A Comparative Study of Meta-Heuristic Algorithms for Solving Quadratic Assignment Problem
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4. Convex Optimization Problems
Convex Optimization — Boyd & Vandenberghe 4. Convex optimization problems optimization problem in standard form • convex optimization problems • quasiconvex optimization • linear optimization • quadratic optimization • geometric programming • generalized inequality constraints • semidefinite programming • vector optimization • 4–1 Optimization problem in standard form minimize f0(x) subject to f (x) 0, i =1,...,m i ≤ hi(x)=0, i =1,...,p x Rn is the optimization variable • ∈ f : Rn R is the objective or cost function • 0 → f : Rn R, i =1,...,m, are the inequality constraint functions • i → h : Rn R are the equality constraint functions • i → optimal value: p⋆ = inf f (x) f (x) 0, i =1,...,m, h (x)=0, i =1,...,p { 0 | i ≤ i } p⋆ = if problem is infeasible (no x satisfies the constraints) • ∞ p⋆ = if problem is unbounded below • −∞ Convex optimization problems 4–2 Optimal and locally optimal points x is feasible if x dom f and it satisfies the constraints ∈ 0 ⋆ a feasible x is optimal if f0(x)= p ; Xopt is the set of optimal points x is locally optimal if there is an R> 0 such that x is optimal for minimize (over z) f0(z) subject to fi(z) 0, i =1,...,m, hi(z)=0, i =1,...,p z x≤ R k − k2 ≤ examples (with n =1, m = p =0) f (x)=1/x, dom f = R : p⋆ =0, no optimal point • 0 0 ++ f (x)= log x, dom f = R : p⋆ = • 0 − 0 ++ −∞ f (x)= x log x, dom f = R : p⋆ = 1/e, x =1/e is optimal • 0 0 ++ − f (x)= x3 3x, p⋆ = , local optimum at x =1 • 0 − −∞ Convex optimization problems 4–3 Implicit constraints the standard form optimization problem has an implicit -
Metaheuristics1
METAHEURISTICS1 Kenneth Sörensen University of Antwerp, Belgium Fred Glover University of Colorado and OptTek Systems, Inc., USA 1 Definition A metaheuristic is a high-level problem-independent algorithmic framework that provides a set of guidelines or strategies to develop heuristic optimization algorithms (Sörensen and Glover, To appear). Notable examples of metaheuristics include genetic/evolutionary algorithms, tabu search, simulated annealing, and ant colony optimization, although many more exist. A problem-specific implementation of a heuristic optimization algorithm according to the guidelines expressed in a metaheuristic framework is also referred to as a metaheuristic. The term was coined by Glover (1986) and combines the Greek prefix meta- (metá, beyond in the sense of high-level) with heuristic (from the Greek heuriskein or euriskein, to search). Metaheuristic algorithms, i.e., optimization methods designed according to the strategies laid out in a metaheuristic framework, are — as the name suggests — always heuristic in nature. This fact distinguishes them from exact methods, that do come with a proof that the optimal solution will be found in a finite (although often prohibitively large) amount of time. Metaheuristics are therefore developed specifically to find a solution that is “good enough” in a computing time that is “small enough”. As a result, they are not subject to combinatorial explosion – the phenomenon where the computing time required to find the optimal solution of NP- hard problems increases as an exponential function of the problem size. Metaheuristics have been demonstrated by the scientific community to be a viable, and often superior, alternative to more traditional (exact) methods of mixed- integer optimization such as branch and bound and dynamic programming. -
Removing the Genetics from the Standard Genetic Algorithm
Removing the Genetics from the Standard Genetic Algorithm Shumeet Baluja Rich Caruana School of Computer Science School of Computer Science Carnegie Mellon University Carnegie Mellon University Pittsburgh, PA 15213 Pittsburgh, PA 15213 [email protected] [email protected] Abstract from both parents into their respective positions in a mem- ber of the subsequent generation. Due to random factors involved in producing “children” chromosomes, the chil- We present an abstraction of the genetic dren may, or may not, have higher fitness values than their algorithm (GA), termed population-based parents. Nevertheless, because of the selective pressure incremental learning (PBIL), that explicitly applied through a number of generations, the overall trend maintains the statistics contained in a GA’s is towards higher fitness chromosomes. Mutations are population, but which abstracts away the used to help