This chapter proposes the use of modern first-order large-scale optimization techniques to manage a cloud-based densely deployed next-generation wireless network. In the first part of the chapter we survey a few popular first-order methods for large-scale optimization, including the block coordinate descent (BCD) method, the block successive upper-bound minimization (BSUM) method and the alternating direction method of multipliers (ADMM). In the second part of the chapter, we show that many difficult problems in managing large wireless networks can be solved efficiently and in a parallel manner, by modern first-order optimization methods. Extensive numerical results are provided to demonstrate the benefit of the proposed approach.

Introduction

Motivation

The ever-increasing demand for rapid access to large amounts of data anywhere anytime has been the driving force in the current development of next-generation wireless network infrastructure. It is projected that within 10 years, the wireless cellular network will offer up to 1000× throughput performance over the current 4G technology [1]. By that time the network should also be able to deliver a fiber-like user experience, boasting 10 Gb/s individual transmission rate for data-intensive cloud-based applications.

Achieving this lofty goal requires revolutionary infrastructure and highly sophisticated resource management solutions. A promising network architecture to meet this requirement is the so-called cloud-based radio access network (RAN), where a large number of networked base stations (BSs) are deployed for wireless access, while powerful cloud centers are used at the back end to perform centralized network management [1–4]. Intuitively, a large number of networked access nodes, when intelligently provisioned, will offer significantly improved spectrum efficiency, real-time load balancing and hotspot coverage. In practice, the optimal network provisioning is extremely challenging, and its success depends on smart joint backhaul provisioning, physical layer transmit/receive schemes, BS/user cooperation and so on.

This chapter proposes the use of modern first-order large-scale optimization techniques to manage a cloud-based densely deployed next-generation wireless network. We show that many difficult problems in this domain can be solved efficiently and in a parallel manner, by advanced optimization algorithms such as the block successive upper-bound minimization (BSUM) method and the alternating direction methods of multipliers (ADMM) method.

The organization of the chapter

To begin with, we introduce a few well-known first-order optimization algorithms. Our focus is on algorithms suitable for solving problems with certain block-structure, where the optimization variables can be divided into (possibly overlapping) blocks.

Many important engineering problems can be cast in the form of a quadratically constrained quadratic program (QCQP) or a fractional QCQP. In general, these problems are nonconvex and NP-hard. This chapter introduces a semidefinite programming (SDP) relaxation procedure for this class of quadratic optimization problems which can generate a provably approximately optimal solution with a randomized polynomial time complexity. We illustrate the use of SDP relaxation in the context of downlink transmit beamforming, and show that the SDP relaxation approach can either generate the global optimum solution, or provide an approximately optimal solution with a guaranteed worst-case approximation performance. Moreover, we describe how the SDP relaxation approach can be used in magnitude filter design and in magnetic resonance imaging systems.

Introduction

In this chapter, we consider several classes of nonconvex quadratic constrained quadratic programs (QCQPs) and a class of nonconvex fractional QCQPs. The importance of these classes of problems lies in their wide-ranging applications in signal processing and communications which include:

• the Boolean least-squares (LS) problem in digital communications [1];

• the noncoherent maximum-likelihood detection problem in multiple-input multipleoutput (MIMO) communications [2, 3];

• the MAXCUT problem in network optimization [4];

• the large-margin parameter estimation problem in automatic speech recognition [5–8];

• the optimum coded waveform design for radar detection [9];

• the image segmentation problem in pattern recognition [10];

• the magnitude filter design problem in digital signal processing [11];

• the transmit B1 shim and specific absorption rate computation in magnetic resonance imaging (MRI) systems [12, 13];

• […]

This is the first of three chapters in which we derive some necessary optimality conditions for the MPEC (1.1.1). This chapter is concerned with the fundamental first-order conditions; Chapter 4 deals with the verification of hypotheses required for these first-order conditions; and Chapter 5 is concerned with the second-order conditions. Although we have seen that the MPEC (1.1.1) can be formulated as a standard nonlinear program via the KKT formulation of the inner VIs (see, e.g., Theorem 1.3.5), for reasons to be given in this chapter a straightforward application of the optimality conditions in classical nonlinear programming is inappropriate.

One approach to deriving optimality conditions of an MPEC is to use a differentiable exact penalty equivalent of the problem, such as (2.4.7) or (2.4.10), and apply classical results to such a formulation. A drawback of this approach is that strong assumptions are required for the MPEC to be equivalent to a differentiable constrained optimization problem; see Theorem 2.4.7. Even in the case of the MPAEC (Theorem 2.4.4), the equivalent formulation requires the restrictive nondegeneracy assumption. For the MPAEC, one could use the less restrictive Theorem 2.4.3; although the min function is nondifferentiable, it is simple enough to make the formulation (2.4.6) a viable candidate for this purpose; see Subsection 2.4.3. Proposition 2.4.9 provides a preliminary set of necessary conditions for a local minimum of the MPAEC.

