In the last three decades, a number of fundamental and practical results have been obtained in the area of convex optimization theory. It is a well-developed area, both in the theoretical and practical aspects, and the engineering community has greatly benefited from these recent advances by finding key applications.

This graduate course introduces convex optimization theory and illustrates its use with many applications where convex and nonconvex formulations arise. The emphasis will be on i) the art of unveiling the hidden convexity of problems by appropriate manipulations, ii) a proper characterization of the solution either analytically or algorithmically, and iii) multiple practical ways to approach nonconvex problems.

The course follows a case-study approach by considering recent successful applications of convex optimization published within the last decade in top scientific journals in the areas of signal processing, finance, machine learning, and big data. Problems covered include portfolio optimization in financial markets, filter design, beamforming design in wireless communications, classificationin machine learning, circuit design, robust designs under uncertainty, sparse optimization, low-rank optimization, graph learning from data, discrete maximum likelihood decoding, network optimization, distributed algorithms, Internet protocol design, etc.