preserve diversity in the population. Muta- crossover operator and redefines the role of tions introduce random changes into the chromosomes. A the population. This results in PBIL being good overview of GAs can be found in [Goldberg, 1989] simpler, both computationally and theoreti- [De Jong, 1975]. cally, than the GA. Empirical results reported elsewhere show that PBIL is faster Although there has recently been some controversy in the and more effective than the GA on a large set GA community as to whether GAs should be used for of commonly used benchmark problems. static function optimization, a large amount of research Here we present results on a problem custom has been, and continues to be, conducted in this direction. designed to benefit both from the GA’s cross- [De Jong, 1992] claims that the GA is not a function opti- over operator and from its use of a popula- mizer, and that typical GAs which are used for function tion. -
Simulated Annealing Hyper-Heuristic for a Shelf Space Allocation on Symmetrical Planograms Problem
S S symmetry Article Simulated Annealing Hyper-Heuristic for a Shelf Space Allocation on Symmetrical Planograms Problem Kateryna Czerniachowska * and Marcin Hernes Department of Process Management, Wroclaw University of Economics and Business, Komandorska 118/120, 53-345 Wroclaw, Poland; [email protected] * Correspondence: [email protected] Abstract: The allocation of products on shelves is an important issue from the point of view of effective decision making by retailers. In this paper, we investigate a practical shelf space allocation model which takes into account the number of facings, capping, and nesting of a product. We divide the shelf into the segments of variable size in which the products of the specific types could be placed. The interconnections between products are modelled with the help of categorizing the products into specific types as well as grouping some of them into clusters. This results in four groups of constraints—shelf constraints, shelf type constraints, product constraints, position allocation constraints—that are used in the model for aesthetic symmetry of a planogram. We propose a simulated annealing algorithm with improvement and reallocation procedures to solve the planogram profit maximization problem. Experiments are based on artificial data sets that have been generated according to real-world conditions. The efficiency of the designed algorithm has been estimated using the CPLEX solver. The computational tests demonstrate that the proposed algorithm gives valuable results in an acceptable time. Citation: Czerniachowska, K.; Hernes, M. Simulated Annealing Keywords: retailing decisions; shelf space allocation; simulated annealing; heuristics; merchandising Hyper-Heuristic for a Shelf Space Allocation on Symmetrical Planograms Problem. -
A Simulated Annealing Based Inexact Oracle for Wasserstein Loss Minimization
A Simulated Annealing Based Inexact Oracle for Wasserstein Loss Minimization Jianbo Ye 1 James Z. Wang 1 Jia Li 2 Abstract ryl(x; ·) is computed in O(m) time, m being the complex- ity of outcome variables x or y. This part of calculation is Learning under a Wasserstein loss, a.k.a. Wasser- often negligible compared with the calculation of full gra- stein loss minimization (WLM), is an emerging dient with respect to the model parameters. But this is no research topic for gaining insights from a large longer the case in learning problems based on Wasserstein set of structured objects. Despite being concep- distance due to the intrinsic complexity of the distance. tually simple, WLM problems are computation- We will call such problems Wasserstein loss minimization ally challenging because they involve minimiz- (WLM). Examples of WLMs include Wasserstein barycen- ing over functions of quantities (i.e. Wasserstein ters (Li & Wang, 2008; Agueh & Carlier, 2011; Cuturi & distances) that themselves require numerical al- Doucet, 2014; Benamou et al., 2015; Ye & Li, 2014; Ye gorithms to compute. In this paper, we intro- et al., 2017b), principal geodesics (Seguy & Cuturi, 2015), duce a stochastic approach based on simulated nonnegative matrix factorization (Rolet et al., 2016; San- annealing for solving WLMs. Particularly, we dler & Lindenbaum, 2009), barycentric coordinate (Bon- have developed a Gibbs sampler to approximate neel et al., 2016), and multi-label classification (Frogner effectively and efficiently the partial gradients of et al., 2015). a sequence of Wasserstein losses. Our new ap- proach has the advantages of numerical stability Wasserstein distance is defined as the cost of matching two and readiness for warm starts. -
