Recent developments and future directions in mathematical programming

Recent advances in mathematical programming methodology have included: development of interior methods competing with the simplex method; improved simplex codes; vastly improved performance for mixed integer programming using strong linear programming formulations; and a renewed interest in decomposition. In addition, use of vector and parallel processing have improved performance and influenced algorithmic developments. Applications areas have been expanding from the traditional refinery planning and distribution models to include finance, scheduling, manufacturing, manpower planning and may other areas. The authors see the acceleration of better methods and improved codes moving together with faster, lower-cost, and more interesting hardware into a variety of application areas and thereby opening up new demands for greater function of optimization codes. These new functions might include, for example, more powerful nonlinear codes, decomposition techniques taking advantage of network and other problem-dependent structures and mixed integer capability in quadratic and general nonlinear problems. Stochastic scenario programming and multi-time period problems are becoming solvable and open up applications and algorithmic challenges. The Optimization Subroutine Library has helped to accelerate these changes, but will have to continue to change and expand in ways that will be touched upon in this paper.