Mixed-integer linear programming approach for global discrete sizing optimization of frame structures

This paper focuses on discrete sizing optimization of frame structures using commercial profile catalogs. The optimization problem is formulated as a mixed-integer linear programming (MILP) problem by including the equations of structural analysis as constraints. The internal forces of the members are taken as continuous state variables. Binary variables are used for choosing the member profiles from a catalog. Both the displacement and stress constraints are formulated such that for each member limit values can be imposed at predefined locations along the member. A valuable feature of the formulation, lacking in most contemporary approaches, is that global optimality of the solution is guaranteed by solving the MILP using branch-and-bound techniques. The method is applied to three design problems: a portal frame, a two-story frame with three load cases and a multiple-bay multiple-story frame. Performance profiles are determined to compare the MILP reformulation method with a genetic algorithm.     

Truss topology optimization with discrete design variables—Guaranteed global optimality and benchmark examples

This paper considers the problem of optimal truss topology design subject to multiple loading conditions. We minimize a weighted average of the compliances subject to a volume constraint. Based on the ground structure approach, the cross-sectional areas are chosen as the design variables. While this problem is well-studied for continuous bar areas, we consider in this study the case of discrete areas. This problem is of major practical relevance if the truss must be built from pre-produced bars with given areas. As a special case, we consider the design problem for a single available bar area, i.e., a 0/1 problem. In contrast to the heuristic methods considered in many other approaches, our goal is to compute guaranteed globally optimal structures. This is done by a branch-and-bound method for which convergence can be proven. In this branch-and-bound framework, lower bounds of the optimal objective function values are calculated by treating a sequence of continuous but non-convex relaxations of the original mixed-integer problem. The main effect of using this approach lies in the fact that these relaxed problems can be equivalently reformulated as convex problems and, thus, can be solved to global optimality. In addition, these convex problems can be further relaxed to quadratic programs for which very efficient numerical solution procedures exist. By exploiting this special problem structure, much larger problem instances can be solved to global optimality compared to similar mixed-integer problems. The main intention of this paper is to provide optimal solutions for single and multiple load benchmark examples, which can be used for testing and validating other methods or heuristics for the treatment of this discrete topology design problem.     

Suspension system performance optimization with discrete design variables

Suspension systems on commercial vehicles have become an important feature meeting the requirements from costumers and legislation. The performance of the suspension system is often limited by available catalogue components. Additionally the suspension performance is restricted by the travel speed which highly influences the ride comfort. In this article a suspension system for an articulated dump truck is optimized in sense of reducing elapsed time for two specified duty cycles without violating a certain comfort threshold level. The comfort threshold level is here defined as a whole-body vibration level calculated by ISO 2631-1. A three-dimensional multibody dynamics simulation model is applied to evaluate the suspension performance. A non-gradient optimization routine is used to find the best possible combination of continuous and discrete design variables including the optimum operational speed without violating a set of side constraints. The result shows that the comfort level converges to the comfort threshold level. Thus it is shown that the operational speed and hence the operator input influences the ride comfort level. Three catalogue components are identified by the optimization routine together with a set of continuous design variables and two operational speeds one for each load case. Thus the work demonstrates handling of human factors in optimization of a mechanical system with discrete and continuous design variables.     

Enhanced colliding bodies optimization for design problems with continuous and discrete variables

Colliding Bodies Optimization (CBO) is a new multi-agent algorithm inspired by a collision between two objects in one-dimension. Each agent is modeled as a body with a specified mass and velocity. A collision occurs between pairs of objects and the new positions of the colliding bodies are updated based on the collision laws. In this paper, Enhanced Colliding Bodies Optimization (ECBO) which uses memory to save some best solutions is developed. In addition, a mechanism is utilized to escape from local optima. The performance of the proposed algorithm is compared to those of standard CBO and some optimization techniques on some benchmark mathematical functions and three standard discrete and continuous structural design problems. Optimization results confirm the validity of the proposed approach.     

