Monotony analysis and sparse-grid integration for nonlinear chance constrained process optimization

The numerical solution of a nonlinear chance constrained optimization problem poses a major challenge. The idea of back-mapping as introduced by M. Wendt, P. Li and G. Wozny in 2002 is a viable approach for transforming chance constraints on output variables (of unknown distribution) into chance constraints on uncertain input variables (of known distribution) based on a monotony relation. Once transformation of chance constraints has been accomplished, the resulting optimization problem can be solved by using a gradient-based algorithm. However, the computation of values and gradients of chance constraints and the objective function involves the evaluation of multi-dimensional integrals, which is computationally very expensive. This study proposes an easy-to-use method for analysing monotonic relations between constrained outputs and uncertain inputs. In addition, sparse-grid integration techniques are used to reduce the computational time decisively. Two examples from process optimization under uncertainty demonstrate the performance of the proposed approach.     

An interior point technique for solving bilevel programming problems

This paper deals with bilevel programs with strictly convex lower level problems. We present the theoretical basis of a kind of necessary and sufficient optimality conditions that involve a single-level mathematical program satisfying the linear independence constraint qualification. These conditions are obtained by replacing the inner problem by their optimality conditions and relaxing their inequality constraints. An algorithm for the bilevel program, based on a well known technique for classical smooth constrained optimization, is also studied. The algorithm obtains a solution of this problem with an effort similar to that required by a classical well-behaved nonlinear constrained optimization problem. Several illustrative problems which include linear, quadratic and general nonlinear functions and constraints are solved, and very good results are obtained for all cases.     

Efficient unmanned aerial vehicle formation rendezvous trajectory planning using Dubins path and sequential convex programming

Trajectory planning of formation rendezvous of multiple unmanned aerial vehicles (UAVs) is formulated as a mixed-integer optimal control problem, and an efficient hierarchical planning approach based on the Dubins path and sequential convex programming is proposed. The proposed method includes the assignment of rendezvous points (high level) and generation of cooperative trajectories (low level). At the high level, the assignment of rendezvous points to UAVs is optimized to minimize the total length of Dubins-path-based approximate trajectories. The assignment results determine the geometric relations between the UAVs’ goals, which are used as equality constraints for generating trajectories. At the low level, trajectory generation is treated as a non-convex optimal control problem, which is transformed to a non-convex parameter optimization and then solved via sequentially performing convex optimization. Numerical experiments demonstrate that the proposed method can generate feasible trajectories and can outperform a typical nonlinear programming method in terms of efficiency.     

Cyclical parthenogenesis algorithm for layout optimization of truss structures with frequency constraints

Structural optimization with frequency constraints is seen as a challenging problem because it is associated with highly nonlinear, discontinuous and non-convex search spaces consisting of several local optima. Therefore, competent optimization algorithms are essential for addressing these problems. In this article, a newly developed metaheuristic method called the cyclical parthenogenesis algorithm (CPA) is used for layout optimization of truss structures subjected to frequency constraints. CPA is a nature-inspired, population-based metaheuristic algorithm, which imitates the reproductive and social behaviour of some animal species such as aphids, which alternate between sexual and asexual reproduction. The efficiency of the CPA is validated using four numerical examples.     

Efficiently solving linear bilevel programming problems using off-the-shelf optimization software

Many optimization models in engineering are formulated as bilevel problems. Bilevel optimization problems are mathematical programs where a subset of variables is constrained to be an optimal solution of another mathematical program. Due to the lack of optimization software that can directly handle and solve bilevel problems, most existing solution methods reformulate the bilevel problem as a mathematical program with complementarity conditions (MPCC) by replacing the lower-level problem with its necessary and sufficient optimality conditions. MPCCs are single-level non-convex optimization problems that do not satisfy the standard constraint qualifications and therefore, nonlinear solvers may fail to provide even local optimal solutions. In this paper we propose a method that first solves iteratively a set of regularized MPCCs using an off-the-shelf nonlinear solver to find a local optimal solution. Local optimal information is then used to reduce the computational burden of solving the Fortuny-Amat reformulation of the MPCC to global optimality using off-the-shelf mixed-integer solvers. This method is tested using a wide range of randomly generated examples. The results show that our method outperforms existing general-purpose methods in terms of computational burden and global optimality.     

