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This paper describes a new instance library for quadratic programming (QP), i.e., the family of continuous and (mixed)-integer optimization problems where the objective function and/or the constraints are quadratic. QP is a very diverse class of problems, comprising sub-classes ranging from trivial to undecidable. This diversity is reflected in the variety of QP solution methods, ranging from entirely combinatorial approaches to completely continuous algorithms, including many methods for which both aspects are fundamental. Selecting a set of instances of QP that is at the same time not overwhelmingly onerous but sufficiently challenging for the different, interested communities is therefore important. We propose a simple taxonomy for QP instances leading to a systematic problem selection mechanism. We then briefly survey the field of QP, giving an overview of theory, methods and solvers. Finally, we describe how the library was put together, and detail its final contents.
The following paper, “On fast trust region methods for quadratic models with linear constraints” by Michael J.D. Powell, was submitted to Mathematical Programming Computation in August 2014, andwas still in review when we became aware of Mike’s deteriorating health. Unfortunately, despite our best efforts and those of two referees, Mike died before we were able to send him our verdict. Both referees had concerns about points that needed further explanation, but in all likelihood—and after a few clarifying iterations—the paper would ultimately have been published. As we believe that the ideas central to the paper are of general interest to the journal’s readership, we have taken the unusual step of publishing the paper “as is”. Readers might, like the referees, question some of the statements made, but we feel it would be wrong tosecond guess Mike’s intentions, and have chosen to leave such statements alone.Mike had a profound effect on computational nonlinear optimization. We dedicate this paper to his memory.