University of Waterloo Faculty of Engineering
University of Waterloo Faculty of Engineering Global Optimization of General Failure Surfaces in Slope Analysis by Hybrid Simulated Annealing Rocscience, Inc. Toronto, Ontario Prepared by Su, Xiao ID 20251755 2B Chemical Engineering May 1st , 2009 4105 Credit Pointe Dr, Mississauga, Ontario L5M 3K7 May 1st, 2009 Dr Thomas Duever, Chair of Chemical Engineering University of Waterloo Waterloo, Ontario N2L 3G1 Dear Dr Duever, I am submitting this document, entitled “Global Optimization of General Failure Surfaces in Slope Analysis by Hybrid Simulated Annealing” as my 2B Work Report for Rocscience, Inc. My project investigated the efficiency of a hybrid simulated annealing (HSA) method for finding the minima (critical failure surfaces and factors of safety) of slope stability equations. The HSA combines a Very Fast Simulated Annealing (VFSA) and a Local Monte-Carlo (LMC) search, the latter being an optimization algorithm designed specifically for solving slope stability problems. This work is an extension as well as a generalization of the project from my summer work-term (2008), also undertaken at Rocscience. During both terms, I worked under the supervision of Dr. Brent Corkum, software development manager at Rocscience. I implemented the simulated annealing algorithm as a new optimization method in SLIDE 5.0, a slope stability analysis program developed by Rocscience. In addition to learning professional C++ and principles of object oriented programming, I had the opportunity to study the theoretical and practical aspects of non-convex optimization, especially the performance of stochastic methods when used on multi- modal functions. I also learned soil mechanics and slope stability principles. This report was written entirely by me and has not received any previous academic credit at Rocscience or the University of Waterloo. -
Arxiv:2101.10154V1 [Cond-Mat.Dis-Nn] 25 Jan 2021
Variational Neural Annealing 1, 2, 3, 1 1, 2 2, 3 1, 2 Mohamed Hibat-Allah, ∗ Estelle M. Inack, Roeland Wiersema, Roger G. Melko, and Juan Carrasquilla 1Vector Institute, MaRS Centre, Toronto, Ontario, M5G 1M1, Canada 2Department of Physics and Astronomy, University of Waterloo, Ontario, N2L 3G1, Canada 3Perimeter Institute for Theoretical Physics, Waterloo, ON N2L 2Y5, Canada (Dated: January 26, 2021) Many important challenges in science and technology can be cast as optimization problems. When viewed in a statistical physics framework, these can be tackled by simulated annealing, where a gradual cooling procedure helps search for groundstate solutions of a target Hamiltonian. While powerful, simulated annealing is known to have prohibitively slow sampling dynamics when the optimization landscape is rough or glassy. Here we show that by generalizing the target distribution with a parameterized model, an analogous annealing framework based on the variational principle can be used to search for groundstate solutions. Modern autoregressive models such as recurrent neural networks provide ideal parameterizations since they can be exactly sampled without slow dynamics even when the model encodes a rough landscape. We implement this procedure in the classical and quantum settings on several prototypical spin glass Hamiltonians, and find that it significantly outperforms traditional simulated annealing in the asymptotic limit, illustrating the potential power of this yet unexplored route to optimization. P I. INTRODUCTION <latexit sha1_base64="Wk+wWdaEIxpmR5GD8uDFICvy2Ig=">AAACNHicfVDLSgMxFL1TX7W+Rl26CRbBVZkRQZdFN4KbCvYBbRkyaaYNzWSGJGMpQz/KjR/iRgQXirj1G8y0s6iteCBwOOfcJPf4MWdKO86rVVhZXVvfKG6WtrZ3dvfs/YOGihJJaJ1EPJItHyvKmaB1zTSnrVhSHPqcNv3hdeY3H6hULBL3ehzTboj7ggWMYG0kz76teSnq9KKRwFJGI9RJ4kUyZ/6Xm5Q8u+xUnCnQMnFzUoYcNc9+NjeSJKRCE46VartOrLsplpoRTielTqJojMkQ92nbUIFDqrrpdOkJOjFKDwWRNEdoNFXnJ1IcKjUOfZMMsR6oRS8T//LaiQ4uuykTcaKpILOHgoQjHaGsQdRjkhLNx4ZgIpn5KyIDLDHRpuesBHdx5WXSOKu4TsW9Oy9Xr/I6inAEx3AKLlxAFW6gBnUg8Agv8A4f1pP1Zn1aX7NowcpnDuEXrO8fzwWstw==</latexit> -
Genetic Algorithm: Reviews, Implementations, and Applications
