Joint Sensing and Power Allocation in Nonconvex Cognitive Radio Games: Quasi-Nash Equilibria
In this paper, we propose a novel class of Nash problems for Cognitive Radio (CR) networks composed of multiple primary users (PUs) and secondary users (SUs) wherein each SU (player) competes against the others to maximize his own opportunistic throughput by choosing jointly the sensing duration, the detection thresholds, and the vector power allocation over a multichannel link. In addition to power budget constraints, several (deterministic or probabilistic) interference constraints can be accommodated in the proposed general formulation, such as constraints on the maximum individual/aggregate (probabilistic) interference tolerable from the PUs. To keep the optimization as decentralized as possible, global interference constraints, when present, are imposed via pricing; the prices are thus additional variables to be optimized. The resulting players’ optimization problems are nonconvex and there are price clearance conditions associated with the nonconvex global interference constraints to be satisfied by the equilibria of the game, which make the analysis of the proposed game a challenging task; none of classical results in the game theory literature can be successfully applied. To deal with the nonconvexity of the game, we introduce a relaxed equilibrium conceptthe Quasi-Nash Equilibrium (QNE)and study its main properties, performance, and connection with local Nash equilibria. Quite interestingly, the proposed game theoretical formulations yield a considerable performance improvement with respect to current centralized and decentralized designs of CR systems, which validates the concept of QNE.
1 Introduction and Motivation
In the last decade, Cognitive Radio (CR) has received considerable attention as a way to improve the efficiency of radio networks [1, 2]. The CR networks adopt a hierarchical access structure where the primary users (PUs) are the legacy spectrum holders while the secondary users (SUs) are the unlicensed users who sense the electromagnetic environment and adapt their transceiver parameters as well as the resource allocation decisions in order to dynamically access temporally unoccupied spectrum regions.
The challenge for a reliable sensing algorithm is to identify suitable transmission opportunities without compromising the integrity of the PUs. One of the design criteria is to make the probability of false alarm as low as possible, since it measures the percentage of vacant spectrum which is misclassified as busy, increasing thus the opportunistic usage of the spectrum from the SUs. On the other hand, in order to limit the probability of interfering with PUs, it is desirable to keep the missed detection probability as low as possible. The detection thresholds are the trade-off factor between the false alarm and the missed detection probabilities; low thresholds will result in high false alarm rate in favor of low missed detection probability and vice versa. Alternatively, the choice of the sensing time offers a trade-off between the quality and speed of sensing: increasing the sensing times permits to reach both low false alarm and detection probability values, thus reducing the time available for secondary transmissions, which would result in low SUs’ throughput. The above trade-off calls naturally for a joint determination of the sensing and transmission parameters of the SUs, under the paradigm of selfish behavior among these users. The modeling and analysis of this competitive multi-agent optimization is the main subject of this work.
1.1 Related work
The joint optimization of the sensing and transmission parameters has been only partially addressed in the literature; current works on the subject can be divided in the following three main classes.
- Optimization of the sensing parameters only [3, 4, 5, 6, 7, 8]. In [3, 4], the authors proposed alternative centralized schemes that optimize the detection thresholds for a bank of energy detectors, in order to maximize the so-called opportunistic throughput, while keeping the sensing time and the transmission parameters of the SU fixed and given a priori. The optimization of the sensing time and the sensing time/detection thresholds for a given missed detection probability and transmission rate of one SU was addressed in  and [7, 8], respectively. A throughput-sensing trade-off for a given transmission rate of the SU was studied in . In the above papers there is no optimization of the SU’s transmission strategies, and the proposed schemes are applicable only to CR scenarios composed by one PU and SU.
- Optimization of the transmission parameters only [9, 10, 11, 12, 13, 14, 15, 16, 17, 18]. These papers address the optimization of the SUs’ transceivers in a multiuser OFDM CR scenario; several centralized [9, 10, 11, 18] schemes or distributed algorithms based on a game theoretic approach [12, 13, 14, 15] were proposed; a recent overview of the latter approaches can be found in [15, 19]. A general framework based on variational inequalities was proposed in [16, 17] to study and solve in a distributed way convex noncooperative games with (possibly) side (i.e., coupling) constraints (e.g., interference constraints). In all the aforementioned papers the sensing process is not considered as part of the optimization; in fact the SUs do not perform any sensing but they are allowed to transmit over the licensed spectrum provided that they satisfy interference constraints imposed by the PUs, no matter if the PUs are active of not.