Multidisciplinary design optimization with discrete and continuous variables of various uncertainties

As a powerful design tool, Reliability Based Multidisciplinary Design Optimization (RBMDO) has received increasing attention to satisfy the requirement for high reliability and safety in complex and coupled systems. In many practical engineering design problems, design variables may consist of both discrete and continuous variables. Moreover, both aleatory and epistemic uncertainties may exist. This paper proposes the formula of RFCDV (Random/Fuzzy Continuous/Discrete Variables) Multidisciplinary Design Optimization (RFCDV-MDO), uncertainty analysis for RFCDV-MDO, and a method of RFCDV-MDO within the framework of Sequential Optimization and Reliability Assessment (RFCDV-MDO-SORA) to solve RFCDV-MDO problems. A mathematical problem and an engineering design problem are used to demonstrate the efficiency of the proposed method.     

A new PSO-based algorithm for multi-objective optimization with continuous and discrete design variables

This paper presents a new multi-objective optimization algorithm called FC-MOPSO for optimal design of engineering problems with a small number of function evaluations. The proposed algorithm expands the main idea of the single-objective particle swarm optimization (PSO) algorithm to deal with constrained and unconstrained multi-objective problems (MOPs). FC-MOPSO employs an effective procedure in selection of the leader for each particle to ensure both diversity and fast convergence. Fifteen benchmark problems with continuous design variables are used to validate the performance of the proposed algorithm. Finally, a modified version of FC-MOPSO is introduced for handling discrete optimization problems. Its performance is demonstrated by optimizing five space truss structures. It is shown that the FC-MOPSO can effectively find acceptable approximations of Pareto fronts for structural MOPs within very limited number of function evaluations.     

Restoration of matrix fields by second-order cone programming

Summary Wherever anisotropic behavior in physical measurements or models is encountered matrices provide adequate means to describe this anisotropy. Prominent examples are the diffusion tensor magnetic resonance imaging in medical imaging or the stress tensor in civil engineering. As most measured data these matrix-valued data are also polluted by noise and require restoration. The restoration of scalar images corrupted by noise via minimization of an energy functional is a well-established technique that offers many advantages. A convenient way to achieve this minimization is second-order cone programming (SOCP). The goal of this article is to transfer this method to the matrix-valued setting. It is shown how SOCP can be applied to minimize various energy functionals defined for matrix fields. These functionals couple the different matrix channels taking into account the relations between them. Furthermore, new functionals for the regularization of matrix data are proposed and the corresponding Euler–Lagrange equations are derived by means of matrix differential calculus. Numerical experiments substantiate the usefulness of the proposed methods for the restoration of matrix fields.      

Structural optimization with several discrete design variables per part by outer approximation

The article proposes an optimal design approach to minimize the mass of load carrying structures with discrete design variables. The design variables are chosen from catalogues, and several variables are assigned to each part of the structure. This allows for more design freedom than only choosing parts from a catalogue. The problems are modelled as mixed 0–1 nonlinear problems with nonconvex continuous relaxations. An algorithm based on outer approximation is proposed to find optimized designs. The capabilities of the approach are demonstrated by optimal design of a space frame (jacket) structure for offshore wind turbines, with requirements on natural frequencies, strength, and fatigue lifetime.     

A second-order cone programming formulation for twin support vector machines

Second-order cone programming (SOCP) formulations have received increasing attention as robust optimization schemes for Support Vector Machine (SVM) classification. These formulations study the worst-case setting for class-conditional densities, leading to potentially more effective classifiers in terms of performance compared to the standard SVM formulation. In this work we propose an SOCP extension for Twin SVM, a recently developed classification approach that constructs two nonparallel classifiers. The linear and kernel-based SOCP formulations for Twin SVM are derived, while the duality analysis provides interesting geometrical properties of the proposed method. Experiments on benchmark datasets demonstrate the virtues of our approach in terms of classification performance compared to alternative SVM methods.     

A trust region SQP-filter method for nonlinear second-order cone programming

In this paper, we consider the nonlinear second-order cone programming problem. By combining an SQP method and filter technique, we present a trust region SQP-filter method for solving this problem. The proposed algorithm avoids using the classical merit function with penalty term. Furthermore, under standard assumptions, we prove that the iterative sequence generated by the presented algorithm converges globally. Preliminary numerical results indicate that the algorithm is promising.     