Paper— Genetic Algorithm: Reviews, Implementation and Applications Genetic Algorithm: Reviews, Implementations, and Applications Tanweer Alam() Faculty of Computer and Information Systems, Islamic University of Madinah, Saudi Arabia [email protected] Shamimul Qamar Computer Engineering Department, King Khalid University, Abha, Saudi Arabia Amit Dixit Department of ECE, Quantum School of Technology, Roorkee, India Mohamed Benaida Faculty of Computer and Information Systems, Islamic University of Madinah, Saudi Arabia How to cite this article? Tanweer Alam. Shamimul Qamar. Amit Dixit. Mohamed Benaida. " Genetic Al- gorithm: Reviews, Implementations, and Applications.", International Journal of Engineering Pedagogy (iJEP). 2020. Abstract—Nowadays genetic algorithm (GA) is greatly used in engineering ped- agogy as an adaptive technique to learn and solve complex problems and issues. It is a meta-heuristic approach that is used to solve hybrid computation chal- lenges. GA utilizes selection, crossover, and mutation operators to effectively manage the searching system strategy. This algorithm is derived from natural se- lection and genetics concepts. GA is an intelligent use of random search sup- ported with historical data to contribute the search in an area of the improved outcome within a coverage framework. Such algorithms are widely used for maintaining high-quality reactions to optimize issues and problems investigation. These techniques are recognized to be somewhat of a statistical investigation pro- cess to search for a suitable solution or prevent an accurate strategy for challenges in optimization or searches. These techniques have been produced from natu- ral selection or genetics principles. For random testing, historical information is provided with intelligent enslavement to continue moving the search out from the area of improved features for processing of the outcomes. -
IEOR 269, Spring 2010 Integer Programming and Combinatorial Optimization
IEOR 269, Spring 2010 Integer Programming and Combinatorial Optimization Professor Dorit S. Hochbaum Contents 1 Introduction 1 2 Formulation of some ILP 2 2.1 0-1 knapsack problem . 2 2.2 Assignment problem . 2 3 Non-linear Objective functions 4 3.1 Production problem with set-up costs . 4 3.2 Piecewise linear cost function . 5 3.3 Piecewise linear convex cost function . 6 3.4 Disjunctive constraints . 7 4 Some famous combinatorial problems 7 4.1 Max clique problem . 7 4.2 SAT (satisfiability) . 7 4.3 Vertex cover problem . 7 5 General optimization 8 6 Neighborhood 8 6.1 Exact neighborhood . 8 7 Complexity of algorithms 9 7.1 Finding the maximum element . 9 7.2 0-1 knapsack . 9 7.3 Linear systems . 10 7.4 Linear Programming . 11 8 Some interesting IP formulations 12 8.1 The fixed cost plant location problem . 12 8.2 Minimum/maximum spanning tree (MST) . 12 9 The Minimum Spanning Tree (MST) Problem 13 i IEOR269 notes, Prof. Hochbaum, 2010 ii 10 General Matching Problem 14 10.1 Maximum Matching Problem in Bipartite Graphs . 14 10.2 Maximum Matching Problem in Non-Bipartite Graphs . 15 10.3 Constraint Matrix Analysis for Matching Problems . 16 11 Traveling Salesperson Problem (TSP) 17 11.1 IP Formulation for TSP . 17 12 Discussion of LP-Formulation for MST 18 13 Branch-and-Bound 20 13.1 The Branch-and-Bound technique . 20 13.2 Other Branch-and-Bound techniques . 22 14 Basic graph definitions 23 15 Complexity analysis 24 15.1 Measuring quality of an algorithm . -
A Simulated Annealing Algorithm for Solving Two-Echelon Vehicle Routing Problem with Locker Facilities
algorithms Article A Simulated Annealing Algorithm for Solving Two-Echelon Vehicle Routing Problem with Locker Facilities A. A. N. Perwira Redi 1 , Parida Jewpanya 2,*, Adji Candra Kurniawan 3, Satria Fadil Persada 4, Reny Nadlifatin 5 and Oki Anita Candra Dewi 6 1 Industrial Engineering Department, BINUS Graduate Program—Master of Industrial Engineering, Bina Nusantara University, Jakarta 11480, Indonesia; [email protected] 2 Department of Industrial Engineering, Rajamangala University of Technology Lanna Tak, Tak 63000, Thailand 3 Department of Logistics Engineering, Universitas Pertamina, Jakarta 12220, Indonesia; [email protected] 4 Department of Business Management, Institut Teknologi Sepuluh Nopember, Surabaya 60111, Indonesia; [email protected] 5 Department of Technology Management, Institut Teknologi Sepuluh Nopember, Surabaya 60111, Indonesia; [email protected] 6 Logistics Engineering Department, Universitas Internasional Semen Indonesia, Gresik 61122, Indonesia; [email protected] * Correspondence: [email protected] Received: 6 August 2020; Accepted: 1 September 2020; Published: 3 September 2020 Abstract: We consider the problem of utilizing the parcel locker network for the logistics solution in the metropolitan area. Two-echelon distribution systems are attractive from an economic standpoint, whereas the product from the depot can be distributed from or to intermediate facilities. In this case, the intermediate facilities are considered as locker facilities present in an accessible location in the vicinity of the final customers. In addition, the utilization of locker facilities can reduce the cost caused by the unattended deliveries. The problem is addressed as an optimization model that formulated into an integer linear programming model denoted as the two-echelon vehicle routing problem with locker facilities (2EVRP-LF). -
