Discrete Lagrangian Method Research Articles

A revised discrete Lagrangian-based search algorithm for the optimal design of skeletal structures using available sections

Issues relating to the application of the discrete Lagrangian method (DLM) to the discrete sizing optimal design of skeletal structures are addressed. The resultant structure, whether truss or rigid frame, is subjected to stress and displacement constraints under multiple load cases. The members’ sections are selected from an available set of profiles. A table that contains sectional properties for all the available profiles is used directly in structural optimization. Each profile in the table is assigned by a unique profile number, which is used as the integer design variable for each of the structural members. It is proposed that we use a revised DLM search algorithm with static weighting to design trusses and rigid frames for minimum weight. Five examples are used to demonstrate the feasibility of the method. It is shown that, for monotonic as well as nonmonotonic constraint functions, the DLM is effective and robust for the discrete sizing design of skeletal structures.

Optimum design of truss structures using discrete Lagrangian method

This paper studies the minimum weight design of truss structures with discrete variables using the discrete Lagrangian method (DLM). The theory of the DLM is briefly presented first, and parameters that influence convergence speed and solution quality for this method are investigated and discussed. A revised searching algorithm is proposed for solving truss optimization problems subject to stress, buckling stress, and displacement constraints. The feasibility of the DLM is validated by three truss design examples. The results from comparative studies of the DLM against other discrete optimization algorithms for representative truss design problems are reported to show the solution quality of the DLM. It is shown that the DLM can often find better solutions for truss optimization problems than methods reported in previous literature by using appropriate weighting for the objective function and changing speeds for the Lagrange multipliers.

A Lagrangian reconstruction of GENET

GENET is a heuristic repair algorithm which demonstrates impressive efficiency in solving some large-scale and hard instances of constraint satisfaction problems (CSPs). In this paper, we draw a surprising connection between GENET and discrete Lagrange multiplier methods. Based on the work of Wah and Shang, we propose a discrete Lagrangian-based search scheme LSDL , defining a class of search algorithms for solving CSPs. We show how GENET can be reconstructed from LSDL . The dual viewpoint of GENET as a heuristic repair method and a discrete Lagrange multiplier method allows us to investigate variants of GENET from both perspectives. Benchmarking results confirm that first, our reconstructed GENET has the same fast convergence behavior as the original GENET implementation, and has competitive performance with other local search solvers DLM, WalkSAT, and W sat( oip), on a set of difficult benchmark problems. Second, our improved variant, which combines techniques from heuristic repair and discrete Lagrangian methods, is always more efficient than the reconstructed GENET, and can better it by an order of magnitude.

Discrete Lagrangian methods for optimizing the design of multiplierless QMF banks

In this paper, we present a new discrete Lagrangian method for designing multiplierless quadrature mirror filter banks. The filter coefficients in these filter banks are in powers-of-two, where numbers are represented as sums or differences of powers of two (also called canonical signed digit representation), and multiplications are carried out as additions, subtractions, and shifts. We formulate the design problem as a nonlinear discrete constrained optimization problem, using reconstruction error as the objective, and stopband and passband energies, stopband and passband ripples, and transition bandwidth as constraints. Using the performance of the best existing designs as constraints, we search for designs that improve over the best existing designs with respect to all the performance metrics. We propose a new discrete Lagrangian method for finding good designs and study methods to improve the convergence speed of Lagrangian methods without affecting their solution quality. This is done by adjusting dynamically the relative weights between the objective and the Lagrangian part. We show that our method can find designs that improve over Johnston's benchmark designs using a maximum of three to six ONE bits in each filter coefficient instead of using floating-point representations. Our approach is general and is applicable to the design of other types of multiplierless filter banks.

Discrete Lagrangian Methods for Designing Multiplierless Two-Channel PR-LP Filter Banks

In this paper, we present a new search method based on the theory of discrete Lagrange multipliers for designing multiplierless PR (perfect reconstruction) LP (linear phase) filter banks. To satisfy the PR constraints, we choose a lattice structure that, under certain conditions, can guarantee the resulting two filters to be a PR pair. Unlike the design of multiplierless QMF filter banks that represents filter coefficients directly using PO2 (powers-of-two) form (also called Canonical Signed Digit or CSD representation), we use PO2 forms to represent the parameters associated with the lattice structure. By representing these parameters as sums or differences of powers of two, multiplications can be carried out as additions, subtractions, and shifts. Using the lattice representation, we decompose the design problem into a sequence of four subproblems. The first two subproblems find a good starting point with continuous parameters using a single-objective, multi-constraint formulation. The last two subproblems first transform the continuous solution found by the second subproblem into a PO2 form, then search for a design in a mixed-integer space. We propose a new search method based on the theory of discrete Lagrange multipliers for finding good designs, and study methods to improve its convergence speed by adjusting dynamically the relative weights between the objective and the Lagrangian part. We show that our method can find good designs using at most four terms in PO2 form in each lattice parameter. Our approach is unique because our results are the first successful designs of multiplierless PR-LP filter banks. It is general because it is applicable to the design of other types of multiplierless filter banks.