- Joint optimization of some of the sensing/transmission parameters [20, 21, 22, 23, 24, 25]. In  (or [21, 22]), the sensing time and the transmit power (or the power allocation [21, 22] over multi-channel links) of one SU were jointly optimized while keeping the detection probability (and thus the decision threshold) fixed to a target value. In [23, 24], the authors focused on the joint optimization of the power allocation and the equi-false alarm rate of one SU  over multi-channel links, for a fixed sensing time. The case of multiple SUs and one PU was considered in  (and more recently in ), under the same assumptions of ; however no formal analysis of the proposed formulation was provided. Moreover, these papers [23, 24, 25] did not consider the sensing overhead as part of the optimization, leaving unclear how to optimally choose the sensing time.
1.2 Main contributions
What emerges from the analysis of existing literature is that, up to now, state-of-the-art approaches have focused on the optimization of specific features and single components of a CR system in isolation. In this paper we move a step ahead and propose a novel class of Nash Equilibrium Problems (NEPs), suitable for designing multiple primary/secondary user CR networks, under different practical settings. Aiming to dynamically access temporally unoccupied primary spectrum regions, in our formulation, the SUs maximize their own opportunistic throughput by jointly optimizing the sensing time, the decision thresholds of a bank of energy detectors, and the power allocation over multi-channel links. Because of sensing errors, it may happen however that the SUs attempt to access part of licensed spectrum that is still occupied by active PUs, causing thus harmful interference. This motivates the introduction of probabilistic (rather than deterministic) interference constraints that are imposed to control the power radiated over the licensed spectrum only when a missed detection event occurs (in a probabilistic sense). The proposed formulation accommodates alternative combinations of power/interference constraints. For instance, in addition to classical (deterministic) transmit power (and possibly spectral masks) constraints, we envisage the use of (probabilistic) average individual (i.e., on each SU) and/or global (i.e., over all the SUs) interference tolerable by the primary receivers. Local interference constraints fit naturally to scenarios where the SUs are not willing to cooperate; whereas the global ones, which are less conservative, are more suitable for settings where SUs may want to trade some signaling for better performance. By imposing a coupling among the power allocations of the SUs, global interference constraints introduce a new challenge in the system design: how to enforce global interference constraints without requiring a centralized optimization? A major contribution of the paper is to address this issue by introducing a pricing mechanism in the game, through a penalization in the players’ objective functions. The prices will be then additional variables to be optimized, while guaranteeing the global interference constraints to be satisfied at any solution of the game.
The resulting game-theoretical formulations belong to the class of nonconvex games, where the nonconvexity occurs at both the objective functions and feasible sets of the SUs’ optimization problems. Indeed the objective functions of the SUs (to be maximized) are not quasi-concave and the local/global interference constraints are bi-convex and thus nonconvex; moreover, in the presence of global interference constraints, there are possibly unbounded price variables to be optimized and pricing clearing conditions associated with these constraints to be satisfied at the equilibrium. All this makes the analysis of the proposed games a challenging task; none of previous results in the game theory literature can be successfully applied (see Sec. 4 for more details); without (quasi-)concavity of the players’ payoff functions or convexity of the strategy sets, a solution of the gamethe Nash Equilibrium (NE)may not even exist [26, 27]. To overcome this issue, alternative refinements of the NE concept have been introduced in the literature, in the form of “local” solutions (see, e.g., [26, 28, 29, 30] and references therein): in contrast to the NE that is resistant to “arbitrarily large” deviations of single player’s strategies, local equilibria require stability only against “small” unilaterally deviations of the players. Among all, we mention here some local solution concepts that have been proposed in the literature: such as the “critical NE” , the “Local NE” (LNE) , and the “generalized equilibrium” . Despite their theoretical interest, those local solution concepts have a very limited applicability: when available, there are only abstract mathematical conditions granting their existence (see, e.g., [26, 30]), whose verification for games arising from realistic applications such as those presented in the present paper remains elusive. A system design based on these concepts may lead to an unpredictable system behavior (e.g., equilibrium existence vs. nonexistence, convergence of algorithms vs. nonconvergence), which is not a desirable feature in practice.
