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Open Access Research Article

Adaptive Modulation with Smoothed Flow Utility

Ekine Akuiyibo* and Stephen Boyd

Author Affiliations

Information Systems Laboratory, Department of Electrical Engineering, Stanford University, Stanford, CA 94305, USA

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EURASIP Journal on Wireless Communications and Networking 2010, 2010:815213  doi:10.1155/2010/815213

The electronic version of this article is the complete one and can be found online at: http://jwcn.eurasipjournals.com/content/2010/1/815213


Received:6 May 2010
Accepted:14 September 2010
Published:21 September 2010

© 2010 Ekine Akuiyibo and Stephen Boyd.

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We consider the problem of choosing the data flow rate on a wireless link with randomly varying channel gain, to optimally trade off average transmit power and the average utility of the smoothed data flow rate. The smoothing allows us to model the demands of an application that can tolerate variations in flow over a certain time interval; we will see that this smoothing leads to a substantially different optimal data flow rate policy than without smoothing. We pose the problem as a convex stochastic control problem. For the case of a single flow, the optimal data flow rate policy can be numerically computed using stochastic dynamic programming. For the case of multiple flows on a single link, we propose an approximate dynamic programming approach to obtain suboptimal data flow rate policies. We illustrate, through numerical examples, that these approximate policies can perform very well.

1. Introduction

We consider the flow rate assignment problem on a wireless link with randomly varying channel gain, to optimally trade off average transmit power and the average utility of the smoothed flow data rate. We pose the multiperiod problem as an infinite-horizon stochastic control problem with linear dynamics and convex objective. For the case of a single flow, the optimal policy is easily found using stochastic dynamic programming (DP) and gridding. For the case of multiple flows, DP becomes intractable, and we propose instead an approximate dynamic programming approach using suboptimal policies developed in the single-flow case. Simulations show that these suboptimal policies perform very well.

In the wireless communications literature, varying a link's transmit rate (and power) depending on channel conditions is called adaptive modulation (AM); see, for example, [15]. One drawback of AM is that it is a physical layer optimization technique with no knowledge of upper layer optimization protocols. Maximizing a total utility function is also very common in various communications and networking problem formulations, where it is referred to as network utility maximization (NUM); see, for example, [610]. In the NUM framework, performance of an upper layer protocol (e.g., TCP) is determined by utility of flow attributes, for example, utility of link flow rate.

Our setup involves both adaptive modulation and utility maximization but is nonstandard in several respects. We consider the utility of the smoothed flows, and we consider multiple flows over the same wireless link [11].

2. Problem Setup

2.1. Average Smoothed Flow Utility

A wireless communication link supports data flows in a channel that varies with time, which we model using discrete-time intervals . We let be the data flow rate vector on the link, where , , is the th flow's data rate at time and denotes the set of nonnegative numbers. We let denote the total flow rate over all flows, where is the vector with all entries one. The flows, and the total flow rate, will depend on the random channel gain (through the flow policy, described below) and so are random variables.

We will work with a smoothed version of the flow rates, which is meant to capture the tolerance of the applications using the data flows to time variations in data rate. This was introduced in [12] using delivery contracts, in which the utility is a function of the total flow over a given time interval; here, we use instead a very simple first-order linear smoothing. At each time , the smoothed data flow rate vector is given by

(1)

where , , , is the smoothing parameter for the th flow, and we take . Thus, we have

(2)

where at time , each smoothed flow rate is the exponentially weighted average of previous flow rates.

The smoothing parameter determines the level of smoothing on flow . Small smoothing parameter values ( close to zero) correspond to light smoothing; large values ( close to one) correspond to heavy smoothing. (Note that means that flow is not smoothed; we have .) The level of smoothing can be related to the time scale over which the smoothing occurs. We define to be the smoothing time associated with flow . Roughly speaking, the smoothing time is the time interval over which the effect of a flow on the smoothed flow decays by a factor . Light smoothing corresponds to short smoothing times, while heavy smoothing corresponds to longer smoothing times.

We associate with each smoothed flow rate a strictly concave nondecreasing differentiable utility function , where the utility of is . The average utility derived over all flows, over all time, is

(3)

where . Here, the expectation is over the smoothed flows , and we are assuming that the expectations and limit above exist.

While most of our results will hold for more general utilities, we will focus on the family of power utility functions, defined for as

(4)

parameterized by and . The parameter sets the curvature (or risk aversion), while sets the overall weight of the utility. (For small values of , approaches a log utility.)

