Duality and a Closer Look at Implementation of Linear Optimization Algorithms

Abstract

Duality played, and continues to play a crucial role in the advancement of solving LinearOptimization (LO) problems. In this thesis, we rst review the history of LO and varioussoftware to solve LO problems. In the next chapter, we discuss Pivot Algorithms, basistableaus, primal and dual Simplex methods and their computational implementation.Then we discuss Interior Point Methods (IPM) and the numerical linear algebra involvedin their implementation. The next chapter discusses duality in signicant detail, andthe role of duality in LO software design. We also describe the dualizing scheme usedto dualize the NETLIB test problems. We then discuss the computational results onspecially constructed problems and the primal and dual NETLIB set using some of theleading LO software packages including CPLEX, GuRoBi and MOSEK.In this thesis, the rst chapter deals with the history of LO and LO software packages.The second chapter talks about basis tableau, pivot algorithms | primal and dual Sim-plex methods and some computational methodology. Chapter 3 discuses about InteriorPoint Methods and the numerical linear algebra involved. In Chapter 4, we exploreduality, the role of duality LO software development, and the techniques used to dualizethe standard LO optimization problems in the NETLIB set. In Chapter 5, we presentthe computational experiments on specially constructed problems. We also present theexperiments on primal and dual NETLIB set. We nally present our conclusions inChapter 6