Disciplined Convex Programming∗

Convex programming is a subclass of nonlinear programming (NLP) that unifies and generalizes least squares (LS), linear programming (LP), and convex quadratic programming (QP). This generalization is achieved while maintaining many of the important, attractive theoretical properties of these predecessors. Numerical algorithms for solving convex programs are maturing rapidly, providing reliability, accuracy, and efficiency. A large number of applications have been discovered for convex programming in a wide variety of scientific and non-scientific fields, and it seems clear that even more remain to be discovered. For these reasons, convex programming arguably has the potential to become a ubiquitous modeling technology alongside LS, LP, and QP. Indeed, efforts are underway to develop and teach it as a distinct discipline [BV04, BNO04, Nes04]. Nevertheless, there remains a significant impediment to the more widespread adoption of convex programming: the high level of expertise required to use it. With mature technologies such as LS, LP, and QP, problems can be specified and solved with relatively little effort, and with at most a very basic understanding of the computations involved. This is not the case with general convex programming. That a user must understand the basics of convex analysis is both reasonable and unavoidable; but in fact, a much deeper understanding is required. Furthermore, a user must find a way to transform his problem into one of the many limited standard forms; or, failing that, develop a custom solver. For potential users whose focus is the application, these requirements can form a formidable “expertise barrier”—especially if it is not yet certain that the outcome will be any better than with other methods. The purpose of the work presented here is to lower this barrier. In this article, we introduce a new modeling methodology called disciplined convex programming. As the term “disciplined” suggests, the methodology imposes a set of conventions that one must follow when constructing convex programs. The conventions are simple and teachable, taken from basic principles of convex analysis, and inspired by the practices of those who regularly study and apply convex optimization today. Conforming problems are