Before proceeding, we make some general comments on our use of smoothed flows. The smoothing can be considered as a type of time averaging; then we apply a concave utility function; finally, we average this utility. The time averaging and utility function operations do not commute, except in the case when the utility is linear (or affine). Jensen's inequality tells us that average smoothed utility is greater than or equal to the average utility applied directly to the flow rates, that is,

(5)

So the time smoothing step does affect our average utility; we will see later that it has a dramatic effect on the optimal flow policy.

2.2. Average Power

We model the wireless channel with time-varying positive gain parameters , , which we assume are independent identically distributed (IID), with known distribution. At each time , the gain parameter affects the power required to support the total data flow rate . The power is given by

(6)

where is increasing and strictly convex in for each value of ( is the set of positive numbers).

While our results will hold for the more general case, we will focus on the more specific power function described here. We suppose that the signal-to-interference-and-noise ratio (SINR) of the channel is given by . (Here includes the effect of time-varying channel gain, noise, and interference.) The channel capacity is then , where is a constant; this must equal at least the total flow rate , so we obtain

(7)

The total average power is

(8)

where, again, we are assuming that the expectations and limit exist.

2.3. Flow Rate Control Problem

The overall objective is to maximize a weighted difference between average utility and average power,

(9)

where is used to trade off average utility and power.

We require that the flow policy is causal; that is, when is chosen, we know the previous and current values of the flows, smoothed flows, and channel gains. Standard arguments in stochastic control (see, e.g., [1317]) can be used to conclude that, without loss of generality, we can assume that the flow control policy has the form

(10)

where . In other words, the policy depends only on the current smoothed flows and the current channel gain value.

The flow rate control problem is to choose the flow rate policy to maximize the overall objective in (9). This is a standard convex stochastic control problem, with linear dynamics.

2.4. Our Results

We let be the optimal overall objective value and let be an optimal policy. We will show that in the general (multiple-flow) case, the optimal policy includes a "no-transmit" zone, that is, a region in the space in which the optimal flow rate is zero. Not surprisingly, the optimal flow policy can be roughly described as waiting until the channel gain is large, or until the smoothed flow has fallen to a low level, at which point we transmit (i.e., choose nonzero ). Roughly speaking, the higher the level of smoothing, the longer we can afford to wait for a large channel gain before transmitting. The average power required to support a given utility level decreases, sometimes dramatically, as the level of smoothing increases.

We show that the optimal policy for the case of a single flow is readily computed numerically, working from Bellman's characterization of the optimal policy, and is not particularly sensitive to the details of the utility functions, smoothing levels, or power functions.

For the case of multiple flows, we cannot easily compute (or even represent) the optimal policy. For this case we propose an approximate policy, based on approximate dynamic programming [18, 19]. By computing an upper bound on , by allowing the flow control policy to use future values of channel gain (i.e., relaxing the causality requirement [20]), we show in numerical experiments that such policies are nearly optimal.

3. Optimal Policy Characterization

3.1. No Smoothing

We first consider the special case , in which there is no smoothing. Then we have , so the average smoothed utility is then the same as the average utility, that is,

(11)

In this case the optimal policy is trivial, since the stochastic control problem reduces to a simple optimization problem at each time step. At time , we simply choose to maximize . Thus, we have

(12)

which does not depend on . A simple and effective approach is to presolve this problem for a suitably large set of values of the channel gain and store the resulting tables of individual flow rates versus ; online we can interpolate between points in the table to find the (nearly) optimal policy. Another option is to fit a simple function to the optimal flow rate data and use this function as our (nearly) optimal policy.

For future reference, we note that the problem can also be solved using a waterfilling method (see, e.g., [21,Section ]). Dropping the time index and using to denote the flow index, we must solve the problem

(13)

with variables and . Introducing a Lagrange multiplier for the equality constraint (which we can show must be nonnegative, using monotonicity of with ), we are to maximize

(14)

over . This problem is separable in and , so we can maximize over and separately. We find that

(15)

(Each of these can be expressed in terms of conjugate functions; (see, e.g., [21,Section ].) We then adjust (say, using bisection) so that . An alternative is to carry out bisection on , defining in terms of as above, until , where refers to the derivative with respect to .

For our particular power law utility functions (4), we can give an explicit formula for in terms of :

(16)

For our particular power function (7), we use bisection to find the value of that yields

(17)

where the flow values come from the equation above. (The left-hand side is decreasing in , while the right-hand side is increasing.)

