# The GOP Primal — Relaxed Dual Decomposition Approach : Theory

• Christodoulos A. Floudas
Part of the Nonconvex Optimization and Its Applications book series (NOIA, volume 37)

## Abstract

This chapter presents the theory of the GOP algorithm introduced by Floudas and Visweswaran (1990, 1993). The GOP algorithm follows the basic ideas of decomposition first developed by Benders (1962) for mixed integer linear programming, and later extended by Geoffrion (1972) for nonlinear programming. The decomposition and duality principles are presented in detail in Floudas (1995). In section 3.1, the problem is stated and the classes of mathematical models that meet the imposed conditions are identified. In section 3.2, the reader is introduced to the basics of duality theory. In section 3.3, the theoretical properties that form the basis for the GOP are presented. Section 3.4 discusses the treatment of infeasible primal problems. Section 3.5 presents the algorithmic steps of the GOP approach and section 3.6 focuses on the convergence to an e-global optimum solution. Section 3.7 presents an illustrative example for the geometrical interpretation of the GOP algorithm. In section 3.8, the theoretical development of the GOP for quadratic programming problems with linear constraints is presented. In section 3.9, the GOP theory for quadratically constrained problems is discussed. In section 3.10, the theoretical analysis of the GOP for univariate polynomial problems is presented. Section 3.11 presents additional properties of the GOP that exploit the mathematical structure of the Lagrange function and enhance the computational performance. Section 3.12 presents the GOP approach in a branch and bound framework and its illustration. Section 3.13 introduces the reformulation of the relaxed dual problems as a single MILP model. Finally, section 3.15 presents a linear branching scheme for the GOP algorithm.

## Keywords

Dual Problem Mixed Integer Linear Program Lagrange Function Previous Iteration Primal Problem
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