In this paper, we propose an alternative formalization of solution in nonconvex games; building on the well-established and accepted definition of stationary solutions of a nonlinear programming, we introduce a (relaxed) equilibrium concept for a nonconvex NEP, which is a solution of the aggregated stationarity conditions (the Karush-Kuhn-Tucker conditions) of the players’ optimization problems. We call such a stationary tuple a quasi-Nash equilibrium (QNE) of the (nonconvex) game; the prefix “quasi” is intended to signify that a NE (if it exists) must be a QNE under mild conditions (constraint qualifications). While the concept of QNE seems quite natural and conceptually similar to the idea of critical NE  and generalized equilibrium , its analysis is not an easy task (cf. Sec. 4), and cannot be addressed using results in previous papers (e.g., [26, 30]); for instance, in the absence of convexity or boundedness of the price variables that could lead to the non-existence of a NE, the existence of a QNE is by no means obvious. Building on our recent results in , a major contribution of the paper is to provide a satisfactory characterization of the QNE, in terms of existence, connection with the (L)NE, and performance. The main features of the QNE are the following: i) it always exists for the proposed class of nonconvex games; ii) it is shown to have some local optimality properties (in the sense described in Sec. 6); and iii) the proposed joint optimization of sensing and transmission strategies based on the QNE yields a considerable performance improvements (more than one hundred per cent in “high” interference regimes) with respect to both centralized  and decentralized [12, 14, 15, 17] CR designs that do not perform any joint optimization. These desired properties validate the use of the QNE both theoretically and practically.
At the QNE of the game, the optimal sensing times of the SUs tend to be different, implying that some SUs may start transmitting while some others are still in the sensing phase, which may introduce a significant performance degradation in the sensing process of the other SUs. To overcome this issue, a further contribution of the paper is to propose a distributed approach to “force” (at least asymptotically) the same optimal sensing time for all the SUs; which is referred to as the equi-sensing case.
In summary, the main contributions of the paper
are the following:
We propose a novel class of NEPs (possibly with pricing), where for the first time a joint optimization of the detection threshold, the sensing time, and the transmission strategies of multiple SUs is performed, under power and several (novel) local and/or global probabilistic interference constraints;
We introduce a relaxed equilibrium conceptthe QNEand develop a novel optimization-based theory for studying its main properties, the connection with the LNE, and its performance;
We show how to modify the original NEPs to impose in a distributed way an optimal equi-sensing time to all the SUs, and study the main properties of the resulting game.
Overall, we introduce a new line of analysis based on Variational Inequalities (VIs) in the literature of nonconvex games (possibly) with pricing that is expected to be broadly applicable for other game models.
Because of the space limitation, we omit here details on how to solve the proposed games, which is the subject of a companion paper , where we propose alternative distributed algorithms along with their convergence properties and a detailed discussion on their practical implementation; and also establish the existence of NE of the games under a set of small-interference assumptions between the SUs and PUs (that are not needed for the QNE).
The rest of the paper is organized as follows. Sec. 2 describes the system model, whereas Sec. 3 introduces the new game-theoretical formulations. The analysis of the game is addressed in Sec. 4, where the LNE and the concept of QNE are introduced, and their properties and connection are studied. Sec. 5 focuses on the equi-sensing case; whereas Sec. 6 outlines some extensions of the proposed formulations covering more general settings. Sec. 7 provides some numerical results showing the quality of the proposed approach; and finally Sec. 8 draws the conclusions.
2 System Model
We consider a scenario composed of active SUs, each formed by a transmitter-receiver pair, coexisting in the same area and sharing the same band. We focus on block transmissions over SISO frequency-selective channels. It is well known that, in such a case, multicarrier transmission is capacity achieving for large block-length. Based on the 802.22-Wireless-Regional-Area-Networks standard (WRAN) , we assume that the Medium Access Control (MAC) frame is divided in two slots: one sensing slot and one data slot. During the sensing interval, the SUs stay silent and sense the electromagnetic environment looking for the spectrum holes, whereas during the data slot they transmit (possibly) simultaneously over the portions of the licensed spectrum detected as available. The sensing and transmission phases are described in Sec. 2.1 and Sec. 2.2, respectively, whereas the proposed joint optimization of the sensing and transmission strategies over the two phases is introduced in Sec. 3.