A parallel bi-level multidisciplinary design optimization architecture with convergence proof for general problem

Quite a number of distributed Multidisciplinary Design Optimization (MDO) architectures have been proposed for the optimal design of large-scale multidisciplinary systems. However, just a few of them have available numerical convergence proof. In this article, a parallel bi-level MDO architecture is presented to solve the general MDO problem with shared constraints and a shared objective. The presented architecture decomposes the original MDO problem into one implicit nonlinear equation and multiple concurrent sub-optimization problems, then solves them through a bi-level process. In particular, this architecture allows each sub-optimization problem to be solved in parallel and its solution is proven to converge to the Karush–Kuhn–Tucker (KKT) point of the original MDO problem. Finally, two MDO problems are introduced to perform a comprehensive evaluation and verification of the presented architecture and the results demonstrate that it has a good performance both in convergence and efficiency.     

A Convex and Exact Approach to Discrete Constrained TV-L1 Image Approximation

We study the TV-L1 image approximation model from primal and dual perspective, based on a proposed equivalent convex formulations. More specifically, we apply a convex TV-L1 based approach to globally solve the discrete constrained optimization problem of image approximation, where the unknown image function $u(x)∈\{f_1 ,... , f_n\}$, $∀x ∈ Ω$. We show that the TV-L1 formulation does provide an exact convex relaxation model to the non-convex optimization problem considered. This result greatly extends recent studies of Chan et al., from the simplest binary constrained case to the general gray-value constrained case, through the proposed rounding scheme. In addition, we construct a fast multiplier-based algorithm based on the proposed primal-dual model, which properly avoids variability of the concerning TV-L1 energy function. Numerical experiments validate the theoretical results and show that the proposed algorithm is reliable and effective.     

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.     

Homotopy methods for constraint relaxation in unilevel reliability based design optimization

In reliability based design optimization, a methodology for finding optimized designs characterized with a low probability of failure the main objective is to minimize a merit function while satisfying the reliability constraints. Traditionally, these have been formulated as a double-loop (nested) optimization problem, which is computationally intensive. A new efficient unilevel formulation for reliability based design optimization was developed by the authors in earlier studies, where the lower-level optimization was replaced by its corresponding first-order Karush–Kuhn–Tucker (KKT) necessary optimality conditions at the upper-level optimization and imposed as equality constraints. But as most commercial optimizers are usually numerically unreliable when applied to problems accompanied by many equality constraints, an optimization framework for reliability based design using the unilevel formulation is developed here. Homotopy methods are used for constraint relaxation and to obtain a relaxed feasible design and heuristic scheme is employed to update the homotopy parameter.     

Optimality Condition and Branch and Bound Algorithm for Constrained Redundancy Optimization in Series Systems

This paper considers the constrained redundancy optimization problem in series systems. This problem can be formulated as a nonlinear integer programming problem of maximizing the overall systems reliability under limited resource constraints. By exploiting special features of the problem, we derive a new necessary condition for optimal redundancy assignments. This condition leads to a new fathoming condition in the branch and bound method that may result in a significant reduction of computational efforts, as evidenced in our numerical calculation for linearly constrained redundancy optimization problems.     

An Efficient Reliability-based Optimization Method for Uncertain Structures Based on Non-probability Interval Model

In this paper, an efficient interval optimization method based on a reliability-based possibility degree of interval (RPDI) is suggested for the design of uncertain structures. A general nonlinear interval optimization problem is studied in which the objective function and constraints are both nonlinear and uncertain. Through an interval order relation and a reliability-based possibility degree of interval, the uncertain optimization problem is transformed into a deterministic one. A sequence of approximate optimization problems are constructed based on the linear approximation technique. Each approximate optimization problem can be changed to a traditional linear programming problem, which can be easily solved by the simplex method. An iterative framework is also created, in which the design space is updated adaptively and a fine optimum can be well reached. Two numerical examples are investigated to demonstrate the effectiveness of the present method. Finally, it is employed to perform the optimization design of a practical automobile frame.     

Multi-Disciplinary Optimization for Multi-Objective Uncertainty Design of Thin Walled Beams

The focus of this paper is concentrated on multi-disciplinary and multi-objective optimization for thin walled beam systems considering safety, normal mode, static loading-bearing and weight, in which the uncertainties of the parameters are described via intervals. The size and shape of the cross-section are treated as design parameters during optimization. Considering the lightweight and safety, the design problem is formulated with two individual objectives to measure structural weight and maximum energy absorption, respectively, constrained by the average force, normal mode and maximum stress. The optimization problem with uncertainties is further transformed into a deterministic optimization based on interval number programming. The approximation models, coupled with the design of experiment (DOE) technique, are employed to construct objective functions and constraints. The uncertain optimization problem characterized with these approximation models is performed and applied to a practical thin walled beam design problems.     

