An improved weighting method with multibounds formulation and convex programming for multicriteria structural optimization

This paper presents an improved weighting method for multicriteria structural optimization. By introducing artificial design variables, here called as multibounds formulation (MBF), we demonstrate mathematically that the weighting combination of criteria can be transformed into a simplified problem with a linear objective function. This is a unified formulation for one criterion and multicriteria problems. Due to the uncoupling of involved criteria after the transformation, the extension and the adaptation of monotonic approximation‐based convex programming methods such as the convex linearization (CONLIN) or the method of moving asymptotes (MMA) are made possible to solve multicriteria problems as efficiently as for one criterion problems. In this work, a multicriteria optimization tool is developed by integrating the multibounds formulation with the CONLIN optimizer and the ABAQUS finite element analysis system. Some numerical examples are taken into account to show the efficiency of this approach. Copyright © 2001 John Wiley & Sons, Ltd.     

Reliability-based design optimization using a generalized subset simulation method and posterior approximation

The evaluation of the probabilistic constraints in reliability-based design optimization (RBDO) problems has always been significant and challenging work, which strongly affects the performance of RBDO methods. This article deals with RBDO problems using a recently developed generalized subset simulation (GSS) method and a posterior approximation approach. The posterior approximation approach is used to transform all the probabilistic constraints into ordinary constraints as in deterministic optimization. The assessment of multiple failure probabilities required by the posterior approximation approach is achieved by GSS in a single run at all supporting points, which are selected by a proper experimental design scheme combining Sobol’ sequences and Bucher’s design. Sequentially, the transformed deterministic design optimization problem can be solved by optimization algorithms, for example, the sequential quadratic programming method. Three optimization problems are used to demonstrate the efficiency and accuracy of the proposed method.     

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.     

Concave programming and piece-wise linear programming

This paper discusses the mathematical foundations of a technique that has been used extensively in structural optimization.^1–6 Two basic problems are considered. The first of these is the concave programming problem which consists of finding the global minimum of ‘piece-wise concave functions’ on ‘piece-wise concave sets’. Since any function can be approximated by a piece-wise concave function, this method could in principle be used to find the global minimum in non-convex optimization problems. The second one is the piece-wise linear programming problem in which the objective function is convex and piece-wise linear. The iterative method outlined for handling this problem is shown to be much more efficient than the standard simplex method of linear programming.     

Multiresponse systems optimization using a goal attainment approach

A goal attainment approach to optimize multiresponse systems is presented. This approach aims to identify the settings of control factors to minimize the overall weighted maximal distance measure with respect to individual response targets. Based on a nonlinear programming technique, a sequential quadratic programming algorithm, the method is proved to be robust and can achieve good performance for multiresponse optimization problems with multiple conflicting goals. Moreover, the optimization formulation may include some prior work as special cases by assigning proper response targets and weights. Fewer assumptions are needed when using the approach as compared to other techniques. Furthermore, the decision-maker's preference and the model's predictive ability can easily be incorporated into the weights' adjustment schemes with explicit physical interpretation. The proposed approach is investigated and compared with other techniques through various classical examples in the literature.     

Minimax multiobjective optimization in structural design

The use of multiobjective optimization techniques, which may be regarded as a systematic sensitivity analysis, for the selection and modification of system parameters is presented. A minimax multiobjective optimization model for structural optimization is proposed. Three typical multiobjective optimization techniques—goal programming, compromise programming and the surrogate worth trade-off method—are used to solve such a problem. The application of multiobjective optimization techniques to the selection of system parameters and large scale structural design optimization problems is the main purpose of this paper.     

Explicit model of dual programming and solving method for a class of separable convex programming problems

An objective function for a dual model of nonlinear programming problems is an implicit function with respect to Lagrangian multipliers. This study aims to address separable convex programming problems. An explicit expression with respect to Lagrangian multipliers is derived for the dual objective function. The exact solution of the dual model can be achieved because an explicit objective function is more exact than an approximated objective function. Then, a set of improved Lagrangian multipliers can be used to obtain the optimal solution of the original nonlinear programming model. A corresponding dual programming and explicit model (DP-EM) method is proposed and applied to the structural topology optimization of continuum structures. The solution efficiency of the DPEM is compared with the dual sequential quadratic programming (DSQP) method and method of moving asymptotes (MMA). The results show that the DP-EM method is more efficient than the DSQP and MMA.     