We wish to point out that, even if we focus on ad-hoc networks and adopt the classical CR terminology (e.g., PUs, SUs, etc.), the proposed model and results are applicable to broader scenarios, wherever there are heterogeneous (small-cell) networks organized in a hierarchical or multi-tier structure and sharing the same spectrum (e.g., due to the spatial reuse of the frequencies). This happens for example in femtocell networks, where femto access points (FAPs) serve femto users, subject to some constraints on the interference radiated toward mobile users served by macro base stations.
2.1 The spectrum sensing problem
For notational simplicity, we preliminarily introduce the sensing problem in a CR scenario composed of one active PU and under some simplified assumptions; a more general setting (e.g., multiple active PUs) is addressed in Sec. 6 within the framework of robust detection.
The spectrum sensing problem of SU on subcarrier is formulated as a binary hypothesis testing, with the following two sets of hypotheses: at time index
where represents the absence of any primary signal over the subcarrier the received baseband complex signal contains only additive background noise and represents the presence of (at least) one PUthe received signal contains the primary signaling corrupted by noiseand is the number of samples, with and denoting the sensing time and the sampling frequency, respectively.
Given the hypothesis testing problem in (1), we make the following standard assumptions in the literature of sensing algorithms [3, 4, 5, 6, 7, 8, 20, 21, 23]: the noise is drawn by a circularly symmetric complex stationary (ergodic) Gaussian process with zero mean and variance ; the primary signaling are samples of a circularly symmetric complex stationary (ergodic) random process with zero mean, variance , and statistically independent of the noise process; the sample frequency is chosen so that the random variables (RVs) are independent, for all and ;111We can relax the constraint on by requiring in (A3) instead block independence of the RVs , and invoke for our derivations the central limit theorem valid for correlated (-indepedend) RVs, as stated, e.g., in [34, Th. 2.8.1]; because of the space limitation, we omit these details here and stay within the original assumption (A3). the system parametersthe spectrum occupancy status of the PUs and the primary/secondary (cross-)channelschange sufficiently slowly such that they can be assumed to be constant over each sensing/transmission interval; and the noise variance and the power of the primary signaling are estimate with no errors at the secondary receiver . Note that can be relaxed by taking explicitly into account estimation errors in the knowledge of and as well as the effect of shadowing and fading, as outlined in Sec. 6; see also .
Within the Neyman-Pearson framework, the test statistic of SU over subcarrier based on the energy detector maximizing the detection probability under a given false alarm rate is :
where is the decision threshold of SU for the carrier , to be chosen to meet the required false alarm rate. Note that if the received samples can be assumed Gaussian distributed (and uncorrelated), the above energy-based decision is optimal; otherwise it is still a valuable choice when no a-priori information is available on the primary signal features. Moreover, it is interesting to report some results in [37, 38], where it has been argued that for several models, if the probability density functions under both hypotheses are perfectly known, energy detection performs close to the optimal detector. The case of partial knowledge of the probability density functions is discussed in Sec. 6 within the context of robust detection; see also .
Invoking the central limit theorem, the random variable can be approximated for sufficiently large by a Gaussian distribution: for , , where
An explicit expression of the expected value above depends on the PUs’ signaling features. For example, if the primary signals are drawn from a PSK modulation, then ; whereas if the PU signaling is assumed to be Gaussian, then .
The performance of the energy detector scheme are measured in terms of the detection probability and false alarm probability that, under the assumptions above, are given by
where is the Q-function. The interpretation of and within the CR scenario is the following: signifies the probability of successfully identifying from the SU a spectral hole over carrier , whereas the missed detection probability represents the probability of SU failing to detect the presence of the PUs on the subchannel and thus generating interference against the PUs. The free variables to optimize are the detection thresholds ’s and the sensing times ’s; ideally, we would like to choose ’s and ’s in order to minimize both and , but (3) shows that there exists a trade-off between these two quantities that will affect both primary and secondary performance. It turns out that, ’s and ’s can not be chosen by focusing only on the detection problem (as in classical decision theory), but the optimal choice of and must be the result of a joint optimization of the sensing and transmission strategies over the two phases; such a novel optimization is formulated in Sec. 3.