Mixed Variable Optimization of a Load-Bearing Thermal Insulation System Using a Filter Pattern Search Algorithm

This paper describes the optimization of a load-bearing thermal insulation system characterized by hot and cold surfaces with a series of heat intercepts and insulators between them. The optimization problem is represented as a mixed variable programming (MVP) problem with nonlinear constraints, in which the objective is to minimize the power required to maintain the heat intercepts at fixed temperatures so that one surface is kept sufficiently cold. MVP problems are more general than mixed integer nonlinear programming (MINLP) problems in that the discrete variables are categorical; i.e., they must always take on values from a predefined enumerable set or list. Thus, traditional approaches that use branch and bound techniques cannot be applied.In a previous paper, a linearly constrained version of this problem was solved numerically using the Audet-Dennis generalized pattern search (GPS) method for MVP problems. However, this algorithm may not work for problems with general nonlinear constraints. A new algorithm that extends that of Audet and Dennis by incorporating a filter to handle nonlinear constraints makes it possible to solve the more general problem. Additional nonlinear constraints on stress, mass, and thermal contraction are added to that of the previous work in an effort to find a more realistic feasible design. Several computational experiments show a substantial improvement in power required to maintain the system, as compared to the previous literature. The addition of the new constraints leads to a very different design without significantly changing the power required. The results demonstrate that the new algorithm can be applied to a very broad class of optimization problems, for which no previous algorithm with provable convergence results could be applied.     

Load Allocation Research in a Manufacturing System

Based on queuing theory, a nonlinear optimization model is proposed in this paper, which has the service load as its objective function and includes three inequality constraints of Work In Progress （WIP）. A novel transformation of optimization variables is also devised and the constraints are properly combined so as to make this model into a convex one from which the Lagrangian function and the Karurh Kuhn Tucker （KKT） conditions are derived. The interior-point method for convex optimization is presented here as a computationaUy efficient tool. Finally, this model is evaluated on a real example, from which such conclusions are reached that the optimum result can ensure the full utilization of machines and the least amount of WIP in manufacturing systems; the interior-point method needs fewer iterations with significant computational savings and it is possible to make nonlinear and complicated optimization problems convexified so as to obtain the optimum.     

Global Optimization of Nonlinear Blend-Scheduling Problems

The scheduling of gasoline-blending operations is an important problem in the oil refining industry. This problem not only exhibits the combinatorial nature that is intrinsic to scheduling problems, but also non-convex nonlinear behavior, due to the blending of various materials with different quality properties. In this work, a global optimization algorithm is proposed to solve a previously published continuous-time mixed-integer nonlinear scheduling model for gasoline blending. The model includes blend recipe optimization, the distribution problem, and several important operational features and constraints. The algorithm employs piecewise McCormick relaxation (PMCR) and normalized multiparametric disaggregation technique (NMDT) to compute estimates of the global optimum. These techniques partition the domain of one of the variables in a bilinear term and generate convex relaxations for each partition. By increasing the number of partitions and reducing the domain of the variables, the algorithm is able to refine the estimates of the global solution. The algorithm is compared to two commercial global solvers and two heuristic methods by solving four examples from the literature. Results show that the proposed global optimization algorithm performs on par with commercial solvers but is not as fast as heuristic approaches.     

A new harmony search algorithm for solving mixed–discrete engineering optimization problems

In general design optimization problems, it is usually assumed that the design variables are continuous. However, many practical problems in engineering design require considering the design variables as integer or discrete values. The presence of discrete and integer variables along with continuous variables adds to the complexity of the optimization problem. Very few of the existing methods can yield a globally optimal solution when the objective functions are non-convex and non-differentiable. This article presents a mixed–discrete harmony search approach for solving these nonlinear optimization problems which contain integer, discrete and continuous variables. Some engineering design examples are also presented to demonstrate the effectiveness of the proposed method.     

