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1. Theory and applications of Robust Optimization Dimitris Bertsimas⁄, David B. Brown y, Constantine Caramanis z July 6, 2007 Abstract In this paper we survey the primary research, both theoretical and applied, in the fleld of Robust.
2. In addition, submodular functions find novel applications in combinatorial auctions, machine learning, and social networks. This workshop aims to provide a forum for researchers from a variety of backgrounds to exchange results, ideas, and problems on submodular optimization and its applications.
3. Optimization: Theory, Algorithms, Applications MSRI - Berkeley SAC, Nov/06 Henry Wolkowicz Department of Combinatorics & Optimization University of Waterloo.
4. In linear algebra and optimization theory. This is a problem because it means investing a great deal of time and energy studying these elds, but we believe that perseverance will be amply rewarded. This second volume covers some elements of optimization theory and applications, espe-cially to machine learning. This volume is divided in ve parts.

• Journal Of Optimization Theory And Applications Ranking
• Journal Of Optimization Theory Application
• Journal Of Optimization Theory And Application
• Optimization Theory And Applications By S. S. Rao
• Optimization Theory And Methods
• Journal Of Optimization Theory And Applications Impact Factor
• Robust Optimization Ppt
• Theory And Application Of Robust Optimization

Modern optimization theory includes traditional optimization theory but also overlaps with game theory and the study of economic equilibria. The Journal of Economic Literature codes classify mathematical programming, optimization techniques, and related topics under JEL:C61-C63. Among the areas of application covered are mathematical economics, mathematical physics and biology, and aerospace, chemical, civil, electrical, and mechanical engineering. The Journal of Optimization Theory and Applications journal publishes five types of contributions: survey papers, contributed papers, technical notes, technical comments.

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Optimization Theory, Methods, and Applications in Engineering

“Optimization Theory, Methods, and Applications in Engineering” is an annual special issue published in “Mathematical Problems in Engineering.” The current issue is the 2014 issue, which is now closed for submissions.

Optimization Theory, Methods, and Applications in Engineering 2014

• Optimization Theory, Methods, and Applications in Engineering 2014, Jung-Fa Tsai, John Gunnar Carlsson, Dongdong Ge, Yi-Chung Hu, and Jianming Shi
  Editorial (3 pages), Article ID 345858, Volume 2015 (2015)
• Metaheuristic Approaches for Solving Truck and Trailer Routing Problems with Stochastic Demands: A Case Study in Dairy Industry, Seyedmehdi Mirmohammadsadeghi and Shamsuddin Ahmed
  Research Article (14 pages), Article ID 494019, Volume 2015 (2015)
• Energy-Saving Generation Dispatch Using Minimum Cost Flow, Zhan’an Zhang and Xingguo Cai
  Research Article (9 pages), Article ID 562462, Volume 2015 (2015)
• A Variable Depth Search Algorithm for Binary Constraint Satisfaction Problems, N. Bouhmala
  Research Article (10 pages), Article ID 637809, Volume 2015 (2015)
• Moving Clusters within a Memetic Algorithm for Graph Partitioning, Inwook Hwang, Yong-Hyuk Kim, and Yourim Yoon
  Research Article (10 pages), Article ID 238529, Volume 2015 (2015)
• A Real-Time Pothole Detection Approach for Intelligent Transportation System, Hsiu-Wen Wang, Chi-Hua Chen, Ding-Yuan Cheng, Chun-Hao Lin, and Chi-Chun Lo
  Research Article (7 pages), Article ID 869627, Volume 2015 (2015)
• Equilibrium Customer Strategies in the Queue with Threshold Policy and Setup Times, Peishu Chen, Wenhui Zhou, and Yongwu Zhou
  Research Article (11 pages), Article ID 361901, Volume 2015 (2015)
• Minimum Time Approach to Emergency Collision Avoidance by Vehicle Handling Inverse Dynamics, Wang Wei, Bei Shaoyi, Yang Hui, Wang Yongzhi, and Zhang Lanchun
  Research Article (9 pages), Article ID 276460, Volume 2015 (2015)
• Configuration, Deployment, and Scheduling Models for Management and Optimization of Patrol Services, Bin Yang, Zhi-Hua Hu, and Jing-Xian Zhou
  Research Article (13 pages), Article ID 738578, Volume 2015 (2015)
• Parallel Control to Fragments of a Cylindrical Structure Driven by Explosive inside, Wenkai Chen, Xiangyu Li, Fangyun Lu, Zhenduo Li, and Zhenyu Zhang
  Research Article (13 pages), Article ID 723463, Volume 2015 (2015)
• An Ant Optimization Model for Unrelated Parallel Machine Scheduling with Energy Consumption and Total Tardiness, Peng Liang, Hai-dong Yang, Guo-sheng Liu, and Jian-hua Guo
  Research Article (8 pages), Article ID 907034, Volume 2015 (2015)
• An Effective Hybrid of Bees Algorithm and Differential Evolution Algorithm in Data Clustering, Mohammad Babrdel Bonab, Siti Zaiton Mohd Hashim, Nor Erne Nazira Bazin, and Ahmed Khalaf Zager Alsaedi
  Research Article (17 pages), Article ID 240419, Volume 2015 (2015)
• Multiple Sparse Measurement Gradient Reconstruction Algorithm for DOA Estimation in Compressed Sensing, Weijian Si, Xinggen Qu, Yilin Jiang, and Tao Chen
  Research Article (6 pages), Article ID 152570, Volume 2015 (2015)
• A Robust Optimization of Capacity Allocation Policies in the Third-Party Warehouse, Xu Xian-hao, Dong Wei-hong, and Peng Hongxia
  Research Article (10 pages), Article ID 810798, Volume 2015 (2015)
• To Make Good Decision: A Group DSS for Multiple Criteria Alternative Rank and Selection, Chen-Shu Wang, Heng-Li Yang, and Shiang-Lin Lin
  Research Article (15 pages), Article ID 186970, Volume 2015 (2015)
• A Risk-Averse Inventory Model with Markovian Purchasing Costs, Sungyong Choi and Kyungbae Park
  Research Article (9 pages), Article ID 925765, Volume 2015 (2015)
• Maintaining Track Continuity for Extended Targets Using Gaussian-Mixture Probability Hypothesis Density Filter, Yulan Han, Hongyan Zhu, and ChongZhao Han
  Research Article (16 pages), Article ID 501915, Volume 2015 (2015)
• Robust Control of the Knee Joint Angle of Paraplegic Patients considering Norm-Bounded Uncertainties, Nilson Moutinho dos Santos, Ruberlei Gaino, Márcio Roberto Covacic, Marcelo Carvalho Minhoto Teixeira, Aparecido Augusto de Carvalho, Edvaldo Assunção, Rodrigo Cardim, and Marcelo Augusto Assunção Sanches
  Research Article (8 pages), Article ID 736246, Volume 2015 (2015)

