Convex separable problems with linear and box constraints

Abstract

In this work, we focus on separable convex optimization problems with linear and box constraints and compute the solution in closed-form as a function of some Lagrange multipliers that can be easily computed in a finite number of iterations. This allows us to bridge the gap between a wide family of power allocation problems of practical interest in signal processing and communications and their efficient implementation in practice.

On solving convex optimization problems with linear ascending constraints

“…If one denotes the arithmetic complexity of solving equations in (32) by O(I ), then the total arithmetic complexity of the P-S algorithm is O(n 2 I ). 4 Therefore, our algorithm works better than the P-S algorithm when solving equations in (32) has similar complexity as solving equations in (18) and (21), but may work relatively worse if (32) can be solved explicitly (see [13] for several examples). This tradeoff is demonstrated in the numerical experiments in Sect.…”

“…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.…”

On the Minimization of Convex Functionals of Probability Distributions Under Band Constraints

“…An approach similar to the one presented here was used in [19] to solve a constrained minimization problem with a separable cost function. However, neither is the problem in [19] a special case of the problem investigated in this paper, nor vice versa.…”

“…The minimization of convex functionals, and their relation to robust decision making, has been addressed by Huber [16], Poor [17], Kassam [6], and Guntuboyina [18] to name just a few. An approach similar to the one presented here was used in [19] to solve a constrained minimization problem with a separable cost function. However, neither is the problem in [19] a special case of the problem investigated in this paper, nor vice versa.…”