The Theory and Applications of Discrete Constrained Optimization Using Lagrange

In this thesis, we present a new theory of discrete constrained optimization using Lagrange multipliers and an associated first-order search procedure (DLM) to solve general constrained optimization problems in discrete, continuous and mixed-integer space. The constrained problems are general in the sense that they do not assume the differentiability or convexity of functions. Our proposed theory and methods are targeted at discrete problems and can be extended to continuous and mixed-integer problems by coding continuous variables using a floating-point representation (discretization). We have characterized the errors incurred due to such discretization and have proved that there exists upper bounds on the errors. Hence, continuous and mixed-integer constrained problems, as well as discrete ones, can be handled by DLM in a unified way with bounded errors. Starting from new definitions on discrete neighborhoods, constrained local minima in discrete space, and new generalized augmented Lagrangian function, we have developed new discrete-space first-order necessary and sufficient conditions that are able to characterize all constrained local minima in discrete space. Our proposed first-order conditions show a one-to-one correspondence between a discrete-space constrained local minimum, a discretespace saddle point, and a solution to the first-order conditions. They are important because they allow us to transform the difficult problem of looking for constrained local minima in the original-variable space to the easier problem of looking for saddle points in the discrete Lagrangian space. They provide a solid foundation for DLM to solve general constrained problems that cannot be achieved in the conventional theory of Lagrange-multipliers for solving continuous constrained nonlinear programming problems. Finally, we demonstrate the efficiency and effectiveness of our proposed theory and methods. DLM is able to solve systematically general discrete, continuous and mixed-integer constrained benchmarks, which is a task not achieved by previous methods. DLM has found