2.2 The transmission phase
We model the set of the active SUs as a frequency-selective -parallel interference channel, where is the number of available subcarriers; no interference cancellation is performed at the secondary receivers and the Multi-User Interference (MUI) is treated as additive colored noise at each receiver. This model is suitable for most of the CR scenarios, where the SUs coexisting in the network operate in a uncoordinated manner without cooperating with each other, and no centralized authority is assumed to handle the network access for the SUs. The transmission strategy of each SU is then the power allocation vector over the subcarriers ( is the power allocated over carrier ), subject to the following transmit power constraints
where we also included (possibly) local spectral mask constraints [the vector inequality in (4) is component-wise].
Opportunistic throughput. The goal of each SU is to optimize his “opportunistic spectral utilization” of the licensed spectrum, having no a priori knowledge of the probabilities on the PUs’ presence. According to this paradigm, each subcarrier is available for the transmission of SU if no primary signal is detected over that frequency band, which happens with probability . This motivates the use of the aggregate opportunistic throughput as a measure of the spectrum efficiency of each SU . Given the power allocation profile of the SUs, the detection thresholds , and sensing time , the aggregate opportunistic throughput of SU is defined as
where () is the portion of the frame duration available for opportunistic transmissions, is the worst-case false alarm rate defined in (3), and is the maximum information rate achievable on secondary link over carrier when no primary signal is detected and the power allocation of the SUs is . Under basic information theoretical assumptions, the maximum achievable rate for a specific power allocation profile is
where is the channel transfer function of the direct link and is the cross-channel transfer function between the secondary transmitter and the secondary receiver ; and is the variance of the background noise over carrier at the receiver (assumed to be Gaussian zero-mean distributed).
Probabilistic interference constraints. Due to the inherent trade-off between and (cf. Sec. 2.1), maximizing the aggregate opportunistic throughput (5) of SUs will result in low and thus large , hence causing harmful interference to PUs (which happens with probability ). To control the interference radiated by the SUs, we propose to impose probabilistic interference constraints in the form of individual and/or global constraints. Individual interference constraints are imposed at the level of each SU to limit the average interference generated at the primary receiver, whereas global interference constraints limit the average aggregate interference generated by all the SUs, which is in fact the average interference experimented by the PU. Examples of such constraints are the following.
Individual interference constraints:
Global interference constraints,
where [or ] is the maximum average interference allowed to be generated by the SU [or all the SU’s], and ’s are a given set of positive weights. If an estimate of the cross-channel transfer functions between the secondary transmitters ’s and the primary receiver is available, then the natural choice of is , so that (7) and (8) become the average interference experienced at the primary receiver. In some scenarios where the primary receivers have a fixed geographical location, it may be possible to install some monitoring devices close to each primary receiver having the functionality of (cross-)channel measurement and estimation of ’s. In scenarios where this option is not feasible and the channel state information cannot be acquired, a different choice of the weights coefficients ’s and the interference threshold in (7) and (8) can be made, based e.g. on worst-case channel/interference statistics; several alternative options are discussed in our companion paper , which is devoted to the design of distributed solution algorithms along with their practical implementation.
Remark 1 (Individual vs. global constraints).
The proposed individual and/or global interference constraints provide enough flexibility to explore the interplay between performance and signaling among the SUs, making thus the proposed model applicable to different CR scenarios. For instance, in the settings where the SUs cannot exchange any signaling, the system design based on individual interference constraints seems to be the most natural formulation (see Sec. 3.1); this indeed leads to totally distributed algorithms with no signaling among the SUs, as we show in the companion paper . On the other hand, there are scenarios where the SUs may want to trade some signaling for better performance; in such cases imposing global interference constraints rather than the more conservative individual constraints is expected to provide larger SUs’ throughputs. This however introduces a new challenge: how to enforce global interference constraints in a distributed way? By imposing a coupling among the power allocations of all the SUs, global interference constraints in principle would call for a centralized optimization. A major contribution of the paper is to propose a pricing mechanism based on the relaxation of the coupling interference constraints as penalty term in the SUs’ objective functions (see Sec. 3.2). In the companion paper , we show that this formulation leads to (fairly) distributed best-response algorithms where the SUs can update the prices via consensus based-schemes, which requires a limited signaling among the SUs, in favor of better performance.