A multiscale strategy for structural optimization

This paper presents a multiscale strategy dedicated to structural optimization. The applications concern the study of geometric details (such as holes, surface profiles, etc) within the structures with frictional contacts. The first characteristic of the method is that it uses a micro–macro approach. This approach is based on a domain decomposition into substructures and interfaces, which involves the resolution of independent ‘micro’ problems in each substructure and transfers ‘macro’ information only through the interfaces. The second characteristic is the use of a multiresolution strategy in order to reduce the computation cost for problems with evolving design parameters. The last characteristic is the capability to model the geometry of details without remeshing thanks to two features: the use of a local enrichment method , and the use of level set functions to easily modify the boundary of the detail during the optimization process. Copyright © 2008 John Wiley & Sons, Ltd.     

An interval full-infinite programming method to supporting environmental decision-making

An interval full-infinite programming (IFIP) method is developed by introducing a concept of functional intervals into an optimization framework. Since the solutions of the problem should be ‘globally’ optimal under all possible levels of the associated impact factors, the number of objectives and constraints is infinite. To solve the IFIP problem, it is converted to two interactive semi-infinite programming (SIP) submodels that can be solved by conventional SIP solution algorithms. The IFIP method is applied to a solid waste management system to illustrate its performance in supporting decision-making. Compared to conventional interval linear programming (ILP) methods, the IFIP is capable of addressing uncertainties arising from not only the imprecise information but also complex relations to external impact factors. Compared to SIP that can only handle problems containing infinite constraints, the IFIP approaches are useful for addressing inexact problems with infinite objectives and constraints.     

MULTIVARIATE HERMITE APPROXIMATION FOR DESIGN OPTIMIZATION

An approximation based on multiple function and gradient information is developed using Hermite interpolation concepts. The goal is to build a high-quality approximation for complex and multidisciplinary design optimization problems employing analysis such as aeroservoelasticity, structural control, probability, etc. The proposed multidimensional approximation utilizes exact analyses data generated during the course of iterative optimization. The approximation possesses the property of reproducing the function and gradient information of known data points. The accuracy of the new approach is compared with linear, reciprocal and other standard approximations. Because the proposed algorithm uses more data points, its efficiency has to be compared in the context of iterative optimization.     

Multi-material topology optimization for the transient heat conduction problem using a sequential quadratic programming algorithm

Transient heat conduction analysis involves extensive computational cost. It becomes more serious for multi-material topology optimization, in which many design variables are involved and hundreds of iterations are usually required for convergence. This article aims to provide an efficient quadratic approximation for multi-material topology optimization of transient heat conduction problems. Reciprocal-type variables, instead of relative densities, are introduced as design variables. The sequential quadratic programming approach with explicit Hessians can be utilized as the optimizer for the computationally demanding optimization problem, by setting up a sequence of quadratic programs, in which the thermal compliance and weight can be explicitly approximated by the first and second order Taylor series expansion in terms of design variables. Numerical examples show clearly that the present approach can achieve better performance in terms of computational efficiency and iteration number than the solid isotropic material with penalization method solved by the commonly used method of moving asymptotes. In addition, a more lightweight design can be achieved by using multi-phase materials for the transient heat conductive problem, which demonstrates the necessity for multi-material topology optimization.     

Soft computing approach for reliability optimization: State-of-the-art survey

In the broadest sense, reliability is a measure of performance of systems. As systems have grown more complex, the consequences of their unreliable behavior have become severe in terms of cost, effort, lives, etc., and the interest in assessing system reliability and the need for improving the reliability of products and systems have become very important. Most solution methods for reliability optimization assume that systems have redundancy components in series and/or parallel systems and alternative designs are available. Reliability optimization problems concentrate on optimal allocation of redundancy components and optimal selection of alternative designs to meet system requirement. In the past two decades, numerous reliability optimization techniques have been proposed. Generally, these techniques can be classified as linear programming, dynamic programming, integer programming, geometric programming, heuristic method, Lagrangean multiplier method and so on. A Genetic Algorithm (GA), as a soft computing approach, is a powerful tool for solving various reliability optimization problems. In this paper, we briefly survey GA-based approach for various reliability optimization problems, such as reliability optimization of redundant system, reliability optimization with alternative design, reliability optimization with time-dependent reliability, reliability optimization with interval coefficients, bicriteria reliability optimization, and reliability optimization with fuzzy goals. We also introduce the hybrid approaches for combining GA with fuzzy logic, neural network and other conventional search techniques. Finally, we have some experiments with an example of various reliability optimization problems using hybrid GA approach.     

