Abstract.
The finite dimensional McCormick second-order sufficiency theory for non-linear programming problems with a finite number of constraints is now a classical part of the optimization literature. It was introduced by McCormick in 1967 and an improved version was given by Fiacco and McCormick in their 1968 award winning book. Later it was learned that Pennisi in 1953 had presented exactly the same theory. Many authors, most notably Maurer and Zowe in a highly visible paper in 1978 argue that the Pennisi- McCormick theory cannot be extended to infinite dimensions without adding further assumptions, by producing a counter example. They then extend the theory to infinite dimensions, allowing for an infinite number of constraints, by adding additional assumptions. In the current work we use a fundamental principle for second-order sufficiency to extend, the Pennisi-McCormick second-order theory as stated in Rn to infinite dimensional normed vector spaces, without adding additional assumptions. The Maurer and Zowe infinite dimensional counter example carried an infinite number of constraints. Hence they seemed to be unaware that the extension of the Pennisi-McCormick theory to infinite dimensions was possible provided the original feature of a finite number of constraints was maintained.
Bio:
Richard A. Tapia, University Professor and Maxfield-Oshman Professor of Engineering, Rice University was born in Los Angeles to parents who emigrated from Mexico when they were children, seeking educational opportunities. He was the first in his family to attend college, earning his B.A., M.A., and Ph.D. degrees in mathematics from the University of California, Los Angeles. Due to his efforts, Rice University has received national recognition for its educational outreach programs, and the Rice Computational and Applied Mathematics Department has become a national leader in producing women and underrepresented minority Ph.D.s in the mathematical sciences.
Dr. Tapia’s major research contributions have been in the area of computational optimization, both linear and nonlinear programming, where he pioneered the exploration and settlement of the important computational methods in numerical optimization known as primal-dual interior point methods. Tapia has authored or co-authored two books and more than 100 mathematical research papers, and is currently authoring a graduate level textbook on the foundations of optimization.
Dr. Tapia’s honors include: election to the National Academy of Engineering (1992) for his seminal work in interior point methods; being the first recipient of the A. Nico Habermann Award from the Computing Research Association (1994) for outstanding contributions in aiding members of underrepresented groups within the computing community; the Presidential Award for Excellence in Science, Mathematics, and Engineering Mentoring from President Bill Clinton (1996); appointment by President Clinton to the National Science Board, the governing body of the National Science Foundation (1996); the Lifetime Mentor Award from the American Association for the Advancement of Science (1997); and the establishment of a lecture series to honor Dr. Tapia and African American mathematician David Blackwell at Cornell University (2000). The Richard Tapia Celebration of Diversity in Computing honors his many contributions to diversity (2001). He received the Hispanic Engineer of the Year Award from Hispanic Engineer Magazine in 1996, and was inducted into the Hispanic Engineer National Achievement Awards Conference Hall of Fame in 1997. Hispanic Engineer & Informational Technology Magazine also selected him as one of the 50 Most Important Hispanics in Technology and Business for 2004. That same year Dr. Tapia was inducted into the Texas Science Hall of Fame.
Dr. Tapia has been named one of 20 most influential leaders in minority math education by the National Research Council; listed as one of the 100 most influential Hispanics in the U.S. by Hispanic Business magazine (2008); and given the “Professor of the Year” award by the Association of Hispanic School Administrators, Houston Independent School District, Houston, Texas. In 2005, Tapia was elected to the Board of Directors for TAMEST, comprising the Texas members of the National Academy of Engineering, National Academy of Sciences and the Institute of Medicine. In 2009, Tapia received the Hispanic Heritage Award for Math and Science. In 2011, President Obama named Dr. Tapia one of the recipients of the National Medal of Science, the highest honor bestowed by the United States government on scientists and engineers.