An individual ant constructs candidate solutions by starting with an empty solution and then iteratively adding solution components until a complete candidate solution is generated.
These solution concepts which allow coalitions to make agreements simultaneously typically suffer from incompatibility of agreements, which can give rise to empty solution sets in games of networking interest.
More precisely, we use the greedy recreate operator (described further) on an empty solution, once per objective.
Then, the inequality (7) has a non-empty solution set S ε for each α > 0 and f δ ∈ X*.
Thus, the more distant the competing transmitters, the lower the interference and, thus, the higher the probability of having a non-empty solution set [21,23].
Sufficient conditions for the feasibility of (17), i.e., there is a non-empty solution set, are derived considering that the QoS constraint must hold true (forall k in {mathcal {K}}).
The proposed best response-based algorithm converges under the same conditions for which the GNE is unique (stated in Proposition 5), which in turn requires the condition stated in Proposition 3 in order to have a non-empty solution set.
Since the objective function in (16) is strictly concave in p q ∈ P q, for any given p−q, and the feasible set P q is compact and convex, game G 1 admits a non-empty solution set for any set of channels and transmit power constraints of the users.
Let ((P_{0}, d_{P_{0}})) be the metric space of data of (SSGVQEP) problems mentioned above, let ((Xtimes Y, d_{Xtimes Y})) be the metric space for the solutions of a problem p in ((P_{0}, d_{P_{0}})) and Γ be the solution mapping from the space ((P_{0}, d_{P_{0}})) of problems to the space (2^{Xtimes Y}) of all non-empty solution subsets in ((Xtimes Y, d_{Xtimes Y})).
This set may be empty (no solution), a singleton (unique solution) or with more than one element (multiple solutions).