Optimal design and verification of temporal and spatial filters using second-order cone programming approach

11 Introduction The filters are widely used in many applications of signal processing. Filter design is an important research problem in many diverse application areas. The filters we usually refer to are temporal filters, which pass the frequency components of interest and attenuate the others. A spatial filter passes the signal radiating from a specific location and attenuates signals from other locations. Beamformer that widely used in radar, sonar,and wireless communications is a kind of …     

A note on truss topology optimization under self-weight load: mixed-integer second-order cone programming approach

This study compares the performance of popular sampling methods for computer experiments using various performance measures to compare them. It is well known that the sample points, in the design space located by a sampling method, determine the quality of the meta-model generated based on expensive computer experiment (or simulation) results obtained at sample (or training) points. Thus, it is very important to locate the sample points using a sampling method suitable for the system of interest to be approximated. However, there is still no clear guideline for selecting an appropriate sampling method for computer experiments. As such, a sampling method, the optimal Latin hypercube design (OLHD), has been popularly used, and quasi-random sequences and the centroidal Voronoi tessellation (CVT) have begun to be noticed recently. Some literature on the CVT asserted that the performance of the CVT was better than that of the LHD, but this assertion seems unfair because those studies only employed space-filling performance measures in favor of the CVT. In this research, we performed the comparison study among the popular sampling methods for computer experiments (CVT, OLHD, and three quasi-random sequences) with employing both space-filling properties and a projective property as performance measures to fairly compare them. We also compared the root mean square error (RMSE) values of Kriging meta-models generated using the five sampling methods to evaluate their prediction performance. From the comparison results, we provided a guideline for selecting appropriate sampling methods for some systems of interest to be approximated.     

A second-order cone programming formulation for nonparallel hyperplane support vector machine

Expert systems often rely heavily on the performance of binary classification methods. The need for accurate predictions in artificial intelligence has led to a plethora of novel approaches that aim at correctly predicting new instances based on nonlinear classifiers. In this context, Support Vector Machine (SVM) formulations via two nonparallel hyperplanes have received increasing attention due to their superior performance. In this work, we propose a novel formulation for the method, Nonparallel Hyperplane SVM. Its main contribution is the use of robust optimization techniques in order to construct nonlinear models with superior performance and appealing geometrical properties. Experiments on benchmark datasets demonstrate the virtues in terms of predictive performance compared with various other SVM formulations. Managerial insights and the relevance for intelligent systems are discussed based on the experimental outcomes.     

Adaptive memory programming for constrained global optimization

The problem of finding a global optimum of a constrained multimodal function has been the subject of intensive study in recent years. Several effective global optimization algorithms for constrained problems have been developed; among them, the multi-start procedures discussed in Ugray et al. [1] are the most effective. We present some new multi-start methods based on the framework of adaptive memory programming (AMP), which involve memory structures that are superimposed on a local optimizer. Computational comparisons involving widely used gradient-based local solvers, such as Conopt and OQNLP, are performed on a testbed of 41 problems that have been used to calibrate the performance of such methods. Our tests indicate that the new AMP procedures are competitive with the best performing existing ones.     

基于相对差商与GA的离散变量结构优化研究

针对离散变量结构优化设计中存在的NP问题,提出了相对差商-遗传算法混合优化设计思想,建立了混合优化教学模型.在隔代映射遗传算法的基础上引入自适应策略,使得IP_GA中的交叉、变异概率根据适应度大小自动调节,提高了收敛速度及解的质量.比较结果表明,该方法在处理目标函数与约束函数具有单调变化性质的离散变量优化问题时,具有很好的可行性.     

