Lorenzo Lampariello

Parallel and Distributed Methods for Constrained Nonconvex Optimization—Part I: Theory

In this two-part paper, we propose a general algorithmic framework for the minimization of a nonconvex smooth function subject to nonconvex smooth constraints, and also consider extensions to some structured, nonsmooth problems. The algorithm solves a sequence of (separable) strongly convex problems and maintains feasibility at each iteration. Convergence to a stationary solution of the original nonconvex optimization is established. Our framework is very general and flexible and unifies several existing successive convex approximation (SCA)-based algorithms. More importantly, and differently from current SCA approaches, it naturally leads to distributed and parallelizable implementations for a large class of nonconvex problems. This Part I is devoted to the description of the framework in its generality. In Part II, we customize our general methods to several (multiagent) optimization problems in communications, networking, and machine learning; the result is a new class of centralized and distributed algorithms that compare favorably to existing ad-hoc (centralized) schemes.

Parallel and Distributed Methods for Nonconvex Optimization-Part II: Applications.

In this two-part paper, we propose a general algorithmic framework for the minimization of a nonconvex smooth function subject to nonconvex smooth constraints, and also consider extensions to some structured, nonsmooth problems. The algorithm solves a sequence of (separable) strongly convex problems and maintains feasibility at each iteration. Convergence to a stationary solution of the original nonconvex optimization is established. Our framework is very general and flexible and unifies several existing successive convex approximation (SCA)-based algorithms. More importantly, and differently from current SCA approaches, it naturally leads to distributed and parallelizable implementations for a large class of nonconvex problems. This Part I is devoted to the description of the framework in its generality. In Part II, we customize our general methods to several (multiagent) optimization problems in communications, networking, and machine learning; the result is a new class of centralized and distributed algorithms that compare favorably to existing ad-hoc (centralized) schemes.

Parallel and distributed methods for nonconvex optimization

We propose a general algorithmic framework for the minimization of a nonconvex smooth function subject to nonconvex smooth constraints. The algorithm solves a sequence of (separable) strongly convex problems. Convergence to a stationary solution of the original nonconvex optimization is established. Our framework is very general and flexible; it unifies several existing Successive Convex Approximation (SCA)-based algorithms such as (proximal) gradient or Newton type methods, block coordinate (parallel) descent schemes, difference of convex functions methods, and improves on their convergence properties. More importantly, and differently from current SCA schemes, it naturally leads to distributed and parallelizable schemes for a large class of nonconvex problems. The new method is applied to the solution of a new rate profile optimization problem over Interference Broadcast Channels (IBCs); numerical results show that it outperforms existing ad-hoc algorithms.

VI-constrained hemivariational inequalities: distributed algorithms and power control in ad-hoc networks

We consider centralized and distributed algorithms for the numerical solution of a hemivariational inequality (HVI) where the feasible set is given by the intersection of a closed convex set with the solution set of a lower-level monotone variational inequality (VI). The algorithms consist of a main loop wherein a sequence of one-level, strongly monotone HVIs are solved that involve the penalization of the non-VI constraint and a combination of proximal and Tikhonov regularization to handle the lower-level VI constraints. Minimization problems, possibly with nonconvex objective functions, over implicitly defined VI constraints are discussed in detail. The methods developed in the paper are then used to successfully solve a new power control problem in ad-hoc networks.

Partial penalization for the solution of generalized Nash equilibrium problems

In this paper we reformulate the generalized Nash equilibrium problem (GNEP) as a nonsmooth Nash equilibrium problem by means of a partial penalization of the difficult coupling constraints. We then propose a suitable method for the solution of the penalized problem and we study classes of GNEPs for which the penalty approach is guaranteed to converge to a solution. In particular, we are able to prove convergence for an interesting class of GNEPs for which convergence results were previously unknown.

D3M: Distributed multi-cell multigroup multicasting

The paper studies the max-min fair multicast multigroup beamforming problem in a multi-cell environment, with perfect (instantaneous or statistical) Channel State Information (CSI). We propose a new general distributed algorithmic framework based on INner Convex Approximations (INCA): the nonsmooth NP-hard problem is replaced by a sequence of smooth strongly convex subproblems, which can be solved in a distributed fashion across the cells, with limited communication overhead. Differently from renowned semidefinite-relaxation-based schemes, the INCA algorithm is proved to always converge to a d-stationary solution of the aforementioned class of problems. Numerical results show that it compares favorably with state-of-the-art algorithms.

Equilibrium selection in power control games on the interference channel

In recent years, game-theoretic tools have been increasingly used to study many important resource allocation problems in communications and networking. One common feature shared by all these approaches is that, when it comes to (distributed) computation of equilibria, assumptions are always made that imply uniqueness of the Nash Equilibrium. This simplifies considerably the analysis of the games under investigation and permits to design distributed solution methods with convergence guarantee. However, requiring the uniqueness of the solution may be too demanding in many practical situations, thus strongly limiting the applicability of current game theoretical methodologies. In this paper, we overcome this limitation and propose novel distributed algorithms for noncooperative games having multiple solutions. The new methods, whose convergence analysis is based on variational inequality techniques, are able to select, among all the equilibria of a game, those which optimize a given performance criterion. We apply the developed methods to a power control problem over parallel Gaussian interference channels and show that they yield a considerable performance improvement over classical power control schemes.

Feasible methods for nonconvex nonsmooth problems with applications in green communications

We propose a general feasible method for nonsmooth, nonconvex constrained optimization problems. The algorithm is based on the (inexact) solution of a sequence of strongly convex optimization subproblems, followed by a step-size procedure. Key features of the scheme are: (i) it preserves feasibility of the iterates for nonconvex problems with nonconvex constraints, (ii) it can handle nonsmooth problems, and (iii) it naturally leads to parallel/distributed implementations. We illustrate the application of the method to an open problem in green communications whereby the energy consumption in MIMO multiuser interference networks is minimized, subject to nonconvex Quality-of-Service constraints.