Convex Separable Problems With Linear Constraints in Signal Processing and Communications

Abstract

In this paper, we focus on separable convex optimization problems with box constraints and a specific set of linear constraints. The solution is given in closed-form as a function of some Lagrange multipliers that can be computed through an iterative procedure in a finite number of steps. Graphical interpretations are given casting valuable insights into the proposed algorithm and allowing to retain some of the intuition spelled out by the waterfilling policy. It turns out that it is not only general enough to compute the solution to different instances of the problem at hand, but also remarkably simple in the way it operates. We also show how some power allocation problems in signal processing and communications can be solved with the proposed algorithm.

Convex separable problems with linear and box constraints

“…The proof of the above lemma can be found in [12] and can be used to find an equivalent form of (1). To see how this comes about, denote by A {1, .…”

Fast Computation of Generalized Waterfilling Problems

“…Simulation results show that the proposed solutions requires significantly less number of flops. Application of the 'equivalence' principle to more complicated waterfilling problems like 'cave waterfilling' and 'multiple waterlevel' problems [5], [32], [33] needs to be studied as it is not straight forward.…”

On solving convex optimization problems with linear ascending constraints

“…For example, they occur in the design of multiple-input multiple-output (MIMO) systems dealing with the minimization of the power consumption while meeting the quality-of-service (QoS) requirements, in the design of optimal training sequences for channel estimation in multi-hop transmissions and in the optimal power allocation in amplify-and-forward multi-hop transmissions under short-term power constraints. We refer the readers to D'Amico et al [4,5], Padakandla and Sundaresan [12,13] and Viswanath and Anantharam [18] for more detailed discussions on the applications in this field. In addition to the above applications, we present another motivation of this model in operations management.…”

“…In the numerical experiments, we assume that o i ∼ U [5,10], u i ∼ U [20, 25] and α i ∼ U [0, 20]. We also assume that each D i follows an exponential distribution with parameter η i ∼ U [0.1, 0.2].…”