His current research is in the area of algorithms for constrained optimization and interior point methods for linear and nonlinear programming.

In 1967, he received his Ph.D. from University of California, Los Angeles, in mathematics with the dissertation: "A Generalization of Newton's Method with an Application to the Euler–Lagrange Equation"[7] under the advisors: Magnus Hestenes, Charles Tompkins[8] Tapia also holds honorary doctorates from Carnegie Mellon University and the Colorado School of Mines.

[10] From 2001 to 2004, Tapia chaired the National Research Council's Board of Higher Education and workforce.

[11] In 2009, he received Hispanic Heritage Award in Math and Science and he was chosen to have a "Celebration of Diversity in Computing" [conference][12] named after him (usually held annually or biennially[13]).

[18] He married Jean Tapia,[19] a Nuyorican (daughter of Puerto Rican who grew up in New York) and had 2 children: Circee and Richard.