Optimization Theory, Methods, and Applications in Engineering 2013

• The Optimal Sampling Period of a Fingerprint Positioning Algorithm for Vehicle Speed Estimation, Ding-Yuan Cheng, Chi-Hua Chen, Chia-Hung Hsiang, Chi-Chun Lo, Hui-Fei Lin, and Bon-Yeh Lin
  Research Article (12 pages), Article ID 306783, Volume 2013 (2013)
• A Slicing Tree Representation and QCP-Model-Based Heuristic Algorithm for the Unequal-Area Block Facility Layout Problem, Mei-Shiang Chang and Ting-Chen Ku
  Research Article (19 pages), Article ID 853586, Volume 2013 (2013)
• An Improved Self-Adaptive PSO Algorithm with Detection Function for Multimodal Function Optimization Problems, YingChao Zhang, Xiong Xiong, and QiDong Zhang
  Research Article (8 pages), Article ID 716952, Volume 2013 (2013)
• The Tractor and Semitrailer Routing Considering Carbon Dioxide Emissions, Hongqi Li, Yanran Li, Qiuhong Zhao, Yue Lu, and Qiang Song
  Research Article (12 pages), Article ID 509160, Volume 2013 (2013)
• A New Improved Quantum Evolution Algorithm with Local Search Procedure for Capacitated Vehicle Routing Problem, Ligang Cui, Lin Wang, Jie Deng, and Jinlong Zhang
  Research Article (17 pages), Article ID 159495, Volume 2013 (2013)
• An Efficient Approximate Algorithm for Disjoint QoS Routing, Zhanke Yu, Feng Ma, Jingxia Liu, Bingxin Hu, and Zhaodong Zhang
  Research Article (9 pages), Article ID 489149, Volume 2013 (2013)
• Price and Delivery Time Analyzing in Competition between an Electronic and a Traditional Supply Chain, M. Narenji, Mohammad Fathian, Ebrahim Teimoury, and Seyed Gholamreze Jalali Naini
  Research Article (12 pages), Article ID 596897, Volume 2013 (2013)
• A Robust Optimization Approach to Emergency Vehicle Scheduling, Xiao-Xia Rong, Yi Lu, Rui-Rui Yin, and Jiang-Hua Zhang
  Research Article (8 pages), Article ID 848312, Volume 2013 (2013)
• A New Model for the Regulation Width of Waterway Based on Hydraulic Geometry Relation, Ni Zhi-hui, Wu Li-chun, Zhang Xu-jin, Zeng Qiang, Wang Ming-hui, and Yi Jing
  Research Article (8 pages), Article ID 182656, Volume 2013 (2013)
• Coordination Scheme for Restructuring Business Operation of the Single Period Newsvendor Problem, Chiuh-Cheng Chyu and I-Ping Huang
  Research Article (11 pages), Article ID 308187, Volume 2013 (2013)
• Determining the Optimum Ordering Policy in Multi-Item Joint Replenishment Problem Using a Novel Method, Wen-Tsung Ho
  Research Article (9 pages), Article ID 469794, Volume 2013 (2013)
• Applying Artificial Neural Network to Predict Semiconductor Machine Outliers, Keng-Chieh Yang, Conna Yang, Pei-Yao Chao, and Po-Hong Shih
  Research Article (10 pages), Article ID 210740, Volume 2013 (2013)
• Optimal Ordering Policy of a Risk-Averse Retailer Subject to Inventory Inaccuracy, Lijing Zhu, Ki-Sung Hong, and Chulung Lee
  Research Article (8 pages), Article ID 951017, Volume 2013 (2013)
• Tightness of Semidefinite Programming Relaxation to Robust Transmit Beamforming with SINR Constraints, Yanjun Wang and Ruizhi Shi
  Research Article (10 pages), Article ID 508014, Volume 2013 (2013)
• An Inventory Model for Perishable Products with Stock-Dependent Demand and Trade Credit under Inflation, Shuai Yang, Chulung Lee, and Anming Zhang
  Research Article (8 pages), Article ID 702939, Volume 2013 (2013)
• A Collaborative Optimization Model for Ground Taxi Based on Aircraft Priority, Yu Jiang, Zhihua Liao, and Honghai Zhang
  Research Article (9 pages), Article ID 854364, Volume 2013 (2013)
• Enterprise Information Security Management Based on Context-Aware RBAC and Communication Monitoring Technology, Mei-Yu Wu and Ming-Hsien Yu
  Research Article (11 pages), Article ID 569562, Volume 2013 (2013)
• Analysis of Positioning Error for Two-Dimensional Location System, Yi Jiang, Qing Hu, and Dongkai Yang
  Research Article (8 pages), Article ID 163958, Volume 2013 (2013)
• Building a Smart E-Portfolio Platform for Optimal E-Learning Objects Acquisition, Chih-Kun Ke, Kai-Ping Liu, and Wen-Chin Chen
  Research Article (8 pages), Article ID 896027, Volume 2013 (2013)
• A Branch and Bound Reduced Algorithm for Quadratic Programming Problems with Quadratic Constraints, Yuelin Gao, Feifei Li, and Siqiao Jin
  Research Article (6 pages), Article ID 594693, Volume 2013 (2013)
• 2D Dubins Path in Environments with Obstacle, Dongxiao Yang, Didong Li, and Huafei Sun
  Research Article (6 pages), Article ID 291372, Volume 2013 (2013)
• A Comparative Analysis of Nature-Inspired Optimization Approaches to 2D Geometric Modelling for Turbomachinery Applications, Amir Safari, Hirpa G. Lemu, Soheil Jafari, and Mohsen Assadi
  Research Article (15 pages), Article ID 716237, Volume 2013 (2013)
• Design Optimization of a Speed Reducer Using Deterministic Techniques, Ming-Hua Lin, Jung-Fa Tsai, Nian-Ze Hu, and Shu-Chuan Chang
  Research Article (7 pages), Article ID 419043, Volume 2013 (2013)
• A Review of Piecewise Linearization Methods, Ming-Hua Lin, John Gunnar Carlsson, Dongdong Ge, Jianming Shi, and Jung-Fa Tsai
  Review Article (8 pages), Article ID 101376, Volume 2013 (2013)
• A Novel Nonadditive Collaborative-Filtering Approach Using Multicriteria Ratings, Yi-Chung Hu
  Research Article (10 pages), Article ID 957184, Volume 2013 (2013)
• Forecasting Electrical Energy Consumption of Equipment Maintenance Using Neural Network and Particle Swarm Optimization, Xunlin Jiang, Haifeng Ling, Jun Yan, Bo Li, and Zhao Li
  Research Article (8 pages), Article ID 194730, Volume 2013 (2013)
• Classification of Hospital Web Security Efficiency Using Data Envelopment Analysis and Support Vector Machine, Han-Ying Kao, Tao-Ku Chang, and Yi-Cheng Chang
  Research Article (8 pages), Article ID 542314, Volume 2013 (2013)
• Calculation Formula Optimization and Effect of Ring Clearance on Axial Force of Multistage Pump, Chuan Wang, Weidong Shi, and Li Zhang
  Research Article (7 pages), Article ID 749375, Volume 2013 (2013)
• An Analytical Framework of a Deployment Strategy for Cloud Computing Services: A Case Study of Academic Websites, Chi-Hua Chen, Hui-Fei Lin, Hsu-Chia Chang, Ping-Hsien Ho, and Chi-Chun Lo
  Research Article (14 pages), Article ID 384305, Volume 2013 (2013)
• Optimization Theory, Methods, and Applications in Engineering 2013, Jung-Fa Tsai, John Gunnar Carlsson, Dongdong Ge, Yi-Chung Hu, and Jianming Shi
  Editorial (5 pages), Article ID 319418, Volume 2014 (2014)
• Path-Wise Test Data Generation Based on Heuristic Look-Ahead Methods, Ying Xing, Yun-Zhan Gong, Ya-Wen Wang, and Xu-Zhou Zhang
  Research Article (19 pages), Article ID 642630, Volume 2014 (2014)
• A Hybrid Multiple Criteria Decision Making Model for Supplier Selection, Chung-Min Wu, Ching-Lin Hsieh, and Kuei-Lun Chang
  Research Article (8 pages), Article ID 324283, Volume 2013 (2013)
• Engineering Design by Geometric Programming, Chia-Hui Huang
  Research Article (8 pages), Article ID 568098, Volume 2013 (2013)
• The Reputation Evaluation Based on Optimized Hidden Markov Model in E-Commerce, Liu Chang, Yacine Ouzrout, Antoine Nongaillard, Abdelaziz Bouras, and Zhou Jiliu
  Research Article (11 pages), Article ID 391720, Volume 2013 (2013)
• Ant Colony Optimization for Social Utility Maximization in a Multiuser Communication System, Ming-Hua Lin, Jung-Fa Tsai, and Lu-Yao Lee
  Research Article (8 pages), Article ID 798631, Volume 2013 (2013)
• An Improved Quantum-Inspired Genetic Algorithm for Image Multilevel Thresholding Segmentation, Jian Zhang, Huanzhou Li, Zhangguo Tang, Qiuping Lu, Xiuqing Zheng, and Jiliu Zhou
  Research Article (12 pages), Article ID 295402, Volume 2014 (2014)
• A Multi-Criterion Analysis of Cross-Strait Co-Opetitive Strategy in the Crystalline Silicon Solar Cell Industry, Hsiao-Chi Chen and Chia-Han Yang
  Research Article (11 pages), Article ID 687942, Volume 2014 (2014)
• Bargaining in Patent Licensing Negotiations under Stochastic Environments: An Experimental Study, Yi-Nung Yang and Yu-Jing Chiu
  Research Article (7 pages), Article ID 976450, Volume 2014 (2014)
• Solving the Bilevel Facility Location Problem under Preferences by a Stackelberg-Evolutionary Algorithm, José-Fernando Camacho-Vallejo, Álvaro Eduardo Cordero-Franco, and Rosa G. González-Ramírez
  Research Article (14 pages), Article ID 430243, Volume 2014 (2014)
• On the Transformation Mechanism for Formulating a Multiproduct Two-Layer Supply Chain Network Design Problem as a Network Flow Model, Mi Gan, Zongping Li, and Si Chen
  Research Article (13 pages), Article ID 480127, Volume 2014 (2014)
• Mediating Dynamic Supply Chain Formation by Collaborative Single Machine Earliness/Tardiness Agents in Supply Mesh, Hang Yang, Simon Fong, and Yan Zhuang
  Research Article (21 pages), Article ID 535890, Volume 2014 (2014)
• A Fairness-Based Access Control Scheme to Optimize IPTV Fast Channel Changing, Junyu Lai, James C. F. Li, Alireza Abdollahpouri, Jianhua Zhang, and Ming Lei
  Research Article (12 pages), Article ID 207402, Volume 2014 (2014)
• Measuring the Productivity of Energy Consumption of Major Industries in China: A DEA-Based Method, Xishuang Han, Xiaolong Xue, Jiaoju Ge, Hengqin Wu, and Chang Su
  Research Article (12 pages), Article ID 121804, Volume 2014 (2014)
• Sensitivity Analysis of the Proximal-Based Parallel Decomposition Methods, Feng Ma, Mingfang Ni, Lei Zhu, and Zhanke Yu
  Research Article (9 pages), Article ID 891017, Volume 2014 (2014)
• Stochastic Separated Continuous Conic Programming: Strong Duality and a Solution Method, Xiaoqing Wang
  Research Article (20 pages), Article ID 896591, Volume 2014 (2014)
• Heuristics for Synthesizing Robust Networks with a Diameter Constraint, Harsha Nagarajan, Peng Wei, Sivakumar Rathinam, and Dengfeng Sun
  Research Article (11 pages), Article ID 326963, Volume 2014 (2014)