Microwave Imaging for Conducting Scatterers by Hybrid Particle Swarm Optimization with Simulated Annealing
2011 8th International Multi-Conference on Systems, Signals & Devices MICROWAVE IMAGING FOR CONDUCTING SCATTERERS BY HYBRID PARTICLE SWARM OPTIMIZATION WITH SIMULATED ANNEALING Bouzid Mhamdi, Khaled Grayaa and Taoufik Aguili Syscom Laboratory, Engineer School of Tunis - BP 37 Belvedere, Tunis - 1002, Tunisia [email protected] ABSTRACT The first is based on gradient searching schemes such as the Newton-Kantorovitch method [3], modified In this paper, a microwave imaging technique for gradient method [4] and Levenberg-Marguart algorithm reconstructing the shape of two-dimensional perfectly [5]. It is well-known that these deterministic techniques conducting scatterers by means of a stochastic used for fast reconstruction of microwave images tend to optimization approach is investigated. get trapped in a local extreme and suffer from a major Based on the boundary condition and the measured drawback, where the final image is highly dependent on scattered field derived by transverse magnetic the initial trial solution. To overcome this obstacle, a illuminations, a set of nonlinear integral equations is second class of population based stochastic methods such obtained and the imaging problem is reformulated in to as the genetic algorithm (GA) [1-2], simulated annealing an optimization problem. (SA) [6] and neural network (NN) [7] has become A hybrid approximation algorithm, called PSO-SA, is attractive alternatives to reconstruct microwave images. developed in this work to solve the scattering inverse These techniques consider the imaging problem as a problem. In the hybrid algorithm, particle swarm optimization (PSO) combines global search and local global optimization problem, and reconstruct the correct search for finding the optimal results assignment with image by searching for the optimal solution through the reasonable time and simulated annealing (SA) uses use of either rivalry or cooperation strategies in the certain probability to avoid being trapped in a local problem space. -
A Sequential Hybridization of Genetic Algorithm and Particle Swarm Optimization for the Optimal Reactive Power Flow
sustainability Article A Sequential Hybridization of Genetic Algorithm and Particle Swarm Optimization for the Optimal Reactive Power Flow Imene Cherki 1,* , Abdelkader Chaker 1, Zohra Djidar 1, Naima Khalfallah 1 and Fadela Benzergua 2 1 SCAMRE Laboratory, ENPO-MA National Polytechnic School of Oran Maurice Audin, Oran 31000, Algeria 2 Departments of Electrical Engineering, University of Science and Technology of Oran Mohamed Bodiaf, Oran 31000, Algeria * Correspondence: [email protected] Received: 21 June 2019; Accepted: 12 July 2019; Published: 16 July 2019 Abstract: In this paper, the problem of the Optimal Reactive Power Flow (ORPF) in the Algerian Western Network with 102 nodes is solved by the sequential hybridization of metaheuristics methods, which consists of the combination of both the Genetic Algorithm (GA) and the Particle Swarm Optimization (PSO). The aim of this optimization appears in the minimization of the power losses while keeping the voltage, the generated power, and the transformation ratio of the transformers within their real limits. The results obtained from this method are compared to those obtained from the two methods on populations used separately. It seems that the hybridization method gives good minimizations of the power losses in comparison to those obtained from GA and PSO, individually, considered. However, the hybrid method seems to be faster than the PSO but slower than GA. Keywords: reactive power flow; metaheuristic methods; metaheuristic hybridization; genetic algorithm; particles swarms; electrical network 1. Introduction The objective of any company producing and distributing electrical energy is to ensure that the required power is available at all points and at all times.