Interior Point Methods for Second-Order Cone Programming and OR Applications

Interior point methods (IPM) have been developed for all types of constrained optimization problems. In this work the extension of IPM to second order cone programming (SOCP) is studied based on the work of Andersen, Roos, and Terlaky. SOCP minimizes a linear objective function over the direct product of quadratic cones, rotated quadratic cones, and an affine set. It is described in detail how to convert several application problems to SOCP. Moreover, a proof is given of the existence of the step for the infeasible long-step path-following method. Furthermore, variants are developed of both long-step path-following and of predictor-corrector algorithms. Numerical results are presented and analyzed for those variants using test cases obtained from a number of application problems.     

A multi-objective mathematical optimization model for process targeting using 100% inspection policy

The selection of the optimal process target is an important problem in production planning and quality control. Such process targeting problems are usually modeled in the literature using a single objective optimization model. In this paper multi-objective optimization is introduced in the process targeting area. The quality characteristic under consideration is normally distributed with unknown mean and known standard deviation, and has two market specification limits. 100% inspection is used as the mean of product quality control. Product satisfies the first specification limit is sold in a primary market at a regular price and products fails the first specification limit and satisfies the second one is sold in a secondary market at a reduced price. The product is reworked if it does not satisfy both specification limits. The developed multi-objective optimization model consists of three objective functions, which are to maximize profit, income and product uniformity using Taguchi quadratic function as a surrogate for product uniformity. An algorithm is proposed to obtain and rank the set of Pareto optimal points. The utility of the model has been demonstrated using a numerical example from the literature with some additional data the new model requires. Sensitivity analysis was conducted and showed that the results of the model are sensitive to changes in process variance. In addition the optimal objectives of the profit function and product uniformity are more sensitive to changes in model parameters than the income function.     

A study of design velocity field computation for shape optimal design

Design velocity field computation is an important step in computing shape design sensitivity coefficients and updating a finite element mesh in the shape design optimization process. Applying an inappropriate design velocity field for shape design sensitivity analysis and optimization will yield inaccurate sensitivity results or a distorted finite element mesh, and thus fail in achieving an optimal solution. In this paper, theoretical regularity and practical requirements of the design velocity field are discussed. The crucial step of using the design velocity field to update the finite element mesh in the design optimization process is emphasized. Available design velocity field computation methods in the literature are summarized and their applicability for shape design sensitivity analysis and optimization is discussed. Five examples are employed to discuss applicability of these methods. It was found that a combination of isoparametric mapping and boundary displacement methods is ideal for the design velocity field computation.     

Bilevel aggregator-prosumers' optimization problem in real-time: A convex optimization approach

This paper proposes a Real-Time Market (RTM) platform for an aggregator and its corresponding prosumers to participate in the electricity wholesale market. The proposed energy market platform is modeled as a bilevel optimization problem where the aggregator and the prosumers are considered as self-interested agents. We present a convex optimization problem which can capture a subset of the set of global optima of the bilevel problem as its optimal solution.     

One-level reformulation of the bilevel Knapsack problem using dynamic programming

In this paper, we consider the Bilevel Knapsack Problem (BKP), which is a hierarchical optimization problem in which the feasible set is determined by the set of optimal solutions for a parametric Knapsack Problem. We introduce a new reformulation of the BKP into a one-level integer programming problem using dynamic programming. We propose an algorithm that allows the BKP to be solved exactly in two steps. In the first step, a dynamic programming algorithm is used to compute the set of follower reactions to leader decisions. In the second step, an integer problem that is equivalent to the BKP is solved using a branch-and-bound algorithm. Numerical results are presented to show the performance of our method.     

