The Discrete Lagrangian Theory and Its Application to Solve Nonlinear Discrete Constrained Optimization Problems

In this research we present new results on discrete Lagrangian methods (DLM) and extend our previous (incomplete and highly simpli ed) theory on the method. Our proposed method forms a strong mathematical foundation for solving general nonlinear discrete optimization problems. Speci cally, we show for continuous Lagrangian methods the relationship among local minimal solutions satisfying constraints, solutions found by the rst-order necessary and second-order sufcient conditions, and saddle points. Since there is no corresponding de nition of gradients in discrete space, we propose a new vector-based de nition of gradient, develop rst-order conditions similar to those in continuous space, propose a heuristic method to nd saddle points, and show the relationship between saddle points and local minimal solutions satisfying constraints. We then show, when all the constraint functions are non-negative, that the set of saddle points is the same as the set of local minimal points satisfying constraints. Our formal results for solving discrete problems are stronger than the continuous counterparts because saddle points in discrete space are equivalent to local minima satisfying constraints, whereas local minimal solutions satisfying constraints in continuous space do not necessarily satisfy the rstand second-order conditions. To verify our method, we apply it to solve discrete benchmark problems, design multiplierless QMF lter banks, and solve some mixed nonlinear integer optimization benchmark problems. We also study methods to speed up convergence while maintaining solution quality. Our experimental results demonstrate that DLM is both e ective and e cient.