Optimization Theory, Methods, and Applications in Engineering 2012

• Optimization Theory, Methods, and Applications in Engineering, Jung-Fa Tsai, John Gunnar Carlsson, Dongdong Ge, Yi-Chung Hu, and Jianming Shi
  Editorial (7 pages), Article ID 759548, Volume 2012 (2012)
• A Stone Resource Assignment Model under the Fuzzy Environment, Liming Yao, Jiuping Xu, and Feng Guo
  Research Article (26 pages), Article ID 265837, Volume 2012 (2012)
• Dynamic Programming and Heuristic for Stochastic Uncapacitated Lot-Sizing Problems with Incremental Quantity Discount, Yuli Zhang, Shiji Song, Cheng Wu, and Wenjun Yin
  Research Article (21 pages), Article ID 582323, Volume 2012 (2012)
• Sparse Signal Recovery via ECME Thresholding Pursuits, Heping Song and Guoli Wang
  Research Article (22 pages), Article ID 478931, Volume 2012 (2012)
• Variable Neighborhood Search for Parallel Machines Scheduling Problem with Step Deteriorating Jobs, Wenming Cheng, Peng Guo, Zeqiang Zhang, Ming Zeng, and Jian Liang
  Research Article (20 pages), Article ID 928312, Volume 2012 (2012)
• Global Sufficient Optimality Conditions for a Special Cubic Minimization Problem, Xiaomei Zhang, Yanjun Wang, and Weimin Ma
  Research Article (16 pages), Article ID 871741, Volume 2012 (2012)
• A Two-Stage DEA to Analyze the Effect of Entrance Deregulation on Iranian Insurers: A Robust Approach, Seyed Gholamreza Jalali Naini and Hamid Reza Nouralizadeh
  Research Article (24 pages), Article ID 423524, Volume 2012 (2012)
• Solving Constrained Global Optimization Problems by Using Hybrid Evolutionary Computing and Artificial Life Approaches, Jui-Yu Wu
  Research Article (36 pages), Article ID 841410, Volume 2012 (2012)
• A Hybrid Algorithm Based on ACO and PSO for Capacitated Vehicle Routing Problems, Yucheng Kao, Ming-Hsien Chen, and Yi-Ting Huang
  Research Article (17 pages), Article ID 726564, Volume 2012 (2012)
• New Bounds for Ternary Covering Arrays Using a Parallel Simulated Annealing, Himer Avila-George, Jose Torres-Jimenez, and Vicente Hernández
  Research Article (19 pages), Article ID 897027, Volume 2012 (2012)
• A VNS Metaheuristic with Stochastic Steps for Max 3-Cut and Max 3-Section, Ai-fan Ling
  Research Article (16 pages), Article ID 475018, Volume 2012 (2012)
• Opposition-Based Barebones Particle Swarm for Constrained Nonlinear Optimization Problems, Hui Wang
  Research Article (12 pages), Article ID 761708, Volume 2012 (2012)
• A New Hybrid Nelder-Mead Particle Swarm Optimization for Coordination Optimization of Directional Overcurrent Relays, An Liu and Ming-Ta Yang
  Research Article (18 pages), Article ID 456047, Volume 2012 (2012)
• Evaluating the Performance of Taiwan Homestay Using Analytic Network Process, Yi-Chung Hu, Jen-Hung Wang, and Ru-Yu Wang
  Research Article (24 pages), Article ID 827193, Volume 2012 (2012)
• Mixed Mortar Element Method for Element and Its Multigrid Method for the Incompressible Stokes Problem, Yaqin Jiang and Jinru Chen
  Research Article (18 pages), Article ID 979307, Volume 2012 (2012)
• Applying Neural Networks to Prices Prediction of Crude Oil Futures, John Wei-Shan Hu, Yi-Chung Hu, and Ricky Ray-Wen Lin
  Research Article (12 pages), Article ID 959040, Volume 2012 (2012)
• A Novel Method for Technology Forecasting and Developing R&D Strategy of Building Integrated Photovoltaic Technology Industry, Yu-Jing Chiu and Tao-Ming Ying
  Research Article (24 pages), Article ID 273530, Volume 2012 (2012)
• Solving the Tractor and Semi-Trailer Routing Problem Based on a Heuristic Approach, Hongqi Li, Yue Lu, Jun Zhang, and Tianyi Wang
  Research Article (12 pages), Article ID 182584, Volume 2012 (2012)
• Combining Diffusion and Grey Models Based on Evolutionary Optimization Algorithms to Forecast Motherboard Shipments, Fu-Kwun Wang, Yu-Yao Hsiao, and Ku-Kuang Chang
  Research Article (10 pages), Article ID 849634, Volume 2012 (2012)
• Adaptive Method for Solving Optimal Control Problem with State and Control Variables, Louadj Kahina and Aidene Mohamed
  Research Article (15 pages), Article ID 209329, Volume 2012 (2012)
• A Hybrid Network Model to Extract Key Criteria and Its Application for Brand Equity Evaluation, Chin-Yi Chen and Chung-Wei Li
  Research Article (14 pages), Article ID 971303, Volume 2012 (2012)
• Solving Packing Problems by a Distributed Global Optimization Algorithm, Nian-Ze Hu, Han-Lin Li, and Jung-Fa Tsai
  Research Article (13 pages), Article ID 931092, Volume 2012 (2012)
• Hybrid Optimization Approach for the Design of Mechanisms Using a New Error Estimator, A. Sedano, R. Sancibrian, A. de Juan, F. Viadero, and F. Egaña
  Research Article (20 pages), Article ID 151590, Volume 2012 (2012)
• Goal-Programming-Driven Genetic Algorithm Model for Wireless Access Point Deployment Optimization, Chen-Shu Wang and Ching-Ter Chang
  Research Article (14 pages), Article ID 780637, Volume 2012 (2012)
• Multithreshold Segmentation Based on Artificial Immune Systems, Erik Cuevas, Valentin Osuna-Enciso, Daniel Zaldivar, Marco Pérez-Cisneros, and Humberto Sossa
  Research Article (20 pages), Article ID 874761, Volume 2012 (2012)
• Quality Improvement and Robust Design Methods to a Pharmaceutical Research and Development, Byung Rae Cho and Sangmun Shin
  Research Article (14 pages), Article ID 193246, Volume 2012 (2012)
• A Label Correcting Algorithm for Partial Disassembly Sequences in the Production Planning for End-of-Life Products, Pei-Fang (Jennifer) Tsai
  Research Article (13 pages), Article ID 569429, Volume 2012 (2012)
• Improved Degree Search Algorithms in Unstructured P2P Networks, Guole Liu, Haipeng Peng, Lixiang Li, Yixian Yang, and Qun Luo
  Research Article (18 pages), Article ID 923023, Volume 2012 (2012)
• The Number of Students Needed for Undecided Programs at a College from the Supply-Chain Viewpoint, Jin-Ling Lin, Jy-Hsin Lin, and Kao-Shing Hwang
  Research Article (12 pages), Article ID 276519, Volume 2012 (2012)
• A Review of Deterministic Optimization Methods in Engineering and Management, Ming-Hua Lin, Jung-Fa Tsai, and Chian-Son Yu
  Review Article (15 pages), Article ID 756023, Volume 2012 (2012)
• A Hybrid Genetic Algorithm for the Multiple Crossdocks Problem, Zhaowei Miao, Ke Fu, and Feng Yang
  Research Article (18 pages), Article ID 316908, Volume 2012 (2012)
• A Nonlinear Multiobjective Bilevel Model for Minimum Cost Network Flow Problem in a Large-Scale Construction Project, Jiuping Xu, Yan Tu, and Ziqiang Zeng
  Research Article (40 pages), Article ID 463976, Volume 2012 (2012)
• A Selection Approach for Optimized Problem-Solving Process by Grey Relational Utility Model and Multicriteria Decision Analysis, Chih-Kun Ke and Mei-Yu Wu
  Research Article (14 pages), Article ID 293137, Volume 2012 (2012)
• Predictor-Corrector Primal-Dual Interior Point Method for Solving Economic Dispatch Problems: A Postoptimization Analysis, Antonio Roberto Balbo, Márcio Augusto da Silva Souza, Edméa Cássia Baptista, and Leonardo Nepomuceno
  Research Article (26 pages), Article ID 376546, Volume 2012 (2012)
• Applying Hierarchical Bayesian Neural Network in Failure Time Prediction, Ling-Jing Kao and Hsin-Fen Chen
  Research Article (11 pages), Article ID 953848, Volume 2012 (2012)
• A Fuzzy Dropper for Proportional Loss Rate Differentiation under Wireless Network with a Multi-State Channel, Yu-Chin Szu
  Research Article (16 pages), Article ID 827137, Volume 2012 (2012)
• A Cost-Effective Planning Graph Approach for Large-Scale Web Service Composition, Szu-Yin Lin, Guan-Ting Lin, Kuo-Ming Chao, and Chi-Chun Lo
  Research Article (21 pages), Article ID 783476, Volume 2012 (2012)
• An Optimal Classification Method for Biological and Medical Data, Yao-Huei Huang, Yu-Chien Ko, and Hao-Chun Lu
  Research Article (17 pages), Article ID 398232, Volume 2012 (2012)
• A Modified PSO Algorithm for Minimizing the Total Costs of Resources in MRCPSP, Mohammad Khalilzadeh, Fereydoon Kianfar, Ali Shirzadeh Chaleshtari, Shahram Shadrokh, and Mohammad Ranjbar
  Research Article (18 pages), Article ID 365697, Volume 2012 (2012)
• Comment on “Highly Efficient Sigma Point Filter for Spacecraft Attitude and Rate Estimation”, Baiqing Hu, Lubin Chang, An Li, and Fangjun Qin
  Letter to the Editor (5 pages), Article ID 170391, Volume 2012 (2012)
• A Filter Algorithm with Inexact Line Search, Meiling Liu, Xueqian Li, and Qinmin Wu
  Research Article (20 pages), Article ID 349178, Volume 2012 (2012)
• Optimal Incentive Pricing on Relaying Services for Maximizing Connection Availability in Multihop Cellular Networks, Ming-Hua Lin and Hao-Jan Hsu
  Research Article (17 pages), Article ID 324245, Volume 2012 (2012)

