c
RoQIP2 :
k,n
k,n
to constraints in (8) c wk,n ≥ 0 ∀k, n, c ρ≥0 xk,n , yn,c ∈ {0, 1}, t ∈ R
Another model can be obtained using Ben Tal and Nemirovski robust approach [2]. Using the linearized Fortet model, we state the following conic model: c ,t} (26) RoNem : min{ϕck,n ,xk,n ,yn,c ,φck,n ,ψk,n t K N M c c K N M c c st: k=1 n=1 c=1 Pk,n ϕk,n + k=1 n=1 c=1 |Pk,n |φk,n +
K N M c c 2 +ϑ (27) k=1 n=1 c=1 (Pk,n ψk,n ) ≤ t
(28) (29) (30)
c ≤ φck,n ∀k, n, c −φck,n ≤ ϕck,n − ψk,n to constraints (3), (4), (12), (14), (15) and (16) c ϕck,n , xk,n , yn,c ∈ [0, 1], φck,n , ψk,n ≥ 0, t ∈ R
Here, a positive root square term multiplied by a parameter ϑ is added to control the uncertainty level. Although it has more constraints and variables than Bertsimas and Sim robust approach, we study both approaches in order to see how the optimal values behave when varying parameters ϑ and Γ respectively.
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3
1229
Semidefinite Relaxations
In this section we derive two SDP relaxations for RoQIP1 and RoQIP2. Let Sn = {Z ∈ Mn , Z = Z T } be the set of all n symmetric matrices and Sn+ = {Z ∈ Sn , a ∈ Rn , aT Za ≥ 0} be the set of all symmetric matrices satisfying the condition of positive semidefiniteness [7]. In order to formulate a SDP relaxation for RoQIP1, we define the (0-1) vector z as: (31) z T = x1,1 · · · x1,N · · · xK,1 · · · xK,N y1,1 · · · y1,M · · · yN,1 · · · yN,M Then, let Z be a symmetric positive semidefinite matrix defined by: ⎞ ⎛ T zz z 0 ⎟ ⎜ .. ⎟ ⎜ T (32) Z=⎜ z 1 .⎟ 0 ⎠ ⎝ 0 ··· t We construct symmetric matrices A, Λs for all s, Uk for all k, [exn ][exn ]T and c [eyn ][eyn ]T for all n, and ζk,n for all k, n, c to have positive terms in Z. The SDP relaxation for RoQIP1 is: (33) RoSDP1 : (34) (35) (36) (37) (38) (39) (40)
min Z
st:
T race{AZ} T race{Λs Z} − T race{AZ} ≤ 0 ∀s ∈ S T race{Uk Z} = Rk ∀k T race{[exn ][exn ]T Z} ≤ 1, ∀n T race{[eyn ][eyn ]T Z} ≤ 1, ∀n c T race{ζk,n Z} ≥ 0 ∀k, n, c diag(D) = z Z 0
Here [exn ] and [eyn ] are coefficient vectors for constraints in (3) and (4) according to vector z as defined above. Hence, the rank-1 matrices we construct with these vectors are used to strength the SDP relaxation as in [7]. In the case of RoQIP2, we use the same vector z to propose a SDP relaxation. In ad1 2 M 1 c M dition, we define vector = (w1,1 , w1,1 , . . . , w1,1 , w1,2 , . . . , wk,n , . . . , wK,N , ρ, t) c (wk,n ≥ 0) and add it to the diagonal of matrix Z to obtain a new matrix Z. The SDP relaxation for RoQIP2 can be stated as: Z} (41) RoSDP2 : min T race{A Z
(42) (43)
st:
Z} − T race{A Z} ≤0 T race{Λ c T race{Vk,n Z} ≤ 0 ∀k, n, c
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k Z} = Rk T race{U
(44)
∀k
≤ 1, ∀n T race{[exn ][exn ] Z} ≤ 1, ∀n T race{[eyn ][eyn ]T Z} c T race{ζk,n Z} ≥ 0 ∀k, n, c T
(45) (46) (47) (48)
diag(D) = z Z 0
(49)
k for all k, ζc for all k, n, c are symmetric Λ, V c for all k, n, c, U where A, k,n k,n input matrices of appropriate size according to matrix Z.