3 Joint Optimization of Sensing and Transmissions via Game Theory
We focus now on the system design and formulate the joint optimization of the sensing parameters and the power allocation of the SUs within the framework of game theory. Motivated by the discussion in the previous section (cf. Remark 1), we propose next two classes of equilibrium problems: i) games with individual constraints only; and ii) games with individual and global constraints. The former formulation is suitable for modeling scenarios where the SUs are selfish users who are not willing to cooperate, whereas the latter class of games is applicable to the design of systems where the SUs can exchange limited signaling in favor of better performance. At the best of our knowledge, both formulations based on the joint optimization of sensing and transmission strategies are novel in the literature. We introduce first the two formulations in Sec. 3.1 and Sec. 3.2 below, and then provide a unified analysis of both games.
3.1 Game with individual constraints
In the proposed game, each SU is modeled as a player who aims to maximize his own opportunistic throughput by choosing jointly a proper power allocation strategy , detection thresholds , and sensing time , subject to power and individual probabilistic interference constraints. Stated in mathematical terms, player ’s optimization problem is to determine, for given , a tuple such that
In (9) we also included additional lower and upper bounds of satisfying and upper bounds on detection and missed detection probabilities and , respectively. These bounds provide additional degrees of freedom to limit the probability of interference to the PUs as well as to maintain a certain level of opportunistic spectrum utilization from the SUs . Note that the constraints and do not represent a real loss of generality, because practical CR systems are required to satisfy even stronger constraints on false alarm and detection probabilities; for instance, in the WRAN standard, .
3.2 Game with pricing
We add now global interference constraints to the game theoretical formulation. In order to enforce coupling constraints while keeping the optimization as decentralized as possible, we propose the use of a pricing mechanism through a penalty in the payoff function of each player, so that the interference generated by all the SUs will depend on these prices. Prices are thus addition variables to be optimized (there is one common price associated to any of the global interference constraints); they must be chosen so that any solution of the game will satisfy the global interference constraints, which requires the introduction of additional constraints on the prices, in the form of price clearance conditions. Denoting by the price variable associated with the global interference constraint (8), we have the following formulation.
In (11), the compact notation means , , and . The price clearance conditions (11) state that global interference constraints (8) must be satisfied together with nonnegative price; in addition, they imply that if the global interference constraint holds with strict inequality then the price should be zero (no penalty is needed). Thus, at any solution of the game, the optimal price is such that the interference constraint is satisfied. In the companion paper , we show that a pricing-based game as (10)-(11) can be solved using best-response iterative algorithms, where the price is updated by the SUs themselves via consensus schemes; which require a limited signaling exchange only among neighboring nodes. In Sec. 7 we show that when the SUs are willing to collaborate in the form described above, they reach higher throughputs, which motivates the formulation (10)-(11).
Note that when the price is set to zero, the game (10)-(11) reduces to the NEP in (9) where only the individual interference constraints (7) are imposed. Therefore, hereafter we focus only on (10)-(11), as a unified formulation including also the game (9) as special case (when ). Note that when we deal with the general formulation (10)-(11), we make the blanket assumption that none of the power/interference constraints are redundant.
3.3 Equivalent reformulation of the games and main notation
Before starting the analysis of the proposed games, we rewrite the players’ optimization problems (10) with the side constraint (11) in a more convenient and equivalent form. The principal advantage of the reformulation is that it converts the nonlinear constraints (b) and (c) into linear constraints; thus (i) facilitating the application of the VI approach to the analysis of the game, and (ii) shedding some light on the interpretation of the new concepts and results that will be introduced later on.
