In the first part [1] of this review the heat exchanger network (HEN) synthesis problem has been described and analyzed. Also, targeting approaches for setting up lower bounds on HEN-s cost prior to the synthesis have been reviewed as well as deep insight based methods of synthesis closely linked with targeting concepts. In this part of the paper the second broad class of synthesis methodologies is reviewed. These are mathematical based methods, using optimization techniques such as mixed linear and/or mixed nonlinear programming. Part II also includes a review of existing methods for HEN retrofit design. The present author assumed that a reader will start with Part I of the review since a knowledge of that part is necessary to understand the methods described in the following.