Reliability of portal frames with discrete design variables

Virtually all structural optimization based on system reliability was conducted considering continuous design variables. The solution of the reliability-based design problem is obtained by solving alternatively a reliability assessment problem and an optimal sizing program until the best reliability-based design occurs. The reliability assessment problem is formulated as a linearly constrained concave quadratic program. By introducing the concept of segmental members, the discrete optimum design is achieved based upon linear programming. Examples are solved by employing the proposed computational technique.Presented at NATO ASI Optimization of Large Structural Systems, Berchtesgaden, Germany, Sept. 23 – Oct. 4, 1991     

Topology optimization using a dual method with discrete variables

This paper deals with topology optimization of continuous structures in static linear elasticity. The problem consists in distributing a given amount of material in a specified domain modelled by a fixed finite element mesh in order to minimize the compliance. As the design variables can only take two values indicating the presence or absence of material (1 and 0), this problem is intrinsicallydiscrete. Here, it is solved by a mathematical programming method working in the dual space and specially designed to handle discrete variables. This method is very wellsuited to topology optimization, because it is particularly efficient for problems with a large number of variables and a small number of constraints. To ensure the existence of a solution, the perimeter of the solid parts is bounded. A computer program including analysis and optimization has been developed. As it is specialized for regular meshes, the computational time is drastically reduced. Some classical 2-D and new 3-D problems are solved, with up to 30,000 design variables. Extensions to multiple load cases and to gravity loads are also examined.     

Methods for optimization of nonlinear problems with discrete variables: A review

The methods for discrete-integer-continuous variable nonlinear optimization are reviewed. They are classified into the following six categories: branch and bound, simulated annealing, sequential linearization, penalty functions, Lagrangian relaxation, and other methods. Basic ideas of each method are described and details of some of the algorithms are given. They are transcribed into a step-by-step format for easy implementation into a computer. Under other methods, rounding-off, heuristic, cutting-plane, pure discrete, and genetic algorithms are described. For nonlinear problems, none of the methods are guaranteed to produce the global minimizer; however, good practical solutions can be obtained.Notation BBM branch and bound method - D set of discrete values for all the discrete variables - D _i set of discrete values for thei-th variable - d _ij j-th discrete value for thei-th variable - f cost function to be minimized - f ^* upper bound for the cost function - g _i i-th constraint function - IP integer programming - ILP integer linear programming - L Lagrangian - LP linear programming - m total number of constraints - MDLP mixed-discrete linear programming - MDNLP mixed-discrete nonlinear programming - n _d number of discrete variables - NLP nonlinear programming - p number of equality constraints; acceptance probability used in simulated annealing - q _i number of discrete values for thei-th variable - SLP sequential linear programming - SQP sequential quadratic programming - x design variable vector of dimension n - x _iL smallest allowed value for thei-th variable - x _iU largest allowed value for thei-th variable - the gradient operator     

Structural topology optimization of high-voltage transmission tower with discrete variables

In order to solve the structural optimization problem of long-span transmission tower, topology combination optimization (TCO) method and layer combination optimization (LCO) method based on discrete variables are presented, respectively. An adaptive genetic algorithm (AGA) is proposed as optimization algorithm. Four methods: cross-section size optimization (CSSO) method, shape combination optimization (SCO) method, the TCO method and the LCO method, are utilized to optimize the transmission steel tower, respectively. The topology optimization rules are presented for the TCO method, and the layering optimization rules are presented for the LCO method. A high-voltage steel tower is analyzed as a numerical example to illustrate the performance of the proposed methods. The simulation results demonstrate that the calculated results of both the proposed TCO method and the LCO method are obviously better than those of the CSSO method and the SCO method.     

A heuristic particle swarm optimization method for truss structures with discrete variables

A heuristic particle swarm optimizer (HPSO) algorithm for truss structures with discrete variables is presented based on the standard particle swarm optimizer (PSO) and the harmony search (HS) scheme. The HPSO is tested on several truss structures with discrete variables and is compared with the PSO and the particle swarm optimizer with passive congregation (PSOPC), respectively. The results show that the HPSO is able to accelerate the convergence rate effectively and has the fastest convergence rate among these three algorithms. The research shows the proposed HPSO can be effectively used to solve optimization problems for steel structures with discrete variables.