Christian Zillober

Numerical comparison of nonlinear programming algorithms for structural optimization

For FE-based structural optimization systems, a large variety of different numerical algorithms is available, e.g. sequential linear programming, sequential quadratic programming, convex approximation, generalized reduced gradient, multiplier, penalty or optimality criteria methods, and combinations of these approaches. The purpose of the paper is to present the numerical results of a comparative study of eleven mathematical programming codes which represent typical realizations of the mathematical methods mentioned. They are implemented in the structural optimization system MBB-LAGRANGE, which proceeds from a typical finite element analysis. The comparative results are obtained from a collection of 79 test problems. The majority of them are academic test cases, the others possess some practicalreal life background. Optimization is performed with respect to sizing of trusses and beams, wall thicknesses, etc., subject to stress, displacement, and many other constraints. Numerical comparison is based on reliability and efficiency measured by calculation time and number of analyses needed to reach a certain accuracy level.

A globally convergent version of the method of moving asymptotes

The method of moving asymptotes (MMA) which is known to work excellently for solving structural optimization problems has one main disadvantage: convergence cannot be guaranteed and in practical use this fact sometimes leads to unsatisfactory results. In this paper we prove a global convergence theorem for a new method which consists iteratively of the solution of the known MMA-subproblem and a line search performed afterwards.

Very large scale optimization by sequential convex programming

We introduce a method for constrained nonlinear programming that is widely used in mechanical engineering and that is known under the name SCP for sequential convex programming. The algorithm consists of solving a sequence of convex and separable subproblems, where an augmented Lagrangian merit function is used for guaranteeing convergence. Originally, SCP methods were developed in structural mechanical optimization, and are particularly applied to solve topology optimization problems. These problems are extremely large and possess dense Hessians of the objective function. The purpose of the article is to show that constrained dense nonlinear programs with 105–106 variables can be solved successfully and that SCP methods can be applied also to optimal control problems based on semilinear elliptic partial differential equations after a full discretization.

Global Convergence of a Nonlinear Programming Method Using Convex Approximations

The method of moving asymptotes (MMA) and its globally convergent extension SCP (sequential convex programming) are known to work well for certain problems arising in structural optimization. In this paper, the methods are extended for a general mathematical programming framework and a new scheme to update certain penalty parameters is defined, which leads to a considerable improvement in the performance. Properties of the approximation functions are outlined in detail. All convergence results of the traditional methods are preserved.

A Combined Convex Approximation—Interior Point Approach for Large Scale Nonlinear Programming

The method of moving asymptotes (MMA) and its globally convergent extension SCP (sequential convex programming) are known to work well in the context of structural optimization. The two main reasons are that the approximation scheme used for the objective function and the constraints fits very well to these applications and that at an iteration point a local optimization model is used such that additional expensive function and gradient evaluations of the original problem are avoided. The subproblems that occur in both methods are special nonlinear convex programs and have traditionally been solved using a dual approach. This is now replaced by an interior point approach. The latter one is more suitable for large problems because sparsity properties of the original problem can be preserved and the separability property of the approximation functions is exploited. The effectiveness of the new method is demonstrated by a few examples dealing with problems of structural optimization.

Nonlinear Programming: Algorithms, Software, and Applications

We introduce some methods for constrained nonlinear programming that are widely used in practice and that are known under the names SQP for sequential quadratic programming and SCP for sequential convex programming. In both cases, convex subproblems are formulated, in the first case a quadratic programming problem, in the second case a separable nonlinear program in inverse variables. The methods are outlined in a uniform way and the results of some comparative performance tests are listed. We especially show the suitability of sequential convex programming methods to solve some classes of very large scale nonlinear programs, where implicitly defined systems of equations seem to support the usage of inverse approximations. The areas of interest are structural mechanical optimization, i.e., topology optimization, and optimal control of partial differential equations after a full discretization. In addition, a few industrial applications and case studies are shown to illustrate practical situations under which the codes implemented by the authors are in use.

Sequential Convex Programming Methods

Sequential convex programming methods became very popular in the past for special domains of application, e.g. the optimal structural design in mechanical engineering. The algorithm uses an inverse approximation of certain variables so that a convex, separable nonlinear programming problem must be solved in each iteration. In this paper the method is outlined and it is shown, how the iteration process can be stabilized by a line search. The convergence results are presented for a special variant called method of moving asymptotes. The algorithm was implemented in FORTRAN and the numerical performance is evaluated by a comparative study, where the test problems are formulated through a finite element analysis.

SQP versus SCP Methods for Nonlinear Programming

We introduce two classes of methods for constrained smooth nonlinear programming that are widely used in practice and that are known under the names SQP for sequential quadratic programming and SCP for sequential convex programming. In both cases, convex subproblems are formulated, in the first case a convex quadratic programming problem, in the second case a convex and separable nonlinear program. An augmented Lagrangian merit function can be applied for stabilization and for guaranteeing convergence. The methods are outlined in a uniform way, convergence results are cited, and the results of a comparative performance evaluation are shown based on a set of 306 standard test problems. In addition a few industrial applications and case studies are listed that are obtained for the two computer codes under consideration, i.e., NLPQLP and SCPIP.