Shape functional optimization with restrictions boosted with machine learning techniques

Shape optimization is a widely used technique in the design phase of a product. Current ongoing improvement policies require a product to fulfill a series of conditions from the perspective of mechanical resistance, fatigue, natural frequency, impact resistance, etc. All these conditions are translated into equality or inequality restrictions which must be satisfied during the optimization process that is necessary in order to determine the optimal shape. This article describes a new method for shape optimization that considers any regular shape as a possible shape, thereby improving on traditional methods limited to straight profiles or profiles established a priori. Our focus is based on using functional techniques and this approach is, based on representing the shape of the object by means of functions belonging to a finite-dimension functional space. In order to resolve this problem, the article proposes an optimization method that uses machine learning techniques for functional data in order to represent the perimeter of the set of feasible functions and to speed up the process of evaluating the restrictions in each iteration of the algorithm. The results demonstrate that the functional approach produces better results in the shape optimization process and that speeding up the algorithm using machine learning techniques ensures that this approach does not negatively affect design process response times.     

A new long-step interior point algorithm for linear programming based on the algebraic equivalent transformation

In this paper, we investigate a new primal-dual long-step interior point algorithm for linear optimization. Based on the step size, interior point algorithms can be divided into two main groups, short-step, and long-step methods. In practice, long-step variants perform better, but usually, a better theoretical complexity can be achieved for the short-step methods. One of the exceptions is the large-update algorithm of Ai and Zhang. The new wide neighborhood and the main characteristics of the presented algorithm are based on their approach. In addition, we use the algebraic equivalent transformation technique of Darvay to determine new modified search directions for our method. We show that the new long-step algorithm is convergent and has the best known iteration complexity of short-step variants. We present our numerical results and compare the performance of our algorithm with two previously introduced Ai-Zhang type interior point algorithms on a set of linear programming test problems from the Netlib library.

     

Certification aspects of the fast gradient method for solving the dual of parametric convex programs

This paper examines the computational complexity certification of the fast gradient method for the solution of the dual of a parametric convex program. To this end, a lower iteration bound is derived such that for all parameters from a compact set a solution with a specified level of suboptimality will be obtained. For its practical importance, the derivation of the smallest lower iteration bound is considered. In order to determine it, we investigate both the computation of the worst case minimal Euclidean distance between an initial iterate and a Lagrange multiplier and the issue of finding the largest step size for the fast gradient method. In addition, we argue that optimal preconditioning of the dual problem cannot be proven to decrease the smallest lower iteration bound. The findings of this paper are of importance in embedded optimization, for instance, in model predictive control.     

Characterizations of set order relations and constrained set optimization problems via oriented distance function

Set-valued optimization problems are important and fascinating field of optimization theory and widely applied to image processing, viability theory, optimal control and mathematical economics. There are two types of criteria of solutions for the set-valued optimization problems: the vector criterion and the set criterion. In this paper, we adopt the set criterion to study the optimality conditions of constrained set-valued optimization problems. We first present some characterizations of various set order relations using the classical oriented distance function without involving the nonempty interior assumption on the ordered cones. Then using the characterizations of set order relations, necessary and sufficient conditions are derived for four types of optimal solutions of constrained set optimization problem with respect to the set order relations. Finally, the image space analysis is employed to study the c-optimal solution of constrained set optimization problems, and then optimality conditions and an alternative result for the constrained set optimization problem are established by the classical oriented distance function.     

NAUTILUS method: An interactive technique in multiobjective optimization based on the nadir point

Most interactive methods developed for solving multiobjective optimization problems sequentially generate Pareto optimal or nondominated vectors and the decision maker must always allow impairment in at least one objective function to get a new solution. The NAUTILUS method proposed is based on the assumptions that past experiences affect decision makers’ hopes and that people do not react symmetrically to gains and losses. Therefore, some decision makers may prefer to start from the worst possible objective values and to improve every objective step by step according to their preferences. In NAUTILUS, starting from the nadir point, a solution is obtained at each iteration which dominates the previous one. Although only the last solution will be Pareto optimal, the decision maker never looses sight of the Pareto optimal set, and the search is oriented so that (s)he progressively focusses on the preferred part of the Pareto optimal set. Each new solution is obtained by minimizing an achievement scalarizing function including preferences about desired improvements in objective function values. NAUTILUS is specially suitable for avoiding undesired anchoring effects, for example in negotiation support problems, or just as a means of finding an initial Pareto optimal solution for any interactive procedure. An illustrative example demonstrates how this new method iterates.     

