John Tomlin

On projected Newton barrier methods for linear programming and an equivalence to Karmarkar's projective method

Interest in linear programming has been intensified recently by Karmarkar’s publication in 1984 of an algorithm that is claimed to be much faster than the simplex method for practical problems. We review classical barrier-function methods for nonlinear programming based on applying a logarithmic transformation to inequality constraints. For the special case of linear programming, the transformed problem can be solved by a “projected Newton barrier” method. This method is shown to be equivalent to Karmarkar’s projective method for a particular choice of the barrier parameter. We then present details of a specific barrier algorithm and its practical implementation. Numerical results are given for several non-trivial test problems, and the implications for future developments in linear programming are discussed.

Optimal delivery of sponsored search advertisements subject to budget constraints

We discuss an auction framework in which sponsored search advertisements are delivered in response to queries. In practice, the presence of bidder budgets can have a significant impact on the ad delivery process. We propose an approach based on linear programming which takes bidder budgets into account, and uses them in conjunction with forecasting of query frequencies, and pricing and ranking schemes, to optimize ad delivery. Simulations show significant improvements in revenue and efficiency.

Online allocation of display advertisements subject to advanced sales contracts

In this paper we propose a utility model that accounts for both sales and branding advertisers. We first study the computational complexity of optimization problems related to both online and offline allocation of display advertisements. Next, we focus on a particular instance of the online allocation problem, and design a simple online algorithm with provable approximation guarantees. Our algorithm is near optimal as is shown by a matching lower bound. Finally, we report on experiments to establish actual case behavior on some real datasets, with encouraging results.

George B. Dantzig and systems optimization

We pay homage to George B. Dantzig by describing a less well-known part of his legacy-his early and dedicated championship of the importance of systems optimization in solving complex real-world problems.

Evelyn Martin Lansdowne Beale

Evelyn Martin Lansdowne Beale, alwaysknown as Martin to his friends and colleagues, was a giant of the Operations Research (OR) profession, especially in the U.K., and an outstanding contributor to all aspects of mathematical programming (MP). He not only made major contributions to theory and algorithms but to the development of practical mathematical-programming computer systems. His pioneering work on developing algorithms for real-world problems, and overseeing their implementation in large-scale commercial software systems, made a major impact on the practice of OR at the time and left a lasting imprint.