3.2. General Case

We now consider the more general case, with smoothing. We can characterize the optimal flow rate policy using stochastic dynamic programming [2225] and a form of Bellman's equation [26]. The optimal flow rate policy has the form

(18)

where is the Bellman (relative) value function. The value function (and optimal value) is characterized via the fixed point equation

(19)

where, for any function , the Bellman operator is given by

(20)

where the expectation is over . The fixed point equation and Bellman operator are invariant under adding a constant; that is, we have , for any constant (function) , and, similarly, satisfies the fixed point equation if and only if does. So without loss of generality we can assume that .

The value function can be found (in principle) by value iteration [14, 26]. We take and repeat the following iteration, for .

(1) (apply Bellman operator).

(2) (estimate optimal value).

(3) (normalize).

For technical conditions under which the value function exists and can be obtained via value iteration, see, for example, [2729]. We will simply assume here that the value function exists, and and converge to and , respectively.

The iterations above preserve several attributes of the iterates, which we can then conclude holds for . First of all, concavity of is preserved; that is, if is concave, so is . It is clear that normalization does not affect concavity, since we simply add a constant to the function. The Bellman operator preserves concavity since partial maximization of a function concave in two sets of variables results in a concave function (see, i.e.,[21, Section ]) and expectation over a family of concave functions yields a concave function; finally, addition (of ) preserves concavity. So we can conclude that is concave.

Another attribute that is preserved in value iteration is monotonicity; if is monotone increasing (in each component of its argument), then so is . We conclude that is monotone increasing.

3.3. No-Transmit Region

From the form of the optimal policy, we see that if and only if is optimal for the (convex) problem

(21)

with variable . This is the case if and only if

(22)

(see, e.g., [21,page 142]). We can rewrite this as

(23)

Using the specific power function (7) associated with the log capacity formula, we obtain

(24)

as the necessary and sufficient condition under which . Since is decreasing (by concavity of ), we can interpret (24) roughly as follows: do not transmit if the channel is bad ( small) or if the smoothed flows are large ( large).

4. Single-Flow Case

4.1. Optimal Policy

In the case of a single flow (i.e., ) we can easily carry out value iteration numerically, by discretizing the argument and values of and computing the expectation and maximization numerically. For the single-flow case, then we can compute the optimal policy and optimal performance (up to small numerical integration errors).

4.2. Power Law Suboptimal Policy

We replace the optimal value function (in the above optimal flow policy expression) with a simple analytic approximation of the value function to get the approximate policy

(25)

where is an approximation of the value function.

Since is increasing, concave, and satisfies , it is reasonable to fit it with a power law function as well, say , with , . For example, we can find the minimax (Chebyshev fit) by varying ; for each we choose to minimize

(26)

where are the discretized values of , with associated value function values . We do this by bisection on .

Experiments show that these power law approximate functions are, in general, reasonable approximations for the value function. For our power law utilities, these approximations yield very good matches to the true value function. For other concave utilities, the approximation is not as accurate, but experiments show that the associated approximate policies still yield nearly optimal performance.

We can derive an explicit expression for the approximate policy (25) for our power function:

(27)

where

(28)

and is the Lambert function; that is, is the solution of [30].

Note that this suboptimal policy is not needed in the single-flow case since we can obtain the optimal policy numerically. However, we found that the difference between our power law policy and the optimal policy (see the example of value functions below) is small enough that in practice they are virtually the same. This approximate policy is needed in the case of multiple flows.

4.3. Numerical Example

In this section we give simple numerical examples to illustrate the effect of smoothing on the resulting flow rate policy in the single-flow case. We consider two examples, with different levels of smoothing. The first flow is lightly smoothed (; ), while the second flow is heavily smoothed (; ). We use utility function , that is, , in our utility (4). The channel gains are IID exponential variables with mean . We use the power function (7), with .

We first consider the case . The value functions are shown in Figure 1, together with the power law approximations, which are (light smoothing) and (heavy smoothing). Figure 2 shows the optimal policies for the lightly smoothed flow (), and the heavily smoothed flow (). We can see that the optimal policies are quite different. As expected, the lightly smoothed flow transmits more often, that is, has a smaller no-transmit region.

thumbnailFigure 1. (a) Comparing (blue) with (red, dashed) for the lightly smoothed flow. (b) Comparing (blue) with (red, dashed) for the heavily smoothed flow.

thumbnailFigure 2. (a) Optimal policy for smoothing time (). (b) Optimal policy for (

Average Power versus Average Utility.