Fully implicit Lagrange–Newton–Krylov–Schwarz algorithms for boundary control of unsteady incompressible flows

We develop a parallel fully implicit domain decomposition algorithm for solving optimization problems constrained by time‐dependent nonlinear partial differential equations. In particular, we study the boundary control of unsteady incompressible Navier–Stokes equations. After an implicit discretization in time, a fully coupled sparse nonlinear optimization problem needs to be solved at each time step. The class of full space Lagrange–Newton–Krylov–Schwarz algorithms is used to solve the sequence of optimization problems. Among optimization algorithms, the fully implicit full space approach is considered to be the easiest to formulate and the hardest to solve. We show that Lagrange–Newton–Krylov–Schwarz, with a one‐level restricted additive Schwarz preconditioner, is an efficient class of methods for solving these hard problems. To demonstrate the scalability and robustness of the algorithm, we consider several problems with a wide range of Reynolds numbers and time step sizes, and we present numerical results for large‐scale calculations involving several million unknowns obtained on machines with more than 1000 processors. Copyright © 2012 John Wiley & Sons, Ltd.     

Continuum structural topological optimizations with stress constraints based on an active constraint technique

Stress‐related problems have not been given the same attention as the minimum compliance topological optimization problem in the literature. Continuum structural topological optimization with stress constraints is of wide engineering application prospect, in which there still are many problems to solve, such as the stress concentration, an equivalent approximate optimization model and etc. A new and effective topological optimization method of continuum structures with the stress constraints and the objective function being the structural volume has been presented in this paper. To solve the stress concentration issue, an approximate stress gradient evaluation for any element is introduced, and a total aggregation normalized stress gradient constraint is constructed for the optimized structure under the r?th load case. To obtain stable convergent series solutions and enhance the control on the stress level, two p‐norm global stress constraint functions with different indexes are adopted, and some weighting p‐norm global stress constraint functions are introduced for any load case. And an equivalent topological optimization model with reduced stress constraints is constructed,being incorporated with the rational approximation for material properties, an active constraint technique, a trust region scheme, and an effective local stress approach like the qp approach to resolve the stress singularity phenomenon. Hence, a set of stress quadratic explicit approximations are constructed, based on stress sensitivities and the method of moving asymptotes. A set of algorithm for the one level optimization problem with artificial variables and many possible non‐active design variables is proposed by adopting an inequality constrained nonlinear programming method with simple trust regions, based on the primal‐dual theory, in which the non‐smooth expressions of the design variable solutions are reformulated as smoothing functions of the Lagrange multipliers by using a novel smoothing function. Finally, a two‐level optimization design scheme with active constraint technique, i.e. varied constraint limits, is proposed to deal with the aggregation constraints that always are of loose constraint (non active constraint) features in the conventional structural optimization method. A novel structural topological optimization method with stress constraints and its algorithm are formed, and examples are provided to demonstrate that the proposed method is feasible and very effective. © 2016 The Authors. International Journal for Numerical Methods in Engineering published by John Wiley & Sons Ltd.     

求解非线性系统的信赖域方法

本文给出了一个求解非线性系统的信赖域方法。通过引入松驰变量将非线性系统问题转化为带非负约束的非线性最优化问题,新算法借助于KKT条件和F-BNCP函数,在每次迭代时,不必求解二次信赖域子问题,只需求解一个线性方程组。在一定的假设条件下,该算法还是全局收敛和局部超线性收敛的。数值试验结果表明该算法是有效的。     

Direct Search Methods for Nonlinearly Constrained Optimization Using Filters and Frames

A direct search method for nonlinear optimization problems with nonlinear inequality constraints is presented. A filter based approach is used, which allows infeasible starting points. The constraints are assumed to be continuously differentiable, and approximations to the constraint gradients are used. For simplicity it is assumed that the active constraint normals are linearly independent at all points of interest on the boundary of the feasible region. An infinite sequence of iterates is generated, some of which are surrounded by sets of points called bent frames. An infinite subsequence of these iterates is identified, and its convergence properties are studied by applying Clarke's non-smooth calculus to the bent frames. It is shown that each cluster point of this subsequence is a Karush-Kuhn-Tucker point of the optimization problem under mild conditions which include strict differentiability of the objective function at each cluster point. This permits the objective function to be non-smooth, infinite, or undefined away from these cluster points. When the objective function is only locally Lipschitz at these cluster points it is shown that certain directions still have interesting properties at these cluster points.