Mixed-Integer Linear Optimization: Primal–Dual Relations and Dual Subgradient and Cutting-Plane Methods
Book chapter, 2020

This chapter presents several solution methodologies for mixed-integer linear optimization, stated as mixed-binary optimization problems, by means of Lagrangian duals, subgradient optimization, cutting-planes, and recovery of primal solutions. It covers Lagrangian duality theory for mixed-binary linear optimization, a problem framework for which ultimate success—in most cases—is hard to accomplish, since strong duality cannot be inferred. First, a simple conditional subgradient optimization method for solving the dual problem is presented. Then, we show how ergodic sequences of Lagrangian subproblem solutions can be computed and used to recover mixed-binary primal solutions. We establish that the ergodic sequences accumulate at solutions to a convexified version of the original mixed-binary optimization problem. We also present a cutting-plane approach to the Lagrangian dual, which amounts to solving the convexified problem by Dantzig–Wolfe decomposition, as well as a two-phase method that benefits from the advantages of both subgradient optimization and Dantzig–Wolfe decomposition. Finally, we describe how the Lagrangian dual approach can be used to find near optimal solutions to mixed-binary optimization problems by utilizing the ergodic sequences in a Lagrangian heuristic, to construct a core problem, as well as to guide the branching in a branch-and-bound method. The chapter is concluded with a section comprising notes, references, historical downturns, and reading tips.

Non-smooth convex function

Mixed-binary linear optimization

Column generation

Core problem

Cutting planes

Ergodic sequences

Convexified problem

Subgradient method

Dantzig-wolfe decomposition

Lagrange dual


Ann-Brith Strömberg

Chalmers, Mathematical Sciences, Applied Mathematics and Statistics

University of Gothenburg

Torbjörn Larsson

Linköping University

Michael Patriksson

University of Gothenburg

Chalmers, Mathematical Sciences, Applied Mathematics and Statistics

Numerical Nonsmooth Optimization: State of the Art Algorithms

978-3-030-34910-3 (ISBN)

Nonsmooth convex optimization—theory and solution methodology

Naturvetenskapliga Forskningsrådet, 1998-07-01 -- 2022-12-31.

Chalmers, 1998-07-01 -- 2020-12-31.

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