Mathematical optimization (alternatively spelled optimisation) or mathematical programming is the selection of a best element (with regard to some criterion) from some set of available alternatives.^[1] Optimization problems of sorts arise in all quantitative disciplines from computer science and engineering to operations research and economics, and the development of solution methods has been of interest in mathematics for centuries.^[2]

In the simplest case, an optimization problem consists of maximizing or minimizing a real function by systematically choosing input values from within an allowed set and computing the value of the function. The generalization of optimization theory and techniques to other formulations constitutes a large area of applied mathematics. More generally, optimization includes finding 'best available' values of some objective function given a defined domain (or input), including a variety of different types of objective functions and different types of domains.

• 2Notation
• 4Major subfields
• 5Classification of critical points and extrema
• 6Computational optimization techniques
• 7Applications

Optimization problems[edit]

An optimization problem can be represented in the following way:

Given: a function${\displaystyle f:A\to \mathbb{R}}$ from some set${\displaystyle A}$ to the real numbers

Sought: an element ${\displaystyle {\mathbf{x}}_0\in A}$ such that ${\displaystyle f\left({\mathbf{x}}_0\right)\le f\left(\mathbf{x}\right)}$ for all ${\displaystyle \mathbf{x}\in A}$ ('minimization') or such that ${\displaystyle f\left({\mathbf{x}}_0\right)\ge f\left(\mathbf{x}\right)}$ for all ${\displaystyle \mathbf{x}\in A}$ ('maximization').

Such a formulation is called an optimization problem or a mathematical programming problem (a term not directly related to computer programming, but still in use for example in linear programming – see History below). Many real-world and theoretical problems may be modeled in this general framework.

Since the following is valid

${\displaystyle f\left({\mathbf{x}}_0\right)\ge f\left(\mathbf{x}\right)\iff \tilde{f}\left({\mathbf{x}}_0\right)\le \tilde{f}\left(\mathbf{x}\right)}$

with

${\displaystyle \tilde{f}\left(\mathbf{x}\right):=-f\left(\mathbf{x}\right),\tilde{f}:A\to \mathbb{R}}$

it is more convenient to solve minimization problems. However, the opposite perspective would be valid, too.Problems formulated using this technique in the fields of physics may refer to the technique as energy minimization, speaking of the value of the function ${\displaystyle f}$ as representing the energy of the system being modeled. In Machine Learning, it is always necessary to continuously evaluate the quality of a data model by using a cost function where a minimum implies a set of possibly optimal parameters with an optimal (lowest) error.

Typically, ${\displaystyle A}$ is some subset of the Euclidean space${\displaystyle {\mathbb{R}}^n}$, often specified by a set of constraints, equalities or inequalities that the members of ${\displaystyle A}$ have to satisfy. The domain${\displaystyle A}$ of ${\displaystyle f}$ is called the search space or the choice set,while the elements of ${\displaystyle A}$ are called candidate solutions or feasible solutions.

The function ${\displaystyle f}$ is called, variously, an objective function, a loss function or cost function (minimization),^[3] a utility function or fitness function (maximization), or, in certain fields, an energy function or energy functional. A feasible solution that minimizes (or maximizes, if that is the goal) the objective function is called an optimal solution.