4
Numerical Results
In this section we present our numerical results concerning the robust approaches above-mentioned. We call RoIP1, RoIP2, the integer linear models obtained from RoQIP1 and RoQIP2 respectively. Similarly, we call RoLP1 and RoLP2 their LP relaxations. We generate power data using a wireless channel [9]. A Matlab program is developed using Cplex 9.1 for solving RoIP1, RoIP2, RoLP1, RoLP2. We use Csdp [4] to solve RoSDP1, RoSDP2 and Cvx [5] for solving RoNem. The numerical experiments have been carried out on a Pentium IV, 1.9GHz with 2 GoBytes of RAM under windows XP. Without loss of generality, we set M = 4 in our test instances. Numerical results are shown in table 1 for the Kouvelis and Yu approach. Table 2 shows results for the Bertsimas and Sim approach. In these tables, column RoLP1
# Users
RoSDP1
Integrality Gaps
RoIP1
Feasible
Lower bound
Cpu time (s)
Feasible
Lower bound
RoSDP1
RoLP1
K=6
0.6538
1.8558
0.4074
7.3440
1.7809
0.6377
Cpu time (s) 4131.80
0.0252
0.6048
K=8
0.6167
1.6796
0.3279
9.3130
0.9983
0.5773
9623.10
0.0683
0.8811
K=10
0.8073
1.7436
0.4858
17.6560
1.4350
0.7805
14818.00
0.0342
0.6618
K=12
1.0664
1.8629
0.5370
28.5780
1.2505
1.0423
15425.00
0.0231
0.9859
K=14
0.9428
2.0687
0.4439
76.6720
1.3376
0.9282
25724.00
0.0157
1.1239
Table 1 Results for Worst Case Scenario Approach for S=10 and N=128
1 gives the number of users, column 2 shows the optimal solution obtained by applying Fortet method to RoQIP1 and RoQIP2. Column 3 and 6 show feasible solutions obtained from the outputs of LP and SDP relaxations when using a simple randomized rounding algorithm [1]. Columns 4 and 7 give lower bounds for LP and SDP relaxations while columns 5 and 8 show the
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cpu times in seconds for Cplex and Csdp. Finally, columns 9 and 10 show 1−RoLP 1 integrality gaps for the LP and SDP relaxations i.e. GapLP = RoIPRoLP 1 RoIP 1−RoSDP 1 RoIP 2−RoLP 2 and GapSDP = = in table 1 and Gap and LP RoSDP 1 RoLP 2 2−RoSDP 2 GapSDP = RoIPRoSDP in table 2, respectively. We observe from these 2 RoLP2
# Users
RoSDP2
RoIP2
Feasible
Lower bound
Cpu time (s)
Feasible
Lower bound
K=6
0.4561
0.7232
0.2485
1.0000
0.4561
0.4561
Integrality Gaps Cpu time (s)
RoSDP2
RoLP2
2619.50
0
0.8356
K=8
0.3227
0.5403
0.1963
1.1410
0.3749
0.3191
6878.60
0.0111
0.6441
K=10
0.5607
0.9884
0.3435
1.3290
0.6537
0.5511
20934.00
0.0174
0.6325
K=12
0.2608
0.4444
0.1593
1.4370
0.2860
0.2599
18737.00
0.0033
0.6370
K=14
0.1816
0.3218
0.1187
1.7190
0.2566
0.1776
23060.00
0.0221
0.5295
Table 2 Results for Bertsimas and Sim Approach with ε = 5%, Γ =0.3KNM and N=64
tables that SDP outperform LP bounds. The feasible solutions obtained from the SDP solutions are close to the optimal values for all the instances. The integrality gaps confirm SDP tightness. Regarding the cpu times, Csdp requires a larger cpu time than Cplex to get a solution. We observe that cpu times are higher in table 2. This can be explained by the size of matrix Z which is larger than Z. Table 3 show numerical results for the OFDMA conic model (RoNem). Here, column 1 gives the instance number. Column 2 to 4 Instance
