In the past decade or so, (deep) neural networks have captured people’s imagination through their empirical success in learning problems involving real-world high-dimensional data such as images, speech, and text [LBH15]. Nevertheless, there is quite a bit of mystery as to how deep networks achieve such striking results. Modern deep networks are typically designed through trial and error.

In the previous theoretical Part I of the book, we showed that under fairly broad conditions on the number of measurements needed, many important classes of structured signals can be recovered via computationally tractable optimization problems, such as ℓ1 minimization for recovering sparse signals and nuclear norm minimization for recovering low-rank matrices.

Chapter 1 describes the main objectives of the book. It argues that uncertainties are omnipresent in all aspects of the design, analysis, construction, operation, and maintenance of structures and infrastructure systems. It sets three goals for engineering of constructed facilities under conditions of uncertainty: safety, serviceability, and optimal use of resources. It then argues that probability theory and Bayesian statistics provide the proper mathematical framework for assessing safety and serviceability and for formulating optimal design under uncertainty. The chapter provides a brief review of the history and key developments of the field during the past 100 years. Also described are commercial and free software that can be used to carry out the kind of analyses that are described in the book. The chapter ends with a description of the organization of the book and outlines of the subsequent chapters.

This topic relates to capital budgeting. The starting point is an explanation of why investment analysis is important in managerial economics, and the different types of investment and investment decision. Cash flow analysis and the principles involved in identifying and measuring relevant cash flows is discussed. The concept of risk and types of risk, stand-alone risk, within-firm risk and market risk, are discussed. The security market line (SML), beta coefficients and the capital asset pricing model (CAPM) are explained. The cost of capital is examined, explaining the calculation of the cost of debt, the cost of equity and the weighted average cost of capital (WACC). Methods of evaluation of individual projects are discussed, with a focus on net present value (NPV) and internal rate of return (IRR). There is a discussion of the determination of the optimal capital budget for a firm, in terms of the investment opportunity schedule (IOS) and the marginal cost of capital (MCC), with the distinction between mutually exclusive projects and independent projects. Case studies include two resource-heavy situations: the HS2 rail link and 5G telecommunications.

As engineering and the sciences become increasingly data- and computation-driven, the role of optimization has expanded to touch almost every stage of the data analysis pipeline, from the signal and data acquisition to modeling, analysis, and prediction.

In this chapter, we present an application of compressive sensing to a crucial problem in modern wireless (radio) communication: How can cognitive radios efficiently identify available spectrum? We will see that this problem can be cast as one of recovering the support of a sparse signal, in the presence of noise. We will see how the methods and algorithms described in this book will allow us to break theoretical limits of conventional approaches, and, once properly implemented in hardware, they can significantly advance the state of the art, by enabling better tradeoffs between energy consumption and scan time. Besides its practical importance, this application is very interesting as it is kind of dual to the situation in the magnetic resonance imaging that we studied in the preceding chapter. In MRI, the measurements are the Fourier transform of the image of interest and the sparse patterns are in the image domain; whereas for spectrum sensing, the sparse patterns are in the Fourier domain which we do not measure directly.

In the previous chapter, we saw many problems for which the goal is to find a sparse solution to an underdetermined linear system of equations y = Ax. This problem is NP-hard in general. However, we also observed that certain well-structured instances can be solved efficiently: in experiments, when y = Axo and xo was sufficiently sparse, tractable ℓ1 minimization