In mathematics, conventional optimization problems are usually stated in terms of minimization.

A local minimum${\displaystyle {\mathbf{x}}^{\ast }}$is defined as an element for which there exists some ${\displaystyle \delta >0}$ such that

${\displaystyle \mathrm{\forall }\mathbf{x}\in A}$ where ${\displaystyle \Vert \mathbf{x}-{\mathbf{x}}^{\ast }\Vert \le \delta ,}$ the expression ${\displaystyle f\left({\mathbf{x}}^{\ast }\right)\le f\left(\mathbf{x}\right)}$ holds;

that is to say, on some region around${\displaystyle {\mathbf{x}}^{\ast }}$all of the function values are greater than or equal to the value at that element. Local maxima are defined similarly.

While a local minimum is at least as good as any nearby elements, a global minimum is at least as good as every feasible element.Generally, unless the objective function is convex in a minimization problem, there may be several local minima.In a convex problem, if there is a local minimum that is interior (not on the edge of the set of feasible elements), it is also the global minimum, but a nonconvex problem may have more than one local minimum not all of which need be global minima.

A large number of algorithms proposed for solving the nonconvex problems – including the majority of commercially available solvers – are not capable of making a distinction between locally optimal solutions and globally optimal solutions, and will treat the former as actual solutions to the original problem. Global optimization is the branch of applied mathematics and numerical analysis that is concerned with the development of deterministic algorithms that are capable of guaranteeing convergence in finite time to the actual optimal solution of a nonconvex problem.

Notation[edit]

Optimization problems are often expressed with special notation. Here are some examples:

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Minimum and maximum value of a function[edit]

Consider the following notation:

${\displaystyle \underset{x\in \mathbb{R}}{min}(x^2+1)}$

This denotes the minimum value of the objective function ${\displaystyle x^2+1}$, when choosing x from the set of real numbers${\displaystyle \mathbb{R}}$. The minimum value in this case is ${\displaystyle 1}$, occurring at ${\displaystyle x=0}$.

Similarly, the notation

${\displaystyle \underset{x\in \mathbb{R}}{max}2x}$

asks for the maximum value of the objective function 2x, where x may be any real number. In this case, there is no such maximum as the objective function is unbounded, so the answer is 'infinity' or 'undefined'.

Optimal input arguments[edit]

Consider the following notation:

${\displaystyle \underset{x\in (-\mathrm{\infty },-1]}{\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{m}\mathrm{i}\mathrm{n}}{x}^2+1,}$

or equivalently

${\displaystyle \underset{x}{\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{m}\mathrm{i}\mathrm{n}}{x}^2+1,\text{subject to:}x\in (-\mathrm{\infty },-1].}$

This represents the value (or values) of the argument${\displaystyle x}$ in the interval${\displaystyle (-\mathrm{\infty },-1]}$ that minimizes (or minimize) the objective function ${\displaystyle x^2+1}$ (the actual minimum value of that function is not what the problem asks for). In this case, the answer is ${\displaystyle x=-1}$, since ${\displaystyle x=0}$ is infeasible, i.e. does not belong to the feasible set.

Similarly,

${\displaystyle \underset{x\in [-5,5],y\in \mathbb{R}}{\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{m}\mathrm{a}\mathrm{x}}x\cos (y),}$

or equivalently

${\displaystyle \underset{x,y}{\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{m}\mathrm{a}\mathrm{x}}x\cos (y),\text{subject to:}x\in [-5,5],y\in \mathbb{R},}$

represents the ${\displaystyle (x,y)}$ pair (or pairs) that maximizes (or maximize) the value of the objective function ${\displaystyle x\cos (y)}$, with the added constraint that ${\displaystyle x}$ lie in the interval ${\displaystyle [-5,5]}$ (again, the actual maximum value of the expression does not matter). In this case, the solutions are the pairs of the form ${\displaystyle (5,2k\pi )}$ and ${\displaystyle (-5,(2k+1)\pi )}$, where ${\displaystyle k}$ ranges over all integers.

Operators ${\displaystyle \mathrm{a}\mathrm{r}\mathrm{g}\mathrm{m}\mathrm{i}\mathrm{n}}$ and ${\displaystyle \mathrm{a}\mathrm{r}\mathrm{g}\mathrm{m}\mathrm{a}\mathrm{x}}$ are sometimes also written as ${\displaystyle \mathrm{argmin}}$ and ${\displaystyle \mathrm{argmax}}$, and stand for argument of the minimum and argument of the maximum.

History[edit]

Fermat and Lagrange found calculus-based formulae for identifying optima, while Newton and Gauss proposed iterative methods for moving towards an optimum.

The term 'linear programming' for certain optimization cases was due to George B. Dantzig, although much of the theory had been introduced by Leonid Kantorovich in 1939. (Programming in this context does not refer to computer programming, but comes from the use of program by the United States military to refer to proposed training and logistics schedules, which were the problems Dantzig studied at that time.) Dantzig published the Simplex algorithm in 1947, and John von Neumann developed the theory of duality in the same year.

Other notable researchers in mathematical optimization include the following:

Major subfields[edit]

• Convex programming studies the case when the objective function is convex (minimization) or concave (maximization) and the constraint set is convex. This can be viewed as a particular case of nonlinear programming or as generalization of linear or convex quadratic programming.
  • Linear programming (LP), a type of convex programming, studies the case in which the objective function f is linear and the constraints are specified using only linear equalities and inequalities. Such a constraint set is called a polyhedron or a polytope if it is bounded.
  • Second order cone programming (SOCP) is a convex program, and includes certain types of quadratic programs.
  • Semidefinite programming (SDP) is a subfield of convex optimization where the underlying variables are semidefinitematrices. It is a generalization of linear and convex quadratic programming.
  • Conic programming is a general form of convex programming. LP, SOCP and SDP can all be viewed as conic programs with the appropriate type of cone.
  • Geometric programming is a technique whereby objective and inequality constraints expressed as posynomials and equality constraints as monomials can be transformed into a convex program.
• Integer programming studies linear programs in which some or all variables are constrained to take on integer values. This is not convex, and in general much more difficult than regular linear programming.
• Quadratic programming allows the objective function to have quadratic terms, while the feasible set must be specified with linear equalities and inequalities. For specific forms of the quadratic term, this is a type of convex programming.
• Fractional programming studies optimization of ratios of two nonlinear functions. The special class of concave fractional programs can be transformed to a convex optimization problem.
• Nonlinear programming studies the general case in which the objective function or the constraints or both contain nonlinear parts. This may or may not be a convex program. In general, whether the program is convex affects the difficulty of solving it.
• Stochastic programming studies the case in which some of the constraints or parameters depend on random variables.
• Robust programming is, like stochastic programming, an attempt to capture uncertainty in the data underlying the optimization problem. Robust optimization aims to find solutions that are valid under all possible realizations of the uncertainties.
• Combinatorial optimization is concerned with problems where the set of feasible solutions is discrete or can be reduced to a discrete one.
• Stochastic optimization is used with random (noisy) function measurements or random inputs in the search process.
• Infinite-dimensional optimization studies the case when the set of feasible solutions is a subset of an infinite-dimensional space, such as a space of functions.
• Heuristics and metaheuristics make few or no assumptions about the problem being optimized. Usually, heuristics do not guarantee that any optimal solution need be found. On the other hand, heuristics are used to find approximate solutions for many complicated optimization problems.
• Constraint satisfaction studies the case in which the objective function f is constant (this is used in artificial intelligence, particularly in automated reasoning).
  • Constraint programming is a programming paradigm wherein relations between variables are stated in the form of constraints.
• Disjunctive programming is used where at least one constraint must be satisfied but not all. It is of particular use in scheduling.
• Space mapping is a concept for modeling and optimization of an engineering system to high-fidelity (fine) model accuracy exploiting a suitable physically meaningful coarse or surrogate model.

