Convex optimization has an increasing impact on many areas of mathematics, applied sciences, and practical applications. It is now being taught at many universities and being used by researchers of different fields. As convex analysis is the mathematical foundation for convex optimization, having deep knowledge of convex analysis helps students and researchers apply its tools more effectively. The main goal of this book is to provide an easy access to the most fundamental parts of convex analysis and its applications to optimization. Modern techniques of variational analysis are employed to clarify and simplify some basic proofs in convex analysis and build the theory of generalized differentiation for convex functions and sets in finite dimensions. We also present new applications of convex analysis to location problems in connection with many interesting geometric problems such as the Fermat-Torricelli problem, the Heron problem, the Sylvester problem, and their generalizations. Of course, we do not expect to touch every aspect of convex analysis, but the book consists of sufficient material for a first course on this subject. It can also serve as supplemental reading material for a course on convex optimization and applications.

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Boris Mordukhovich is Distinguished University Professor of Mathematics at Wayne State University holding also Chair Professorship of Mathematics and Statistics at the King Fahd University of Petroleum and Minerals. He has more than 350 publications including several monographs. Among his best known achievements are the introduction and development of powerful constructions of generalized differentiation and their applications to broad classes of problems in variational analysis, optimization, equilibrium, control, economics, engineering, and other fields. Mordukhovich is a SIAM Fellow, an AMS Fellow, and a recipient of many international awards and honors including Doctor Honoris Causa degrees from six universities over the world. He is named a Highly Cited Researcher in Mathematics by the Institute of Scientific Information (ISI). His research has been supported by continued grants from the National Science Foundations.Nguyen Mau Nam received his B.S. from Hue University, Vietnam, in1998 and a Ph.D. from Wayne State University in 2007. He is currently Assistant Professor of Mathematics at Portland State University. He has around 30 papers on convex and variational analysis, optimization, and their applications published in or accepted by high-level mathematical journals. His research is supported by a collaboration grant from the Simons Foundation.