求解图像分割CV模型的BB算法

给出图像分割的一种新算法——BB算法.该方法的优点在于利用迭代过程中当前点和前一点的信息确定搜索步长,从而更有效地搜索最优解.为此,首先通过变分水平集方法将CV模型转化为最优化问题;其次,将BB算法引入该优化问题进行求解;然后,对BB算法进行收敛性分析,为该算法应用在CV模型中提供了理论依据;最后将该方法与已有的最速下降法、共轭梯度法的分割结果进行比较.结果表明,跟其他两种方法相比,BB算法在保证较好分割效果的前提下,提高了算法的速度和性能.     

Two level hierarchical time minimizing transportation problem

A two level hierarchical balanced time minimizing transportation problem is considered in this paper. The whole set of source-destination links consists of two disjoint partitions namely Level-I links and Level-II links. Some quantity of a homogeneous product is first shipped from sources to destinations by Level-I decision maker using only Level-I links, and on its completion the Level-II decision maker transports the remaining quantity of the product in an optimal fashion using only Level-II links. Transportation is assumed to be done in parallel in both the levels. The aim is to find that feasible solution for Level-I decision maker corresponding to which the optimal feasible solution for Level-II decision maker is such that the sum of shipment times in Level-I and Level-II is the least. To obtain the global optimal feasible solution of this non-convex optimization problem, related balanced time minimizing transportation problems are defined. Based upon the optimal feasible solutions of these related problems, standard cost minimizing transportation problems are constructed whose optimal feasible solutions provide various pairs for shipment times for Level-I and Level-II decision makers. The best out of these pairs is finally selected. Being dependent upon solutions of a finite number of balanced time minimizing and cost minimizing transportation problems, the proposed algorithm is a polynomial bound algorithm. The developed algorithm has been implemented and tested on a variety of test problems and performance is found to be quite encouraging.     

Quadratic optimization problems in robust beamforming

This paper presents a class of constrained optimization problems whereby a quadratic cost function is to be minimized with respect to a weight vector subject to an inequality quadratic constraint on the weight vector. This class of constrained optimization problems arises as a result of a motivation for designing robust antenna array processors in the field of adaptive array processing. The constrained optimization problem is first solved by using the primal-dual method. Numerical techniques are presented to reduce the computational complexity of determining the optimal Lagrange multiplier and hence the optimal weight vector. Subsequently, a set of linear constraints or at most linear plus norm constraints are developed for approximating the performance achievable with the quadratic constraint. The use of linear constraints is very attractive, since they reduce the computational burden required to determine the optimal weight vector.     

Joint optimization of product family configuration and scaling design by Stackelberg game

Product family design is generally characterized by two types of approaches: module-based and scale-based. While the former aims to enable product variety based on module configuration, the latter is to variegate product design by scaling up or down certain design parameters. The prevailing practice is to treat module configuration and scaling design as separate decisions or aggregate two design problems as a single-level, all-in-one optimization problem. In practice, optimization of scaling variables is always enacted within a specific modular platform; and meanwhile an optimal module configuration depends on how design parameters are to be scaled. The key challenge is how to deal with explicitly the coupling of these two design optimization problems.     

Optimal policies for inventory systems with two types of product sharing common hardware platforms: Single period and finite horizon

In this paper, we consider an inventory system whose products share a common hardware platform but are differentiated by two types of software. Choice of different software results in different installation cost and different selling price of the whole product. Product with different software also faces different customer demand. We investigate the optimal proportion of an order to be installed with software 1 or 2, that maximizes expected profit in the single and multiple period scenarios, respectively. The optimal policy is analytically obtained and proved to be an order-up-to policy in each scenario. Our investigation reveals that whether to replenish, and how much to replenish each product depend not only on its own initial inventory level, and system parameters, but also the initial inventory level of the other product. We perform numerical experiments using the optimal policies we have derived in the paper.     

