Aggregate Objective Functions and Pareto Frontiers: Required Relationships and Practical Implications

This paper addresses the problem of capturing Pareto optimal points on non-convex Pareto frontiers, which are encountered in nonlinear multiobjective optimization problems in computational engineering design optimization. The emphasis is on the choice of the aggregate objective function (AOF) of the objectives that is employed to capture Pareto optimal points. A fundamental property of the aggregate objective function, the admissibility property, is developed and its equivalence to the coordinatewise increasing property is established. Necessary and sufficient conditions for such an admissible aggregate objective function to capture Pareto optimal points are derived. Numerical examples illustrate these conditions in the biobjective case. This paper demonstrates in general terms the limitation of the popular weighted-sum AOF approach, which captures only convex Pareto frontiers, and helps us understand why some commonly used AOFs cannot capture desirable Pareto optimal points, and how to avoid this situation in practice. Since nearly all applications of optimization in engineering design involve the formation of AOFs, this paper is of direct theoretical and practical usefulness.     

Global optimization for signomial discrete programming problems in engineering design

Signomial discrete programming (SDP) problems arise frequently in a variety of real applications. Although many optimization techniques have been developed to solve an SDP problem, they use too many binary variables to reformulate the problem for finding a globally optimal solution or can only derive a local or an approximate solution. This article proposes a global optimization method to solve an SDP problem by integrating an efficient linear expression of single variable discrete functions and convexification techniques. An SDP problem can be converted into a convex mixed-integer programming problem solvable to obtain a global optimum. Several illustrative examples are also presented to demonstrate the usefulness and effectiveness of the proposed method.     

基于仿真优化的驳运空箱调运系统研究

运用仿真优化方法对驳运空箱调运系统进行了研究.根据系统作业中存在的不确定性、随机性活动的特点,建立了空箱调运系统的离散事件仿真模型;针对空箱调度规则,建立了数学模型;并运用仿真模型与启发式算法相融合的仿真优化方法进行求解.规划目标是在满足系统作业要求的前提下,以最大化计划渡柜数为目标,优化渡柜规则.即,根据各个码头对空...     

Durga: A heuristically-optimized data collection strategy for volumetric magnetic resonance imaging

A heuristic design method for rapid volumetric magnetic resonance imaging data acquisition trajectories is presented, using a series of second-order cone optimization subproblems. Other researchers have considered non-raster data collection trajectories and under-sampled data patterns. This work demonstrates that much higher rates of under-sampling are possible with an asymmetric set of trajectories, with very little loss in resolution, but the addition of noise-like artefacts. The proposed data collection trajectory, Durga, further minimizes collection time by incorporating short un-refocused excitation pulses, resulting in above 98% collection efficiency for balanced steady state free precession imaging. The optimization subproblems are novel, in that they incorporate all requirements, including data collection (coverage), physicality (device limits), and signal generation (zeroth- and higher- moment properties) in a single convex problem, which allows the resulting trajectories to exhibit a higher collection efficiency than any existing trajectory design.     

Economic design of water distribution systems in buildings

This article discusses an engineering optimization problem which arises in hydraulics and is related to the use of a new criterion for sizing water distribution piping in large buildings. The optimization model aims to find the most suitable interior pipe diameters for the various pipes in the system, using commercial sizes and minimizing the overall installation cost according to some boundary conditions. The problem is formulated as a non-convex nonlinear program and a branch-and-bound algorithm is introduced for its solution. A procedure is proposed to obtain a feasible solution with standard values from the optimal solution of the non-convex program. The performance of the algorithm is analysed for a real-life problem and the cost of the computed solution is assessed, showing the appropriateness of the model and the optimization techniques.     

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.     

D.C.集(凸集的差)约束的非凸二次规划的最优解集

本文研究D.C.集(凸集的差)上极小化非凸二次规划问题的最优解。我们首先证明了该问题的Lagrange对偶的稳定性,即不存在对偶间隙;接着利用该性质得到问题的全局最优性条件和最优解集,它可以像凸规划那样,借助它的对偶问题的解集精确地描述出来。最后,通过一个例子来说明这些结论。     

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.     

Turning-lane and signal optimization at intersections with multiple objectives

Traffic congestion at intersections is a serious problem in cities. In order to discharge turning vehicles efficiently at intersections to relieve traffic jams, multiple left-turn and right-turn lanes are often used. This article proposes a novel multi-objective optimization method for signal setting and multiple turning-lane assignment at intersections based on microscopic traffic simulations and a cell-mapping method. Vehicle conflicts and pedestrian interference are considered. The intersection multi-objective optimization problem (MOP) is formulated. The cell-mapping method is adopted to solve the MOP. Three measures of traffic performance are studied including transportation efficiency, energy consumption and road safety. The influence of turning-lane assignment on intersection performance is investigated in the optimization. Significant impacts of the number of turning lanes on the traffic are observed. An algorithm is proposed to assist traffic engineers to select and implement the optimal designs. In general, more turning lanes help increase turning traffic efficiency and lower fuel consumption in most cases. Remarkable improvement in traffic performance can be achieved with combined optimization of lane assignment and signal setting, which cannot be obtained with signal setting optimization alone. The studies reported in this article provide general guidance for intersection planning and operation. The proposed optimization methodology represents a promising emerging technology for traffic applications.     

A new method to generate the ground structure in truss topology optimization

The aim of this article is to provide an effective method to generate the ground structure in truss topology optimization. The core of this method is to place nodal points for the ground structure at the intersection of the first and third principal stress trajectories, which are obtained by solving the equivalent static problem in the design domain with a homogeneous isotropic material property. It is applicable to generate the ground structure for arbitrary regular and irregular geometric design domains. The proposed method is tested on some benchmark examples in truss topology optimization. The optimization model is a standard linear programming problem based on plastic design and solved by the interior point algorithm. Compared with other methods, the proposed method may use a well-defined ground structure with fewer nodes and bars, resulting in faster solution convergence, which shows it to be efficient.