Semi-Markov decision models for real-time scheduling

We first present a class of real-time scheduling problems and show that these can be formulated as semi-Markov decision problems. Then we discuss two practical difficulties in solving such problems. The first is that the resulting model requires a large amount of data that is difficult to obtain; the second is that the resulting model usually has a state space that is too large for analytic consideration. Finally, we present a non-intrusive ‘knowledge acquisition’ method that identifies the states and transition probabilities that an expert uses in solving these problems. This information is then used in the semi-Markov optimization problem. A circuit board production line is used to demonstrate the feasibility of this method. The size of the state space is reduced from 2035 states to 308 by an empirical procedure. An ‘optimal’ solution is derived based on the model with the reduced state space and estimated transition probabilities. The resulting schedule is significantly better than the one used by the observed expert.     

Second-order cone programming formulation of discontinuous deformation analysis

In classic discontinuous deformation analysis (DDA), artificial springs must be employed to enforce the contact condition through the open-close iteration. However, improper stiffness parameters might cause numerical problems. The main goal of this paper is to propose a new framework of DDA using second-order cone programming. The complementarity relationship at contacts can be formulated directly; thus, artificial springs are avoided. Stemming from the equations of momentum conservation of each block, the governing equations of DDA can be cast as convex optimization problems. The basic variables in the formulations can be either block displacements or contact forces. The derived optimization problems can be reformulated into a standard second-order cone programming program, which can be solved using standard efficient optimization solvers. The proposed approach is validated by a series of numerical examples.     

Difference of convex functions algorithms (DCA) for image restoration via a Markov random field model

In this paper, we introduce a novel approach in the nonconvex optimization framework for image restoration via a Markov random field (MRF) model. While image restoration is elegantly expressed in the language of MRF’s, the resulting energy minimization problem was widely viewed as intractable: it exhibits a highly nonsmooth nonconvex energy function with many local minima, and is known to be NP-hard. The main goal of this paper is to develop fast and scalable approximation optimization approaches to a nonsmooth nonconvex MRF model which corresponds to an MRF with a truncated quadratic (also known as half-quadratic) prior. For this aim, we use the difference of convex functions (DC) programming and DC algorithm (DCA), a fast and robust approach in smooth/nonsmooth nonconvex programming, which have been successfully applied in various fields in recent years. We propose two DC formulations and investigate the two corresponding versions of DCA. Numerical simulations show the efficiency, reliability and robustness of our customized DCAs with respect to the standard GNC algorithm and the Graph-Cut based method—a more recent and efficient approach to image analysis.     

Structural topology and shape optimization for a frequency response problem

A topology and shape optimization technique using the homogenization method was developed for stiffness of a linearly elastic structure by Bendsøe and Kikuchi (1988), Suzuki and Kikuchi (1990, 1991), and others. This method has also been extended to deal with an optimal reinforcement problem for a free vibration structure by Diaz and Kikuchi (1992). In this paper, we consider a frequency response optimization problem for both the optimal layout and the reinforcement of an elastic structure. First, the structural optimization problem is transformed to an Optimal Material Distribution problem (OMD) introducing microscale voids, and then the homogenization method is employed to determine and equivalent averaged structural analysis model. A new optimization algorithm, which is derived from a Sequential Approximate Optimization approach (SAO) with the dual method, is presented to solve the present optimization problem. This optimization algorithm is different from the CONLIN (Fleury 1986) and MMA (Svanderg 1987), and it is based on a simpler idea that employs a shifted Lagrangian function to make a convex approximation. The new algorithm is called Modified Optimality Criteria method (MOC) because it can be reduced to the traditional OC method by using a zero value for the shift parameter. Two sensitivity analysis methods, the Direct Frequency Response method (DFR) and the Modal Frequency Response method (MFR), are employed to calculate the sensitivities of the object functions. Finally, three examples are given to show the feasibility of the present approach.     