Figure 3 further illustrates the difference between the two flow rate policies. Using values of , we computed (via simulation) the average power-average utility tradeoff curve for each flow. As expected, we can see that the heavily smoothed flow achieves more average utility, for a given average power, than the lightly smoothed flow. (The heavily smoothed flow requires less average power to achieve a target average utility.)

thumbnailFigure 3. Average utility versus average power: heavily smoothed flow (top, dashed), and lightly smoothed flow (bottom).

Comparing Average Power.

We compare the average power required by each flow to generate a given average utility. Given a target average utility, we can estimate the average power required roughly from Figure 3, or more precisely via simulation as follows: choose a target average utility, and then run each controller, adjusting separately, until we reach the target utility. In our example, we chose and found for the lightly smoothed flow, and for the heavily smoothed flow. Figure 4 shows the associated power trajectories for each flow, along with the corresponding flow and smoothed flow trajectories. The dashed (horizontal) line indicates the average power, average flow, and averaged smoothed flow for each trajectory. Clearly the lightly smoothed flow requires more power than the heavily smoothed flow, by around : the heavily smoothed flow requires , compared to for the lightly smoothed flow.

thumbnailFigure 4. Sample power, flow, and smoothed flow trajectories; lightly smoothed flow (a, c, e), heavily smoothed flow (b, d, f).

Utility Curvature.

Table 1 shows results from similar experiments using different values of , . We see that for each value, as expected, the heavily smoothed flow requires less power. Note also that decreases as increases. This is not surprising as lower curvature (higher ) corresponds to lower risk aversion.

Table 1. Average power required for target , lightly smoothed flow , heavily smoothed flow .

5. A Suboptimal Policy for the Multiple-Flow Case

5.1. Approximate Dynamic Programming (ADP) Policy

In this section we describe a suboptimal policy that can be used in the multiple-flow case. Our proposed policy has the same form as the optimal policy, with the true value function replaced with an approximation or surrogate :

(29)

A policy obtained by replacing with an approximation is called an approximate dynamic programming (ADP) policy [18, 19, 31]. (Note that by this definition (25) is an ADP policy for .)

We construct in a simple way. Let denote the power law approximate function for the associated single-flow problem with only the th flow. (This can be obtained numerically as described above.) We then take

(30)

This approximate value function is separable, that is, a sum of functions of the individual flows, whereas the exact value function is (in general) not. The approximate policy, however, is not separable; the optimization problem solving to assign flow rates couples the different flow rates.

In the literature on approximate dynamic programming, would be considered basis functions [3234]; however, we fix the coefficients of the basis functions as one. (We have found that very little improvement in the policy is obtained by optimizing over the coefficients.)

Evaluating the approximate policy, that is, solving (29), reduces to solving the resource allocation problem

(31)

with optimization variables , . This is a convex optimization problem; its special structure allows it to be solved extremely efficiently, via waterfilling.

5.2. Solution via Waterfilling

We can solve (31) using the waterfilling method (described earlier). At each time , we are to maximize

(32)

over variables , whereas before, is a Lagrange multiplier associated with the equality constraint. For our particular power law approximate functions we can express in terms of :

(33)

We then use bisection on to find the value of for which

(34)

Since our surrogate value function is only approximate, there is no reason to solve this to great accuracy; experiments show that around 5–10 bisection iterations are more than enough.

Each iteration of the waterfilling algorithm has a cost that is which means that we can solve (31) very fast. An interior point method that exploits the structure would also yield a very efficient method; see, for example, [35].

5.3. Upper-Bound Policies

In this section we describe two heuristic data flow rate policies: a steady-state flow policy and a prescient flow policy. We show that both policies result in upper bounds on (the optimal objective value). These upper bounds give us a way to measure the performance of our suboptimal flow policy : if we obtain a from that is close to an upper bound, then we know that our suboptimal flow policy is nearly optimal.

5.3.1. Steady-State Policy

The steady-state policy is given by

(35)

where is channel gain at time and is the steady-state flow rate vector (independent of time) obtained by solving the optimization problem

(36)

with optimization variable , and being known. Let be our steady-state upper bound on obtained using the policy (35) to solve (9). Note that in the above optimization problem, we ignore time (and hence, smoothing) and variations in channel gains, and so, for each , is the optimal (steady-state) flow vector. (This is sometimes called the certainty equivalent problem associated with the stochastic programming problem [36, 37].)