In a number of subfields, the techniques are designed primarily for optimization in dynamic contexts (that is, decision making over time):

• Calculus of variations seeks to optimize an action integral over some space to an extremum by varying a function of the coordinates.
• Optimal control theory is a generalization of the calculus of variations which introduces control policies.
• Dynamic programming is the approach to solve the stochastic optimization problem with stochastic, randomness, and unknown model parameters. It studies the case in which the optimization strategy is based on splitting the problem into smaller subproblems. The equation that describes the relationship between these subproblems is called the Bellman equation.
• Mathematical programming with equilibrium constraints is where the constraints include variational inequalities or complementarities.

Multi-objective optimization[edit]

Adding more than one objective to an optimization problem adds complexity. For example, to optimize a structural design, one would desire a design that is both light and rigid. When two objectives conflict, a trade-off must be created. There may be one lightest design, one stiffest design, and an infinite number of designs that are some compromise of weight and rigidity. The set of trade-off designs that cannot be improved upon according to one criterion without hurting another criterion is known as the Pareto set. The curve created plotting weight against stiffness of the best designs is known as the Pareto frontier.

A design is judged to be 'Pareto optimal' (equivalently, 'Pareto efficient' or in the Pareto set) if it is not dominated by any other design: If it is worse than another design in some respects and no better in any respect, then it is dominated and is not Pareto optimal.

The choice among 'Pareto optimal' solutions to determine the 'favorite solution' is delegated to the decision maker. In other words, defining the problem as multi-objective optimization signals that some information is missing: desirable objectives are given but combinations of them are not rated relative to each other. In some cases, the missing information can be derived by interactive sessions with the decision maker.

Multi-objective optimization problems have been generalized further into vector optimization problems where the (partial) ordering is no longer given by the Pareto ordering.

Multi-modal optimization[edit]

Optimization problems are often multi-modal; that is, they possess multiple good solutions. They could all be globally good (same cost function value) or there could be a mix of globally good and locally good solutions. Obtaining all (or at least some of) the multiple solutions is the goal of a multi-modal optimizer.

Classical optimization techniques due to their iterative approach do not perform satisfactorily when they are used to obtain multiple solutions, since it is not guaranteed that different solutions will be obtained even with different starting points in multiple runs of the algorithm. Evolutionary algorithms, however, are a very popular approach to obtain multiple solutions in a multi-modal optimization task.

Classification of critical points and extrema[edit]

Feasibility problem[edit]

The satisfiability problem, also called the feasibility problem, is just the problem of finding any feasible solution at all without regard to objective value. This can be regarded as the special case of mathematical optimization where the objective value is the same for every solution, and thus any solution is optimal.

Many optimization algorithms need to start from a feasible point. One way to obtain such a point is to relax the feasibility conditions using a slack variable; with enough slack, any starting point is feasible. Then, minimize that slack variable until slack is null or negative.

Existence[edit]

The extreme value theorem of Karl Weierstrass states that a continuous real-valued function on a compact set attains its maximum and minimum value. More generally, a lower semi-continuous function on a compact set attains its minimum; an upper semi-continuous function on a compact set attains its maximum point or view.

Necessary conditions for optimality[edit]

One of Fermat's theorems states that optima of unconstrained problems are found at stationary points, where the first derivative or the gradient of the objective function is zero (see first derivative test). More generally, they may be found at critical points, where the first derivative or gradient of the objective function is zero or is undefined, or on the boundary of the choice set. An equation (or set of equations) stating that the first derivative(s) equal(s) zero at an interior optimum is called a 'first-order condition' or a set of first-order conditions.

Optima of equality-constrained problems can be found by the Lagrange multiplier method. The optima of problems with equality and/or inequality constraints can be found using the 'Karush–Kuhn–Tucker conditions'.

Sufficient conditions for optimality[edit]

While the first derivative test identifies points that might be extrema, this test does not distinguish a point that is a minimum from one that is a maximum or one that is neither. When the objective function is twice differentiable, these cases can be distinguished by checking the second derivative or the matrix of second derivatives (called the Hessian matrix) in unconstrained problems, or the matrix of second derivatives of the objective function and the constraints called the bordered Hessian in constrained problems. The conditions that distinguish maxima, or minima, from other stationary points are called 'second-order conditions' (see 'Second derivative test'). If a candidate solution satisfies the first-order conditions, then satisfaction of the second-order conditions as well is sufficient to establish at least local optimality.

Sensitivity and continuity of optima[edit]

The envelope theorem describes how the value of an optimal solution changes when an underlying parameter changes. The process of computing this change is called comparative statics.

The maximum theorem of Claude Berge (1963) describes the continuity of an optimal solution as a function of underlying parameters.

Calculus of optimization[edit]

For unconstrained problems with twice-differentiable functions, some critical points can be found by finding the points where the gradient of the objective function is zero (that is, the stationary points). More generally, a zero subgradient certifies that a local minimum has been found for minimization problems with convexfunctions and other locallyLipschitz functions.

Further, critical points can be classified using the definiteness of the Hessian matrix: If the Hessian is positive definite at a critical point, then the point is a local minimum; if the Hessian matrix is negative definite, then the point is a local maximum; finally, if indefinite, then the point is some kind of saddle point.

Constrained problems can often be transformed into unconstrained problems with the help of Lagrange multipliers. Lagrangian relaxation can also provide approximate solutions to difficult constrained problems.

Journal Of Optimization Theory And Applications Ranking

When the objective function is Convex function(Convex), then any local minimum will also be a global minimum. There exist efficient numerical techniques for minimizing convex functions, such as interior-point methods.

Computational optimization techniques[edit]

To solve problems, researchers may use algorithms that terminate in a finite number of steps, or iterative methods that converge to a solution (on some specified class of problems), or heuristics that may provide approximate solutions to some problems (although their iterates need not converge).

Optimization algorithms[edit]

• Simplex algorithm of George Dantzig, designed for linear programming.
• Extensions of the simplex algorithm, designed for quadratic programming and for linear-fractional programming.
• Variants of the simplex algorithm that are especially suited for network optimization.

Optimization algorithms in machine learning

Introduction

An optimization algorithm is a procedure which is executed iteratively by comparing various solutions until an optimum or a satisfactory solution is found. Optimization algorithms help us to minimize or maximize an objective function E(x) with respect to the internal parameters of a model mapping a set of predictors (X) to target values(Y). There are three types of optimization algorithms which are widely used; Zero order algorithms, First Order Optimization Algorithms and Second Order Optimization Algorithms.^[4]

Zero-order algorithms

Zero-order (or derivative-free) algorithms use only the criterion value at some positions.^[4] It is popular when the gradient and Hessian information are difficult to obtain, e.g., no explicit function forms are given.^[5]

First Order Optimization Algorithms

These algorithms minimize or maximize a Loss function E(x) using its Gradient values with respect to the parameters.^[4] Most widely used First order optimization algorithm is Gradient Descent. The First order derivative displays whether the function is decreasing or increasing at a particular point. First order Derivative basically will provide us a line which is tangential to a point on its Error Surface.^[6]

Example

Gradient descent

It is a first-order optimization algorithm for finding the minimum of a function.

θ=θ−η⋅∇J(θ) – this is the formula of the parameter updates, where ‘η’ is the learning rate, ’∇J(θ)’ is the Gradient of Loss function-J(θ) w.r.t parameters-‘θ’.

It is the most popular optimization algorithm used in optimizing a Neural Network. Gradient descent is used to update Weights in a Neural Network Model, i.e. update and tune the Model's parameters in a direction so that we can minimize the Loss function. A Neural Network trains via a technique called Back-propagation, in which propagating forward calculating the dot product of Inputs signals and their corresponding Weights and then applying an activation function to those sum of products, which transforms the input signal to an output signal and also is important to model complex Non-linear functions and introduces Non-linearity to the Model which enables the Model to learn almost any arbitrary functional mapping.^[7]

Second Order Optimization Algorithms

Second-order methods use the second order derivative which is also called Hessian to minimize or maximize the loss function.^[4] The Hessian is a matrix of Second Order Partial Derivatives. Since the second derivative is costly to compute, the second order is not used much. The second order derivative informs us whether the first derivative is increasing or decreasing which hints at the function's curvature. It also provides us with a quadratic surface which touches the curvature of the Error Surface.^[8]

Iterative methods[edit]

The iterative methods used to solve problems of nonlinear programming differ according to whether they evaluateHessians, gradients, or only function values. While evaluating Hessians (H) and gradients (G) improves the rate of convergence, for functions for which these quantities exist and vary sufficiently smoothly, such evaluations increase the computational complexity (or computational cost) of each iteration. In some cases, the computational complexity may be excessively high.