1
2
K
10
10
N
32
64
M
4
4
Γ
RoIP2
RoLP2
Cplex cpu time (s)
ϑ
RoNem
Cvx cpu time (s)
0
0.2496
0.1508
2.4
0
0.1507
55.6
512
0.2621
0.1583
1.1
2
0.1583
45.7
1280
0.2621
0.1583
0.8
5
0.1583
49.4
0
0.5274
0.2948
8.8
0
0.2947
323.6
1024
0.5381
0.3095
2.8
2
0.3094
262.4
2560
0.5381
0.3095
2.4
5
0.3094
260.2
Table 3 Results for RoNem and RoLP2 Relaxations
give instances sizes. Column 5 gives values for Γ ∈ [0, KN M ] while column 6 shows the optimal values obtained by RoIP2. Columns 7 and 10 give the lower bounds obtained by RoLP2 and RoNem, respectively. Columns 8 and 11 show the cpu time in seconds for Cplex and Cvx [5]. Finally, column 9 gives values for the input parameter ϑ used to control the uncertainty level in RoNem. We arbitrarily set three levels for parameters Γ and ϑ. The latter is varied within the interval [0, 5]. We observe that RoLP2 and RoNem give almost the same lower bounds. These lower bounds are weak when compared
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to the optimal solution of RoIP2. Moreover, Cvx requires larger cpu time to solve RoNem than Cplex does for solving RoLP2.
5
Conclusions
In this paper, we proposed three robust models for resource allocation in OFDMA networks. The study is made on the basis of a BQCP [1] in which Fortet method is applied as an intermediate step to derive (0-1) robust linear counterparts. In particular, we transform back the second linear robust counterpart into a quadratic one. Then, the robust models are handled by building SDP and conic relaxations. Our numerical results show tight bounds for the SDP relaxations.
References [1] Adasme Pablo, Abdel Lisser and Ismael Soto, Robust Semidefinite Relaxations for a New Quadratic OFDMA Resource Allocation Approach, Working Paper Number 1522, LRI, University of Paris Sud, France. [2] Ben-Tal Aharon, and Arkadi Nemirovski, Robust solutions of Linear Programming problems contaminated with uncertain data, Mathematical Programming, Springer Berlin-Heidelberg, 88 (2000), 411-424. [3] Bertsimas Dimitris, and Melvyn Sim, The Price of Robustness, Operations Research, 52 (2004). [4] CSDP is a Brian Borchers C code project of COIN-OR for SDP. https://projects.coin-or.org/Csdp/ . [5] CVX was designed by Michael Grant and Stephen Boyd, with input from Yinyu Ye. http://www.stanford.edu/∼boyd/cvx/. [6] Fortet R.,Applications de l algebre de boole en recherche operationelle, Revue Francaise de Recherche Operationelle, 4 (1960). [7] Helmberg, C., F. Rendl, and R. Weismantel, A Semidefinite Programming Approach to the Quadratic Knapsack Problem, Journal of combinatorial optimization, 4 (2000), 197-215. [8] Kouvelis, and G. Yu, Robust discrete optimization and its applications, Kluwer Academic Publishers, 1997. [9] Wong Ian C., and Brian L. Evans, Optimal OFDMA Subcarrier, Rate, and Power Allocation for Ergodic Rates Maximixation with Imperfect Channel Knowledge, Acoustics, Speech and Signal Processing, ICASSP, IEEE International Conference, 2007.