In this chapter, we abandon the exact penalty function approach for the derivation of optimality conditions of MPEC. Instead, we resort to first principles to understand the stationarity conditions and the tangent cone of the feasible region of MPEC. The chapter contains four sections.

As for a standard NLP, we can derive some second-order necessary and second-order sufficient conditions for a local minimum of an MPEC. This chapter is a foray into these second-order conditions. For convenience, throughout we require polyhedrality of the upper-level feasible region Z; initially, we also assume the same for the constraint set C(x) in the lower-level VI for all x ∈ dom(C). In standard nonlinearly constrained NLPs, second-order conditions at boundary points must generally account for the curvature of the boundary. Such curvature requirement is usually contained in the positive definiteness properties of the partial Hessian matrix of the Lagrangean function of the nonlinear program, which contains not only the Hessian matrix of the objective function, but also the sum of the Hessian matrices of the active constraint functions, using the KKT multipliers as weights; cf. the discussion on SCOC in Subsection 4.2.7. Initially we confine our interest to polyhedral sets to keep the ideas and analysis relatively simple. Even in this situation, the treatment is complicated by the equilibrium constraint (x,y) ∈ Gr S which, as we have seen in the case of the first-order analysis, leads to some combinatorial considerations that are not present in standard nonlinear programming. Indeed, such complications become more pronounced as we work with the second-order conditions in this chapter.

This chapter discusses a multiplier-based approach, an implicit programming approach, and a piecewise programming approach to the derivation of second-order optimality conditions of MPECs; the treatment here extends ideas of the previous two chapters. Overall, the development here is parallel to that of Chapter 4. Specifically, besides the next section which reviews known NLP theory, there are five sections in the chapter.

In Section 3.3, we have introduced the full, extreme, and basic CQs for an MPEC. Under any one of these CQs, stationarity of a feasible solution to an MPEC can be characterized; see Corollary 3.3.1, Theorem 3.3.4, and Remark 3.3.5. In this chapter, we consider some important cases of the general MPEC (3.1.1) and introduce certain assumptions in order to verify such CQs, thus completing the task of deriving first-order optimality conditions for the MPEC.

This chapter has four main sections. Section 4.1 considers the AVI constrained mathematical program, i.e., the MPAEC. The main goal of this section is to demonstrate that the extreme CQ must hold without any particular assumptions. This conclusion is consistent with a standard nonlinear program with linear constraints for which the KKT conditions must be necessary for local optimality. The next two sections, 4.2 and 4.3, discuss two general approaches to deal with the MP with nonlinear VI constraints. The approach in Section 4.2 is of interest because of the bilevel nature of the MPEC and the special significance of the variables x and y, whereas the approach in Section 4.3 is more along the line of traditional nonlinear programming, but with an important modification to deal with the combinatorial feature of the MPEC. The chapter concludes with a discussion of an exact penalization of order 1, using the assumptions set forth in Section 4.2.

The two approaches discussed in Sections 4.2 and 4.3 together provide a powerful tool for the study of the MPEC. They are based on different assumptions and are therefore applicable to different classes of MPECs.

In the previous three chapters, we studied the necessary optimality conditions and sufficient optimality conditions of the MPEC. This chapter discusses three general classes of iterative algorithms for computing a stationary point of an MPEC. The first class of algorithms is based on a penalty interior point approach (PIPA); the second class of algorithms is based on an implicit programming approach. The third class of algorithms is of the Newton type for solving MPEC, based on the piecewise programming formulation. For the latter algorithms, some conditions pertaining to the relaxed nonlinear program introduced at the end of Subsection 4.3.1 turn out to provide useful conditions for their convergence. In essence, the interior point approach is applicable to MPECs where the lower-level problems possess certain generalized monotonicity properties; the second approach relies on the implicit program formulation of the MPEC; in particular, the SCOC assumption introduced in Subsection 4.2.7 will play an important role in this approach. In both approaches, we establish that any accumulation point of the iterates produced by the algorithms must be a stationary point of MPEC provided that the point satisfies a strict complementarity assumption. The third class of algorithms is an extension of some locally convergent Newton methods for solving smooth nonlinear programs extended to a piecewise smooth setting; thus these algorithms are locally convergent in the sense that a closeness assumption is required on the initial iterate in order to guarantee convergence. Like their smooth counterpart, the piecewise smooth Newton algorithms have superlinear (and even quadratic) rates of convergence under mild assumptions. The chapter ends with a section that reports some preliminary computational results with a MATLAB implementation of PIPA; see Section 6.5.