One major criterion for optimizers is just the number of required function evaluations as this often is already a large computational effort, usually much more effort than within the optimizer itself, which mainly has to operate over the N variables.The derivatives provide detailed information for such optimizers, but are even harder to calculate, e.g. approximating the gradient takes at least N+1 function evaluations. For approximations of the 2nd derivatives (collected in the Hessian matrix), the number of function evaluations is in the order of N². Newton's method requires the 2nd order derivatives, so for each iteration, the number of function calls is in the order of N², but for a simpler pure gradient optimizer it is only N. However, gradient optimizers need usually more iterations than Newton's algorithm. Which one is best with respect to the number of function calls depends on the problem itself.

• Methods that evaluate Hessians (or approximate Hessians, using finite differences):
  • Sequential quadratic programming: A Newton-based method for small-medium scale constrained problems. Some versions can handle large-dimensional problems.
  • Interior point methods: This is a large class of methods for constrained optimization. Some interior-point methods use only (sub)gradient information and others of which require the evaluation of Hessians.
• Methods that evaluate gradients, or approximate gradients in some way (or even subgradients):
  • Coordinate descent methods: Algorithms which update a single coordinate in each iteration
  • Conjugate gradient methods: Iterative methods for large problems. (In theory, these methods terminate in a finite number of steps with quadratic objective functions, but this finite termination is not observed in practice on finite–precision computers.)
  • Gradient descent (alternatively, 'steepest descent' or 'steepest ascent'): A (slow) method of historical and theoretical interest, which has had renewed interest for finding approximate solutions of enormous problems.
  • Subgradient methods - An iterative method for large locallyLipschitz functions using generalized gradients. Following Boris T. Polyak, subgradient–projection methods are similar to conjugate–gradient methods.
  • Bundle method of descent: An iterative method for small–medium-sized problems with locally Lipschitz functions, particularly for convex minimization problems. (Similar to conjugate gradient methods)
  • Ellipsoid method: An iterative method for small problems with quasiconvex objective functions and of great theoretical interest, particularly in establishing the polynomial time complexity of some combinatorial optimization problems. It has similarities with Quasi-Newton methods.
  • Conditional gradient method (Frank–Wolfe) for approximate minimization of specially structured problems with linear constraints, especially with traffic networks. For general unconstrained problems, this method reduces to the gradient method, which is regarded as obsolete (for almost all problems).
  • Quasi-Newton methods: Iterative methods for medium-large problems (e.g. N<1000).
  • Simultaneous perturbation stochastic approximation (SPSA) method for stochastic optimization; uses random (efficient) gradient approximation.
• Methods that evaluate only function values: If a problem is continuously differentiable, then gradients can be approximated using finite differences, in which case a gradient-based method can be used.
  • Interpolation methods
  • Pattern search methods, which have better convergence properties than the Nelder–Mead heuristic (with simplices), which is listed below.

Global convergence[edit]

More generally, if the objective function is not a quadratic function, then many optimization methods use other methods to ensure that some subsequence of iterations converges to an optimal solution. The first and still popular method for ensuring convergence relies on line searches, which optimize a function along one dimension. A second and increasingly popular method for ensuring convergence uses trust regions. Both line searches and trust regions are used in modern methods of non-differentiable optimization. Usually a global optimizer is much slower than advanced local optimizers (such as BFGS), so often an efficient global optimizer can be constructed by starting the local optimizer from different starting points.

Journal Of Optimization Theory Application

Heuristics[edit]

Besides (finitely terminating) algorithms and (convergent) iterative methods, there are heuristics. A heuristic is any algorithm which is not guaranteed (mathematically) to find the solution, but which is nevertheless useful in certain practical situations. List of some well-known heuristics:

• Hill climbing with random restart
• Nelder-Mead simplicial heuristic: A popular heuristic for approximate minimization (without calling gradients)
• Reactive Search Optimization (RSO)^[9] implemented in LIONsolver

Applications[edit]

Mechanics[edit]

Problems in rigid body dynamics (in particular articulated rigid body dynamics) often require mathematical programming techniques, since you can view rigid body dynamics as attempting to solve an ordinary differential equation on a constraint manifold;^[10] the constraints are various nonlinear geometric constraints such as 'these two points must always coincide', 'this surface must not penetrate any other', or 'this point must always lie somewhere on this curve'. Also, the problem of computing contact forces can be done by solving a linear complementarity problem, which can also be viewed as a QP (quadratic programming) problem.

Many design problems can also be expressed as optimization programs. This application is called design optimization. One subset is the engineering optimization, and another recent and growing subset of this field is multidisciplinary design optimization, which, while useful in many problems, has in particular been applied to aerospace engineering problems.

This approach may be applied in cosmology and astrophysics.^[11]

Economics and finance[edit]

Economics is closely enough linked to optimization of agents that an influential definition relatedly describes economics qua science as the 'study of human behavior as a relationship between ends and scarce means' with alternative uses.^[12] Modern optimization theory includes traditional optimization theory but also overlaps with game theory and the study of economic equilibria. The Journal of Economic Literaturecodes classify mathematical programming, optimization techniques, and related topics under JEL:C61-C63.

In microeconomics, the utility maximization problem and its dual problem, the expenditure minimization problem, are economic optimization problems. Insofar as they behave consistently, consumers are assumed to maximize their utility, while firms are usually assumed to maximize their profit. Also, agents are often modeled as being risk-averse, thereby preferring to avoid risk. Asset prices are also modeled using optimization theory, though the underlying mathematics relies on optimizing stochastic processes rather than on static optimization. International trade theory also uses optimization to explain trade patterns between nations. The optimization of portfolios is an example of multi-objective optimization in economics.

Since the 1970s, economists have modeled dynamic decisions over time using control theory.^[13] For example, dynamic search models are used to study labor-market behavior.^[14] A crucial distinction is between deterministic and stochastic models.^[15]Macroeconomists build dynamic stochastic general equilibrium (DSGE) models that describe the dynamics of the whole economy as the result of the interdependent optimizing decisions of workers, consumers, investors, and governments.^[16][17]

Electrical engineering[edit]

Some common applications of optimization techniques in electrical engineering include active filter design,^[18] stray field reduction in superconducting magnetic energy storage systems, space mapping design of microwave structures,^[19] handset antennas,^[20][21][22] electromagnetics-based design. Electromagnetically validated design optimization of microwave components and antennas has made extensive use of an appropriate physics-based or empirical surrogate model and space mapping methodologies since the discovery of space mapping in 1993.^[23][24]

Civil engineering[edit]

Optimization has been widely used in civil engineering. The most common civil engineering problems that are solved by optimization are cut and fill of roads, life-cycle analysis of structures and infrastructures,^[25]resource leveling,^[26]water resource allocation, and schedule optimization.

Operations research[edit]

Journal Of Optimization Theory And Application

Another field that uses optimization techniques extensively is operations research.^[27] Operations research also uses stochastic modeling and simulation to support improved decision-making. Increasingly, operations research uses stochastic programming to model dynamic decisions that adapt to events; such problems can be solved with large-scale optimization and stochastic optimization methods.

Control engineering[edit]

Optimization Theory And Applications By S. S. Rao

Mathematical optimization is used in much modern controller design. High-level controllers such as model predictive control (MPC) or real-time optimization (RTO) employ mathematical optimization. These algorithms run online and repeatedly determine values for decision variables, such as choke openings in a process plant, by iteratively solving a mathematical optimization problem including constraints and a model of the system to be controlled.

