An Augmented Lagrange Multiplier Based Method for Mixed Integer Discrete Continuous Optimization and Its Applications to Mechanical Design

Abstract

An algorithm for solving nonlinear optimization problems involving discrete, integer, zero-one, and continuous variables is presented. The augmented Lagrange multiplier method combined with Powell’s method and Fletcher and Reeves Conjugate Gradient method are used to solve the optimization problem where penalties are imposed on the constraints for integer/discrete violations. The use of zero-one variables as a tool for conceptual design optimization is also described with an example. Several case studies have been presented to illustrate the practical use of this algorithm. The results obtained are compared with those obtained by the Branch and Bound algorithm. Also, a comparison is made between the use of Powell’s method (zeroth order) and the Conjugate Gradient method (first order) in the solution of these mixed variable optimization problems.

Extended Multipoint Approximation Method

“…The design variables consist of the shell thickness , the spherical head thickness , the radius of cylindrical shell , and the shell length . The detailed problem formulation could be seen in [14]. It can be seen from Table 4 that the present algorithms found the best design that is 0.8% less than the best-known design quoted in literature (5885.268 vs. 5889.911).…”

Fish swarm optimization algorithm applied to engineering system design

“…In the literature, this problem has been solved by using an augmented Lagrangian multiplier approach (Kannan and Kramer, 1994), a genetic adaptive search (Deb, 1997), and a co-evolutionary particle swarm optimization (He and Wang, 2007).…”

“…Among the four variables, T s and T h are integer multiples of 0.0625 in that are the available thicknesses of rolled steel plates, and R and L are continuous variables. The problem can be formulated as follows (Kannan and Kramer, 1994):…”

“…The pressure vessel design problem was proposed by Kannan and Kramer (1994) and is devoted to the minimization of the total cost of the specimen studied, including the cost of the material, forming and welding. A cylindrical vessel is capped at both ends by hemispherical heads as shown in Figure 4.…”

A Method for the Mixed Discrete Non-Linear Problems by Particle Swarm Optimization

“…have searched an optimum by regarding all design variables as the continuous at the initial stage, the penalty parameter s in Eq. (12) has been set as zero. After the optimum obtained by regarding all design variables as the continuous has been found, the penalty function of Eq.…”