He is notable for his work on mathematical programming/optimization, “Nonlinear programs”, the proposal of the linear complementarity problem, and the general field of operations research.

After that, admitted to Harvard, Cottle began by studying government (political science) and taking premedical courses.

After the first semester, he changed his major to mathematics in which he earned his bachelor's (cum laude) and master's degrees.

He joined the Mathematics Department at the Middlesex School in Concord, Massachusetts, where he spent two years.

[1] While teaching at Middlesex School, he applied and was admitted to the PhD program in mathematics at the University of California at Berkeley, with the intention of focusing on geometry.

Meanwhile, he also received an offer from the Radiation Laboratory at Berkeley as a part-time computer programmer.

Soon thereafter he became a member of Dantzig's team at UC Berkeley Operations Research Center (ORC).

Cottle's first research contribution, "Symmetric Dual Quadratic Programs," was published in 1963.

This was soon generalized in the joint paper "Symmetric Dual Nonlinear Programs," co-authored with Dantzig and Eisenberg.

This led to the consideration of what is called a "composite problem," the first-order optimality conditions for symmetric dual programs.

A special case of this, called "the linear complementarity problem",[4] is a major part of Cottle's research output.

Also in 1963, he was a summer consultant at the RAND Corporation working under the supervision of Philip Wolfe.

This resulted in the RAND Memo, RM-3858-PR, "A Theorem of Fritz John in Mathematical Programming."

In 1964, upon completion of his doctorate at Berkeley, he worked for Bell Telephone Laboratories in Holmdel, New Jersey.

During 39 years on the active faculty at Stanford he had over 30 leadership roles in national and international conferences.

During his sabbatical year at Harvard and MIT (1970-1971), he wrote “Manifestations of the Schur Complement’’, one of his most cited papers.

In the mid 1980s, two of his former students, Jong-Shi Pang and Richard E. Stone, joined him as co-authors of this book which was published in 1992.

“The Linear Complementarity Problem” won the Frederick W. Lanchester Prize of the Institute for Operations Research and the Management Sciences (INFORMS) in 1994.

There he wrote the paper “Observations on a Class of Nasty Linear Complementarity Problems’’ which relates the celebrated Klee-Minty result on the exponential time behavior of the simplex method of linear programming with the same sort of behavior in Lemke's algorithm for the LCP and hamiltonian paths on the n-cube with the binary Gray code representation of the integers from 0 to 2^n - 1.

In 2006 he was appointed a fellow of INFORMS[5] and in 2018 received the Saul I. Gass Expository Writing Award.

Much of this is an outgrowth of his doctoral dissertation supervised by George Dantzig, with whom he collaborated in some of his earliest papers.

A system of the form (1) in which f is not affine is called a nonlinear complementarity problem and is denoted NCP(