By Jensen's inequality (and convexity of the max) it is easy to see that is an upper bound on . Note that once is determined, we can evaluate (35) using the waterfilling algorithm described earlier.

5.3.2. Prescient Policy

To obtain a prescient upper bound on , we relax the causality requirement imposed earlier on the flow policy in (10) and assume complete knowledge of the channel gains for all . (For more on prescient bounds, see, e.g., [20].) For each realization of channel gains, the flow rate control problem reduces to the optimization problem

(37)

where the optimization variables are the flow rates ,, and smoothed flow rates ,,. (The problem data are and ,,.) The optimal value of (37) is a random variable parameterized by . Let denote our prescient upper bound on . We obtain by using Monte Carlo simulation: we take large and solve (37) for independent realizations of the channel gains. The mean is our prescient upper bound.

5.4. Numerical Example

In this section we compare the performance of our ADP policy to the above prescient policy using a numerical example.

We construct a simple two-flow problem using the previous problem instance from Section 4.3 with , where, now, both flows share the single link, that is, . Our approximate value function is

(38)

(Note that this is easily extended to a problem with more than two flows.)

Let denote the objective obtained using our ADP policy. Each obtains an ADP controller, a point in the plane. Using the same , we can compute the corresponding prescient bound giving the point . (Every feasible controller must lie on or below the line, with slope , that passes through .)

We carried out Monte Carlo simulation (100 realizations, each with time steps) for several values of , computing as described in Section 5.2 and our prescient upper bound as described above.

Figure 5 shows our ADP controllers and the associated upper bounds. We can see that the ADP controllers are clearly feasible and perform very well depending on . For example, for , and , so we know that . So in this example, for , is not more than suboptimal.

thumbnailFigure 5. ADP controllers (red), and prescient upper bound (blue).

6. Conclusion

In this paper we present a variation on a multiperiod stochastic network utility maximization problem as a constrained convex stochastic control problem. We show that judging flow utilities dynamically, that is, with a utility function and a smoothing time scale, is a good way to account for network applications with heterogenous rate demands.

For the case of a single flow, our numerically computed value functions obtain flow policies that optimally trad off average utility and average power. We show that simple power law functions are reasonable approximations of the optimal value functions and that these simple functions obtain near optimal performance.

For the case of multiple flows on a single link (where the value function is not practically computable using dynamic programming), we approximate the value function with a combination of the simple one-dimensional power law functions. Simulations, and comparison with upper bounds on the optimal value, show that the resulting ADP policy can obtain very good performance.

Acknowledgments

This material is based upon work supported by AFOSR Grant FA9550-09-0130 and by Army contract W911NF-07-1-0029. The authors thank Yang Wang and Dan O'Neill for helpful discussions.

References

  1. J Hayes, Adaptive feedback communications. IEEE Transactions on Communication Technology 16(1), 29–34 (1968). Publisher Full Text OpenURL

  2. J Cavers, Variable-rate transmission for Rayleigh fading channels. IEEE Transactions on Communications 20(1), 15–22 (1972). Publisher Full Text OpenURL

  3. VO Hentinen, Error performance for adaptive transmission on fading channels. IEEE Transactions on Communications 22(9), 1331–1337 (1974). Publisher Full Text OpenURL

  4. WT Webb, R Steele, Variable rate QAM for mobile radio. IEEE Transactions on Communications 43(7), 2223–2230 (1995). Publisher Full Text OpenURL

  5. C Soon-Ghee, AJ Goldsmith, Variable-rate variable-power MQAM for fading channels. IEEE Transactions on Communications 45(10), 1218–1230 (1997). Publisher Full Text OpenURL

  6. FP Kelly, AK Maulloo, D Tan, Rate control for communication networks: shadow prices, proportional fairness and stability. Journal of the Operational Research Society 49(3), 237–252 (1997)

  7. SH Low, DE Lapsley, Optimization flow control—I: basic algorithm and convergence. IEEE/ACM Transactions on Networking 7(6), 861–874 (1999). Publisher Full Text OpenURL

  8. M Chiang, SH Low, AR Calderbank, JC Doyle, Layering as optimization decomposition: a mathematical theory of network architectures. Proceedings of the IEEE 95(1), 255–312 (2007)