Iterative scenario based reduction technique for stochastic optimization using conditional value-at-risk

In the last decades, several tools for managing risks in competitive markets, such as the conditional value-at-risk, have been developed. These techniques are applied in stochastic programming models primarily based on scenarios and/or finite sampling, which in case of large-scale models increase considerably their size according to the number of scenarios, sometimes resulting in intractable problems. This shortcoming is solved in the literature using (i) scenario reduction methods, and/or (ii) speeding up optimization techniques. However, when reducing the number of scenarios, part of the stochastic information is lost. In this paper, an iterative scheme is proposed to get the solution of a stochastic problem representing the stochastic processes via a set of scenarios and/or finite sampling, and modeling risk via conditional value-at-risk. This iterative approach relies on the fact that the solution of a stochastic programming problem optimizing the conditional value-at risk only depends on the scenarios on the upper tail of the loss distribution. Thus, the solution of the stochastic problem is obtained by solving, within an iterative scheme, problems with a reduced number of scenarios (subproblems). This strategy results in an important reduction in the computational burden for large-scale problems, while keeping all the stochastic information embedded in the original set of scenarios. In addition, each subproblem can be solved using speeding-up optimization techniques. The proposed method is very easy to implement and, as numerical results show, the reduction in computing time can be dramatic, and more pronounced as the number of initial scenarios or samples increases.     

A LOOK AT TWO UNDERUTILIZED METHODS FOR OPTIMUM STRUCTURAL DESIGN

The paper presents a closer examination of two infrequently used optimization algorithms, in relation to their applicability in large-scale structural synthesis. The geometric programming approach and the Gauss constrained method are recognized as extremely efficient solution strategies. The former approach reduces the solution of a nonlinearly constrained problem to one with strictly linear constraints, and, under special conditions, to a solution of linear algebraic equations. The Gauss constrained method provides a robust approach for optimum design that does not require a step-size determination, and is thus useful in problems where functional evaluations are computationally demanding. An efficient implementation of these methods for structural synthesis can be attained by a recourse to well-documented approximation concepts.     

A MODIFIED DEMPSTER-SHAFER THEORY FOR MULTICRITERIA OPTIMIZATION

Abstract

A new methodology, based on a modified Dempster-Shafer (DS) theory, is proposed for solving multicriteria design optimization problems. It is well known that considerable amount of computational information is acquired during the iterative process of optimization. Based on the computational information generated in each iteration, an evidence-based approach is presented for solving a multiobjective optimization problem. The method handles the multiple design criteria, which are often conflicting and non-commensurable, by constructing belief structures that can quantitatively evaluate the effectiveness of each design in the range 0 to 1. An overall satisfaction function is then defined for converting the original multicriteria design problem into a single-criterion problem so that standard single-objective programming techniques can be employed for the solution. The design of a mechanism in the presence of seven design criteria and eighteen design variables is considered to illustrate the computational details of the approach. This work represents the first attempt made in the literature at applying DS theory for numerical engineering optimization.     

A SEQUENTIAL QUADRATIC PROGRAMMING ALGORITHM UTILIZING QR MATRIX FACTORIZATION

Some recent studies indicate that the sequential quadratic programming (SQP) approach has a sound theoretical basis and promising empirical results for solving general constrained optimization problems. This paper presents a variant of the SQP method which utilizes QR matrix factorization to solve the quadratic programming subproblem which result from taking a quadratic approximation of the original problem. Theoretically, the QR factorization method is more robust and computationally efficient in solving quadratic programs. To demonstrate the validity of this variant, a computer program named SQR is coded in Fortran to solve twenty-eight test problems. By comparing with three other algorithms: one multiplier method, one GRG-type method, and another SQP-type method, the numerical results show that, in general, SQR as devised in this paper is the best method as far as robustness and speed of convergence are concerned in solving general constrained optimization problems.