At the heart of the aforementioned equivalence is a proper change of variables: instead of working with the original variables ’s and ’s, we introduce the new variables ’s and ’s, defined as
Note that the above transformation is a one-to-one mapping for any and ; we can thus reformulate the games using the new variables without loss of generality. Moreover, a property of (12) is that the constraints on and in each player’s optimization problem [see (9)] become linear in the new variables and : for each , we have
where denotes the inverse of the Q-function [ is a strictly decreasing function on ]. Using the above transformation, we can equivalently rewrite the false-alarm rate , the missed detection probability , and the throughput of each player in terms of the new variables ’s and ’s, denoted by , , and , respectively; the explicit expression of these quantities is:
Note that (even in the transformed domain), players’ optimization problems in (17) are still nonconvex, with the nonconvexity occurring in the objective function and the interference constraints (); the constraints () and () are instead linear and thus convex; the polyhedra in () may be however empty. To explore such a structure, as final step, we rewrite the game (17)-(18) in a more compact form, making explicit in the feasible set of each player the polyhedral (convex) part [constraints () and ()] and the nonconvex part [constraint ()]. Let us denote by the feasible set of player in (17), given by
where we have separated the convex part and the nonconvex part; the convex part is given by the polyhedron corresponding to the constraints () and ()
whereas the nonconvex part is given by the constraint () that we have rewritten introducing the local interference violation function
which measure the violation of the local interference constraint () at . Similarly, it is convenient to introduce also the global interference violation function :
which measures the violation of the shared constraint at .
Based on the above definitions, throughout the paper, we will use the following notation. The convex part of the joint strategy set is denoted by , whereas the set containing all the (convex part of) players’ strategy sets except the -th one is denoted by ; similarly, we define and . For notational simplicity, when it is needed, we will use interchangeably either or to denote the strategy tuple of player ; similarly, will denote the strategy profile of all the players, with , and , whereas the strategy profile of all the players except the -th one will be . Using the above definitions, game (17)-(18) can be rewritten as
Given the equivalence between (17)-(18) and (23)-(24), in the following we will focus on the game in the form (23)-(24) w.l.o.g.. For future convenience, Table 1 collects the main definitions and symbols used in (23)-(24).
|sensing time of SU|
|detection thresholds of SU over carriers|
|power allocation vector of SU|
|scalar price variable|
|(transformed) sensing time of SU [(12)]|
|(transformed) detection thresholds of SU over carriers [(12)]|
|(transformed) missed detection probability of SU on carrier [cf. (15)]|
|strategy tuple of SU (in the transformed variables)|
|strategy profile of all the SUs (in the transformed variables) except the -th one|
|strategy profile of all the SUs (in the transformed variables)|
|opportunistic throughput of SU in the transformed variables [cf. (16)]|
|local interference constraint violation of SU [cf. (21)]|
|global interference constraint violation of SU [cf. (22)]|
|,||feasible set of SU [cf. (19)], joint feasible strategy set|
|joint strategy set of the SUs except the -th one|
|,||convex part of [cf. (20)], Cartesian product of all ’s|
4 Solution Analysis
This section is devoted to the solution analysis of the game (23)-(24). It should be noted that except for the recent work , no existing theory to date is applicable to analyze this game. We start our analysis by studying the feasibility of each optimization problem in (23). We then introduce the definitions of NE, LNE, and the relaxed concept QNE, and establish their main properties.
4.1 Feasibility conditions
The feasibility of the players’ problems (23) requires that all the polyhedra defined in (20) be nonempty; the following are necessary and sufficient conditions for this to hold. Introducing the SNR detection experimented by the SU over carrier and using the definitions of and as given in Sec. 2.1, the feasibility requirements are:
The conditions above have an interesting physical interpretation that sheds light on the relationship between the achievable sensing performance and sensing/system parameters. It quantifies the trade-off between the sensing time (the product “time-bandwidth” of the system) and detection accuracy: the smaller the required false alarm and missed detection probabilities, the larger the number of samples and thus to be taken.
4.2 NE and local NE
The definitions of NE and local NE of a game with price equilibrium conditions as (23)-(24) are the natural generalization of the same concepts introduced for classical noncooperative games having no side constraints (see, e.g., [27, 29, 39]) and are given next.
Note that a NE is a LNE, but the converse is not necessarily true. The feasibility of the optimization problems in (23) [cf. (25)] is not enough to guarantee the existence of a (L)NE. The classical case where a NE exists is when the players’ objective functions (to be maximized) are (quasi-)concave in their own variables given the other players’ strategies, and the players’ constraint sets are compact and convex and independent of their rivals’ strategies (see, e.g., [27, 40]); other cases are when the game has a special structure, like potential or supermodular games. The game (23) with side constraints (24) has none of such properties; indeed, each player’s optimization problem in (23) is nonconvex with the nonconvexity occurring in the objective function and the local/global interference constraints; moreover, the set of feasible prices [satisfying (24)] is not explicitly bounded [note that these prices cannot be normalized due to the lack of homogeneity in the players’ optimization problem (23)]. Figure 1 shows an example of feasible set of the detection threshold and power allocation of a SU, for a given sensing time and ; the set is nonconvex, implying the nonconvexity of the SUs’ optimization problems in (23).