  9. MJ Neely, E Modiano, C-P Li, Fairness and optimal stochastic control for heterogeneous networks. IEEE/ACM Transactions on Networking 16(2), 396–409 (2008)

  10. J Chen, W Xu, S He, Y Sun, P Thulasiraman, X Shen, Utility-based asynchronous flow control algorithm for wireless sensor networks. IEEE Journal on Selected Areas in Communications 28(7), 1116–1126 (2010)

  11. D O'Neill, E Akuiyibo, S Boyd, AJ Goldsmith, Optimizing adaptive modulation in wireless networks via multi-period network utility maximization. Proceedings of the IEEE International Conference on Communications, 2010

  12. N Trichakis, A Zymnis, S Boyd, Dynamic network utility maximization with delivery contracts. Proceedings of the IFAC World Congress, 2008, 2907–2912

  13. D Bertsekas, Dynamic Programming and Optimal Control: Volume 1 (Athena Scientific, 2005)

  14. D Bertsekas, Dynamic Programming and Optimal Control: Volume 2 (Athena Scientific, 2007)

  15. K Åström, Introduction to Stochastic Control Theory (Dover, New York, NY, USA, 1970)

  16. P Whittle, Optimization Over Time: Dynamic Programming and Stochastic Control (John Wiley & Sons, New York, NY, USA, 1982)

  17. D Bertsekas, S Shreve, Stochastic Optimal Control: The Discrete-Time Case (Athena Scientific, 1996)

  18. D Bertsekas, J Tsitsiklis, Neuro-Dynamic Programming (Athena Scientific, 1996)

  19. W Powell, Approximate Dynamic Programming: Solving the Curses of Dimensionality (John Wiley & Sons, New York, NY, USA, 2007)

  20. DB Brown, JE Smith, P Sun, Information relaxations and duality in stochastic dynamic programs. Operations Research 58(4), 785–801 (2010). Publisher Full Text OpenURL

  21. S Boyd, L Vandenberghe, Convex Optimization (Cambridge University Press, Cambridge, UK, 2004)

  22. M Puterman, Markov Decision Processes: Discrete Stochastic Dynamic Programming (John Wiley & Sons, New York, NY, USA, 1994)

  23. S Ross, Introduction to Stochastic Dynamic Programming: Probability and Mathematical (Academic Press, 1983)

  24. E Denardo, Dynamic Programming: Models and Applications (Prentice-Hall, New York, NY, USA, 1982)

  25. Y Wang, S Boyd, Performance bounds for linear stochastic control. Systems and Control Letters 58(3), 178–182 (2009). Publisher Full Text OpenURL

  26. R Bellman, Dynamic Programming (Courier Dover, New York, NY, USA, 1957)

  27. C Derman, Finite State Markovian Decision Processes (Academic Press, 1970)

  28. D Blackwell, Discrete dynamic programming. The Annals of Mathematical Statistics 33, 719–726 (1962). Publisher Full Text OpenURL

  29. A Arapostathis, V Borkar, E Fernández-Gaucherand, MK Ghosh, SI Marcus, Discrete-time controlled Markov processes with average cost criterion: a survey. SIAM Journal on Control and Optimization 31(2), 282–344 (1993). Publisher Full Text OpenURL

  30. RM Corless, GH Gonnet, DEG Hare, DJ Jeffrey, DE Knuth, On the Lambert W function. Advances in Computational Mathematics 5(4), 329–359 (1996)

  31. A Manne, Linear programming and sequential decisions. Management Science 6(3), 259–267 (1960). Publisher Full Text OpenURL

  32. PJ Schweitzer, A Seidmann, Generalized polynomial approximations in Markovian decision processes. Journal of Mathematical Analysis and Applications 110(2), 568–582 (1985). Publisher Full Text OpenURL

  33. MA Trick, SE Zin, Spline approximations to value functions: linear programming approach. Macroeconomic Dynamics 1(1), 255–277 (1997)

  34. DP De Farias, B Van Roy, The linear programming approach to approximate dynamic programming. Operations Research 51(6), 850–865 (2003). Publisher Full Text OpenURL

  35. R Madan, SP Boyd, S Lall, Fast algorithms for resource allocation in wireless cellular networks. IEEE/ACM Transactions on Networking 18(3), 973–984 (2010)

  36. J Birge, F Louveaux, Introduction to Stochastic Programming (Springer, New York, NY, USA, 1997)

  37. A Prekopa, Stochastic Programming (Kluwer Academic Publishers, New York, NY, USA, 1995)