Geophysics[edit]

Optimization techniques are regularly used in geophysical parameter estimation problems. Given a set of geophysical measurements, e.g. seismic recordings, it is common to solve for the physical properties and geometrical shapes of the underlying rocks and fluids.

Molecular modeling[edit]

Nonlinear optimization methods are widely used in conformational analysis.

Computational systems biology[edit]

Optimization techniques are used in many facets of computational systems biology such as model building, optimal experimental design, metabolic engineering, and synthetic biology.^[28]Linear programming has been applied to calculate the maximal possible yields of fermentation products,^[28] and to infer gene regulatory networks from multiple microarray datasets^[29] as well as transcriptional regulatory networks from high-throughput data.^[30]Nonlinear programming has been used to analyze energy metabolism^[31] and has been applied to metabolic engineering and parameter estimation in biochemical pathways.^[32]

Optimization Theory And Methods

Machine Learning[edit]

Solvers[edit]

See also[edit]

• Mathematical Optimization Society (formerly Mathematical Programming Society)

Notes[edit]

1. ^'The Nature of Mathematical ProgrammingArchived 2014-03-05 at the Wayback Machine,' Mathematical Programming Glossary, INFORMS Computing Society.
2. ^Du, D. Z.; Pardalos, P. M.; Wu, W. (2008). 'History of Optimization'. In Floudas, C.; Pardalos, P. (eds.). Encyclopedia of Optimization. Boston: Springer. pp. 1538–1542.
3. ^W. Erwin Diewert (2008). 'cost functions,' The New Palgrave Dictionary of Economics, 2nd Edition Contents.
4. ^ ^abcdWalia, Anish (2017). 'Types of Optimization Algorithms used in Neural Networks and Ways to Optimize Gradient Descent'. towardsdatascience.com.
5. ^Ruffio, E.; Saury, D.; Petit, D.; Girault, M. 'Zero-Order optimization algorithms'(PDF).
6. ^Ye.Y. Zero-Order and First-Order Optimization Algorithms I. Stanford University: Department of Management Science and Engineering. Retrieved from https://web.stanford.edu/class/msande311/lecture10.pdf
7. ^Evans.J (1992). Optimization algorithms for networks and graphs. CRC Press 2nd edition.
8. ^Manson, L.; Baxter, J.; Bartlett, P.; Fream, M. (1999). 'Boosting Algorithms as Gradient Descent'. Advances in Neural Information Processing Systems. 12: 512–518. chapterurl= ignored (help)
9. ^Battiti, Roberto; Mauro Brunato; Franco Mascia (2008). Reactive Search and Intelligent Optimization. Springer Verlag. ISBN978-0-387-09623-0. Archived from the original on 2012-03-16.
10. ^Vereshchagin, A.F. (1989). 'Modelling and control of motion of manipulation robots'. Soviet Journal of Computer and Systems Sciences. 27 (5): 29–38.
11. ^Haggag, S.; Desokey, F.; Ramadan, M. (2017). 'A cosmological inflationary model using optimal control'. Gravitation and Cosmology. 23 (3): 236–239. Bibcode:2017GrCo..23.236H. doi:10.1134/S0202289317030069. ISSN1995-0721.
12. ^Lionel Robbins (1935, 2nd ed.) An Essay on the Nature and Significance of Economic Science, Macmillan, p. 16.
13. ^Dorfman, Robert (1969). 'An Economic Interpretation of Optimal Control Theory'. American Economic Review. 59 (5): 817–831. JSTOR1810679.
14. ^Sargent, Thomas J. (1987). 'Search'. Dynamic Macroeconomic Theory. Harvard University Press. pp. 57–91.
15. ^A.G. Malliaris (2008). 'stochastic optimal control,' The New Palgrave Dictionary of Economics, 2nd Edition. AbstractArchived 2017-10-18 at the Wayback Machine.
16. ^Rotemberg, Julio; Woodford, Michael (1997). 'An Optimization-based Econometric Framework for the Evaluation of Monetary Policy'(PDF). NBER Macroeconomics Annual. 12: 297–346. doi:10.2307/3585236. JSTOR3585236.
17. ^From The New Palgrave Dictionary of Economics (2008), 2nd Edition with Abstract links:
    • 'numerical optimization methods in economics' by Karl Schmedders
    • 'convex programming' by Lawrence E. Blume
    • 'Arrow–Debreu model of general equilibrium' by John Geanakoplos.
18. ^De, Bishnu Prasad; Kar, R.; Mandal, D.; Ghoshal, S.P. (2014-09-27). 'Optimal selection of components value for analog active filter design using simplex particle swarm optimization'. International Journal of Machine Learning and Cybernetics. 6 (4): 621–636. doi:10.1007/s13042-014-0299-0. ISSN1868-8071.
19. ^Koziel, Slawomir; Bandler, John W. (January 2008). 'Space Mapping With Multiple Coarse Models for Optimization of Microwave Components'. IEEE Microwave and Wireless Components Letters. 18 (1): 1–3. CiteSeerX10.1.1.147.5407. doi:10.1109/LMWC.2007.911969.
20. ^Tu, Sheng; Cheng, Qingsha S.; Zhang, Yifan; Bandler, John W.; Nikolova, Natalia K. (July 2013). 'Space Mapping Optimization of Handset Antennas Exploiting Thin-Wire Models'. IEEE Transactions on Antennas and Propagation. 61 (7): 3797–3807. Bibcode:2013ITAP..61.3797T. doi:10.1109/TAP.2013.2254695.
21. ^N. Friedrich, “Space mapping outpaces EM optimization in handset-antenna design,” microwaves&rf, Aug. 30, 2013.
22. ^Cervantes-González, Juan C.; Rayas-Sánchez, José E.; López, Carlos A.; Camacho-Pérez, José R.; Brito-Brito, Zabdiel; Chávez-Hurtado, José L. (February 2016). 'Space mapping optimization of handset antennas considering EM effects of mobile phone components and human body'. International Journal of RF and Microwave Computer-Aided Engineering. 26 (2): 121–128. doi:10.1002/mmce.20945.
23. ^Bandler, J.W.; Biernacki, R.M.; Chen, Shao Hua; Grobelny, P.A.; Hemmers, R.H. (1994). 'Space mapping technique for electromagnetic optimization'. IEEE Transactions on Microwave Theory and Techniques. 42 (12): 2536–2544. Bibcode:1994ITMTT.42.2536B. doi:10.1109/22.339794.
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25. ^Piryonesi, Sayed Madeh; Tavakolan, Mehdi (9 January 2017). 'A mathematical programming model for solving cost-safety optimization (CSO) problems in the maintenance of structures'. KSCE Journal of Civil Engineering. 21 (6): 2226–2234. doi:10.1007/s12205-017-0531-z.
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Further reading[edit]

• Boyd, Stephen P.; Vandenberghe, Lieven (2004). Convex Optimization. Cambridge: Cambridge University Press. ISBN0-521-83378-7.
• Gill, P. E.; Murray, W.; Wright, M. H. (1982). Practical Optimization. London: Academic Press. ISBN0-12-283952-8.
• Lee, Jon (2004). A First Course in Combinatorial Optimization. Cambridge University Press. ISBN0-521-01012-8.
• Nocedal, Jorge; Wright, Stephen J. (2006). Numerical Optimization (2nd ed.). Berlin: Springer. ISBN0-387-30303-0.
• Snyman, J. A.; Wilke, D. N. (2018). Practical Mathematical Optimization : Basic Optimization Theory and Gradient-Based Algorithms (2nd ed.). Berlin: Springer. ISBN978-3-319-77585-2.

Journal Of Optimization Theory And Applications Impact Factor

External links[edit]

Robust Optimization Ppt

Theory And Application Of Robust Optimization

• 'Decision Tree for Optimization Software'. Links to optimization source codes
• 'Global optimization'.
• 'EE364a: Convex Optimization I'. Course from Stanford University.
• Varoquaux, Gaël. 'Mathematical Optimization: Finding Minima of Functions'.