The Southwest evokes images of dusty desert landscapes beset with narrow mountain ranges, of the vast and colorful expanses of the Colorado Plateau and land of the Diné, of Monument Valley, and perhaps of the Spanish and Mexican cultural heritage. Terrains range from barren, seemingly lifeless, deserts to verdant, forested mountains, and vegetation zones from Sonoran to Alpine. Its varied landscapes have challenged explorers and settlers; beckoned artists and adventurers. They may elicit wonderment and awe but can be haunting, even intimidating. The beauty of the Southwest is often stark, typically subtle.

The Southwest of the United States is a region that defies precise geographic definition, eschewing neatly defined physiographic subdivisions. Most usages of the term ‘Southwest’ include the arid and semiarid region stretching from west Texas across New Mexico and Arizona to southern California (Fig. I.1; see also Plate 1). This vast expanse of desert, the heart of the Southwest, comprises dominantly the Basin and Range and Colorado Plateau physiographic provinces. The Basin and Range province, typifying the southern parts of New Mexico and Arizona and northern Mexico, is that region characterized by a distinctive physiography of narrow mountain ranges separated by broad, sediment-filled desert basins. In contrast, the Plateau is that region of northern New Mexico, northern Arizona, western Colorado, and much of Utah characterized by broad plateaus, deeply incised canyons, and mainly flat-lying sedimentary strata.

Introduction

The latest chapter in the geological development of the Southwest encompasses the most recent 35 Myr or so of the Earth's history. For perspective, recall that 35 Myr is less than one percent of the age of the Earth. Despite this short period of time, it is the time about which geologists know the most. The present chapter is defined principally by events following cessation of widespread crustal shortening in the western Cordillera. Extension and vertical crustal adjustments replaced shortening.

Despite the fact that the Earth's tectonic plates were nearly in their present relative positions in the Oligocene (Fig. 8.1), a profound change in plate geometry occurred during the latest 35 Myr along the western margin of North America. The Farallon plate, whose subduction beneath the North American plate for more than 140 Myr created the series of nearly continuous orogenies in western North America, was gradually and nearly completely consumed. Only fragments of the original Farallon plate, all carrying different names, remain along the margins of North and South America. Disappearance of the Farallon plate had major implications for the tectonics in the interior of the plate, dramatically affecting the geology and landscape of the Southwest. The change in plate geometry and relative plate motions resulted in massive magmatism and a different kind of orogeny, an ‘extensional orogeny,’ which radically reshaped the landscape.

Introduction: supercontinent Rodinia

From a variety of evidence, including the global pattern of Proterozoic rocks, geologists now recognize that Mesoproterozoic to Neoproterozoic Laurentia was actually part of a supercontinent, i.e. a ‘continent’ comprising continental-scale crustal blocks assembled separately and brought together by plate-tectonic processes. The Southwest and all of present North America lay interior to this vast supercontinent. The present chapter describes the demise of this supercontinent by rifting, and the transition from rifting to drifting to form a passive margin. Part of the evidence that a continent once larger than the present North America existed is the abrupt westward termination of southwest-trending Paleoproterozoic to Mesoproterozoic age provinces (Fig. 1.3). Much of the evidence is provided by Neoproterozoic strata, which are interpreted to indicate a west-facing passive margin with no sign of a continent to the west. Because Mesoproterozoic rocks of the western USA and Canada (including those of the Grand Canyon Supergroup) are interpreted as intracratonic in their setting, i.e. deposited within the interior of a continent, formation of a passive margin implies that such a continent was subsequently rifted apart. Some contention exists concerning priorities for naming the Mesoproterozoic to Neoproterozoic supercontinent (Young, 1995), but ‘Rodinia’ seems to be the most accepted. In addition, the timing of breakup is not as well determined as geologists might like. Very possibly a second supercontinent – Pannotia – formed as the pieces of Rodinia recollided in a different configuration.

Mesoproterozoic and Neoproterozoic stratified rocks

Following the formation of the crust in the Southwest, completed during the Grenville orogeny 1.2–1.0 Ga, an enormous expanse of time was to pass before the widespread, relatively continuous and well-preserved rocks of the Phanerozoic Era were deposited. The present chapter covers part of that interval of time, from roughly 1.35 Ga to about 780 Ma, a length of time approximately equal to the entire Phanerozoic.

This chapter represents a considerable overlap in time with the previous chapter, but focuses on a completely different group of rocks. The rocks covered in the present chapter consist almost entirely of stratified rocks, mostly sedimentary but in part igneous, which were laid down on the previously formed crust. They rode high upon that crust and therefore escaped the great crushing stresses and high temperatures that accompanied the processes of initial crustal formation. They are somewhat arbitrarily distinguished from the little-metamorphosed stratified rocks of the previous chapter by having been deposited in intracratonic basins far from the continental margins. The beginning of this chapter at about 1.35 Ga is the earliest time for which such stratified rocks are preserved; the cutoff at 780 Ma, although it may appear to be arbitrary, signifies a major change in the tectonic events of the Southwest.

The overlap in time with the previous chapter comes about because, as discussed there, the formation of the crust in the Southwest occurred over an interval of 700 Myr, from assembly of the Yavapai province at roughly 1.7 Ga to the Grenville at about 1 Ga.

This book is about convex optimization, a special class of mathematical optimization problems, which includes least-squares and linear programming problems. It is well known that least-squares and linear programming problems have a fairly complete theory, arise in a variety of applications, and can be solved numerically very efficiently. The basic point of this book is that the same can be said for the larger class of convex optimization problems.

While the mathematics of convex optimization has been studied for about a century, several related recent developments have stimulated new interest in the topic. The first is the recognition that interior-point methods, developed in the 1980s to solve linear programming problems, can be used to solve convex optimization problems as well. These new methods allow us to solve certain new classes of convex optimization problems, such as semidefinite programs and second-order cone programs, almost as easily as linear programs.

The second development is the discovery that convex optimization problems (beyond least-squares and linear programs) are more prevalent in practice than was previously thought. Since 1990 many applications have been discovered in areas such as automatic control systems, estimation and signal processing, communications and networks, electronic circuit design, data analysis and modeling, statistics, and finance. Convex optimization has also found wide application in combinatorial optimization and global optimization, where it is used to find bounds on the optimal value, as well as approximate solutions.