The existence of a (L)NE under the nonconvexity of the players’ optimization problems and the unboundedness of the price variables is in jeopardy; in fact, the game may not even have a NE, or the NE may exist under some abstract technical conditions (that are not easy to be checked222Conditions for the existence of a NE are established in [26, Th. 1] for NEPs (with no pricing), which may have discontinuous and/or nonquasi-concave payoff functions, but satisfying the so-called “diagonal transfer quasi-concavity” and “diagonal transfer continuity”; the case of noncompact strategy sets have been addressed in [41, Th. 5] (see also references therein). Despite their theoretical importance, the verification of these conditions for payoff functions and sets arising from realistic applications such those in this paper does not appear possible. ) requiring a particular profile of the system parameters (e.g., channel conditions, interference level, SNR detection, etc…). This would make the system behavior (e.g., equilibrium existence versus nonexistence, convergence of algorithms versus nonconvergence) unpredictable, which is not a desirable feature in practical applications. To overcome this issue, in this paper, we propose the use of a (relaxed) equilibrium conceptthe QNEwhich will be show to alway exist (even when the NE fails). The QNE along with its main properties and optimality interpretation is introduced next.
4.3 Quasi-Nash equilibrium
For a nonlinear program constrained by finitely many algebraic equations and inequalities and a differentiable objective function, stationarity is defined by the well-known Karush-Kuhn-Tucker (KKT) conditions that are necessarily satisfied by a locally optimal solution under an appropriate Constraint Qualification (CQ), see [42, Prop. 1.3.4]; solutions of the KKT system are called stationary solutions of the associated optimization problem. In the context of nonconvex problems, the common approach well-accepted in practice, is then to look for a stationary (possibly locally optimal) solution. Here we apply this approach to nonconvex games and introduce the concept of QNE of the nonconvex game (23)-(24), defined as a stationary solution of the game, which is a tuple that satisfies the KKT conditions of all the players’ optimization problems (23) along with the price equilibrium constraints (24). The prefix “quasi” is intended to signify that a NE (if it exists) must be a QNE under mild CQs.333Note that without some CQs being satisfied, the KKT conditions may not be even valid necessary conditions of optimality for the optimization problems (23), making the QNE meaningless. A direct study of the main properties of the QNE (e.g., existence and uniqueness) based on the KKT conditions of the game is not an easy task; thus to simplify the analysis, we rewrite first the aforementioned KKT conditions as a proper VI problem ; then, building on VI tools, we provide a satisfactory characterization of the QNE. It should be noted that unlike a single optimization problem where a stationary solution must exist (under a CQ) if an optimal solution exists, this is not the case for a game because a QNE is the result of concatenating the KKT conditions of several optimization problems.
At the basis of our approach there is an equivalent and nontrivial reformulation of the necessary conditions for a tuple to be a (L)NE of (23)-(24), which explores the structure of the players’ feasible sets , as described next. The classical approach to write the KKT conditions of each player’s optimization problem would be introducing multipliers associated with all the constraints in the set both the convex part and the nonconvex part [cf. (19)]and then maximizing the resulting Lagrangian function over the whole space (i.e., considering an unconstrained optimization problem for the Lagrangian maximization). The approach we follow here is different: instead of explicitly accounting all the multipliers as variables of the KKT system, for each player’s optimization problem, we introduce multipliers only for the nonconvex constraints [i.e., ()], and retain the convex part as explicit constraints in the maximization of the resulting Lagrangian function. The mathematical formulation of this idea is given next. Denoting by the multiplier associated with the nonconvex constraint of player , the Lagrangian function of player is:
which depends also on the strategies of the other players and the price . Building on Definition 2 it is not difficult to see that if is a (L)NE of (23)-(24) and some CQ holds at , there exist multipliers associated with the local nonconvex constraints such that the tuple satisfies