A robust data envelopment analysis model with different scenarios

Input and output data, under uncertainty, must be taken into account as an essential part of data envelopment analysis (DEA) models in practice. Many researchers have dealt with this kind of problem using fuzzy approaches, DEA models with interval data or probabilistic models. This paper presents an approach to scenario-based robust optimization for conventional DEA models. To consider the uncertainty in DEA models, different scenarios are formulated with a specified probability for input and output data instead of using point estimates. The robust DEA model proposed is aimed at ranking decision-making units (DMUs) based on their sensitivity analysis within the given set of scenarios, considering both feasibility and optimality factors in the objective function. The model is based on the technique proposed by Mulvey et al. (1995) for solving stochastic optimization problems. The effect of DMUs on the product possibility set is calculated using the Monte Carlo method in order to extract weights for feasibility and optimality factors in the goal programming model. The approach proposed is illustrated and verified by a case study of an engineering company.     

全局优化问题的一类积分型最优性条件 (英)

本文通过构造水平集辅助函数对一类积分全局最优性条件进行研究. 所构造的辅助函数仅含有一个参数变量与一个控制变量,该参数变量用以表征对原问题目标函数最优值的估计,而控制变量用以控制积分型全局最优性条件的精度. 对参数变量做极限运算即可得到积分型全局最优性条件.继而给出了用该辅助函数所刻画的全局最优性的充要条件, 从而将原全局优化问题的求解转化为寻找一个非线性方程根的问题.更进一步地,若所取测度为勒贝格测度且积分区域为自然数集合的一个有限子集, 则该积分最优性条件便化为有限极大极小问题中利用凝聚函数对极大值函数进行逼近的近似系统.从而积分型全局最优性条件可以看作是该近似系统从离散到连续的一种推广.     

Simulation optimization for an emergency department healthcare unit in Kuwait

This paper integrates simulation with optimization to design a decision support tool for the operation of an emergency department unit at a governmental hospital in Kuwait. The hospital provides a set of services for different categories of patients. We present a methodology that uses system simulation combined with optimization to determine the optimal number of doctors, lab technicians and nurses required to maximize patient throughput and to reduce patient time in the system subject to budget restrictions. The major objective of this decision supporting tool is to evaluate the impact of various staffing levels on service efficiency. Experimental results show that by using current hospital resources, the optimization simulation model generates optimal staffing allocation that would allow 28% increase in patient throughput and an average of 40% reduction in patients’ waiting time.     

基于银行下侧风险规避的第三方B2B平台激励策略研究

以电子仓单融资为例, 基于银行下侧风险规避角度, 研究联合授信和委托授信下当第三方B2B平台存在行为隐匿的道德风险时, 银行对B2B平台的激励策略设计问题。研究发现:B2B平台的最优努力水平随收益分配比例、回购比例的增大而减小, 随质押率、贷款利率、产品采购量、损失补偿比例的增大而增大;同时银行为规避违约风险, 需设置质押率、贷款利率和贷款额上限及回购比例下限, 并且银行最优收益分配比例与损失补偿比例、最优损失补偿比例与贷款损失率均成正相关关系。此外, 随着B2B平台工作效率的提高, 联合授信下最优收益分配比例将减小, 最优损失补偿比例将增大, 最终近似于委托授信下的最优损失补偿比例。最后给出数值分析。     

基于车身装配结构优化的改进图分解算法

在考虑车身制造和装配成本的前提下对车身装配结构优化方法进行了研究,提出一种改进的图分解算法将车身装配结构最优地分解为一组部件.以白车身侧围的装配模型为例,将结构的几何图形转化为与之对应的关系拓扑图,再分割该关系拓扑图为一组工程约束下的单连通不交叉子图集,结合遗传算法中的算子操作,利用有限单元法分析并计算得到产品几何图形的最优分割,采用NSGA-Ⅱ算法并实现该装配体综合性能最优的目标.