Abstract: Minimization with orthogonality constraints (e.g., $$X^\top X = I$$) and/or spherical constraints (e.g., $$\Vert x\Vert _2 = 1$$) has wide applications in polynomial optimization, combinatorial optimization, eigenvalue problems, sparse PCA, p-harmonic flows, 1-bit compressive sensing, matrix rank minimization, etc. These problems are difficult because the constraints are not only non-convex but numerically expensive to preserve during iterations. To deal with these difficulties, we apply the Cayley transform--a Crank-Nicolson-like update scheme--to preserve the constraints and based on it, develop curvilinear search algorithms with lower flops compared to those based on projections and geodesics. The efficiency of the proposed algorithms is demonstrated on a variety of test problems. In particular, for the maxcut problem, it exactly solves a decomposition formulation for the SDP relaxation. For polynomial optimization, nearest correlation matrix estimation and extreme eigenvalue problems, the proposed algorithms run very fast and return solutions no worse than those from their state-of-the-art algorithms. For the quadratic assignment problem, a gap 0.842 % to the best known solution on the largest problem "tai256c" in QAPLIB can be reached in 5 min on a typical laptop.

Abstract: The roots of Multiple Criteria Decision Making and Multiple Criteria Optimization were laid by Pareto at the end of the 19th century, and since then the discipline has prospered and grown, especially during the last three decades. Today, many decision support systems incorporate methods to deal with conflicting objectives. The foundation for such systems is a mathematical theory of optimization under multiple objectives. Since its beginnings, there have been a vast number of books, journal issues, papers and conferences that have brought the field to its present state. Despite this vast body of literature, there is no reliable guide to provide an access to this knowledge. Over the years, many literature surveys and bibliographies have been published. With the ever rapidly increasing rate of publications in the area and the development of subfields, these were mostly devoted to particular aspects of multicriteria optimization: Multiobjective Integer Programming, Multi-objective Combinatorial Optimization, Vector Optimization, Multiobjective Evolutionary Methods, Applications of MCDM, MCDM Software, Goal Programming. Hence the need for a comprehensive overview of the literature in multicriteria optimization that could serve as a state of the art survey and guide to the vast amount of publications. Multiple Criteria Optimization: State of the Art Annotated Bibliographic Surveys is precisely this book. Experts in various areas of multicriteria optimization have contributed to the volume. The chapters in this book roughly follow a thread from most general to more specific. Some of them are about particular types of problems (Theory of Vector Optimization, Nonlinear Multiobjective Programming, Fuzzy Multiobjective Programming, Multiobjective Combinatorial Optimization, Multicriteria Scheduling Problems), while the others are focused on multi-objective methodologies (Goal Programming, Interactive Methods, Evolutionary Algorithms, Data Envelopment Analysis). All contributing authors invested great effort to produce comprehensive overviews and bibliographies and to have references that are as precise as possible.

Abstract: Submodular functions are relevant to machine learning for at least two reasons: (1) some problems may be expressed directly as the optimization of submodular functions and (2) the Lovsz extension of submodular functions provides a useful set of regularization functions for supervised and unsupervised learning. In Learning with Submodular Functions: A Convex Optimization Perspective, the theory of submodular functions is presented in a self-contained way from a convex analysis perspective, presenting tight links between certain polyhedra, combinatorial optimization and convex optimization problems. In particular, it describes how submodular function minimization is equivalent to solving a wide variety of convex optimization problems. This allows the derivation of new efficient algorithms for approximate and exact submodular function minimization with theoretical guarantees and good practical performance. By listing many examples of submodular functions, it reviews various applications to machine learning, such as clustering, experimental design, sensor placement, graphical model structure learning or subset selection, as well as a family of structured sparsity-inducing norms that can be derived and used from submodular functions. Learning with Submodular Functions: A Convex Optimization Perspective is an ideal reference for researchers, scientists, or engineers with an interest in applying submodular functions to machine learning problems.

Abstract: Exact optimization algorithms are not able to provide an appropriate solution in solving optimization problems with a high-dimensional search space. In these problems, the search space grows exponentially with the problem size therefore; exhaustive search is not practical. Also, classical approximate optimization methods like greedy-based algorithms make several assumptions to solve the problems. Sometimes, the validation of these assumptions is difficult in each problem. Hence, meta-heuristic algorithms which make few or no assumptions about a problem and can search very large spaces of candidate solutions have been extensively developed to solve optimization problems these days. Among these algorithms, population-based meta-heuristic algorithms are proper for global searches due to global exploration and local exploitation ability. In this paper, a survey on meta-heuristic algorithms is performed and several population-based meta-heuristics in continuous (real) and discrete (binary) search spaces are explained in details. This covers design, main algorithm, advantages and disadvantages of the algorithms.

Abstract: Acknowledgments xxi Acronyms xxiii List of Algorithms xxvii Pert I: Introduction to Evolutionary Optimization 1 Introduction 1 2 Optimization 11 Part II: Classic Evoluntionary Algorithms 3 Generic Algorithms 35 4 Mathematical Models of Genetic Algorithms 63 5 Evolutionary Programming 95 6 Evolution Strategies 117 7 Genetic Programming 141 8 Evolutionary Algorithms Variations 179 Part III: More Recent Evolutionary Algorithms 9 Simulated Annealing 223 10 Ant Colony Optimization 241 11 Particle Swarm Optimization 265 12 Differential Evolution 293 13 Estimation of Distribution Algorithms 313 14 Biogeography-Based Optimization 351 15 Cultural Algorithms 377 16 Oppostion-Based Learning 397 17 Other Evolutionary Algorithms 421 Part IV: Special Type of Optimization Problems 18 Combinatorial Optimization 449 19 Constrained Optimization 481 20 Multi-Objective Optimization 517 21 Expensive, Noisy and Dynamic Fitness Functions 563 Appendices A Some Practical Advice 607 B The No Free Luch Therorem and Performance Testing 613 C Benchmark Optimization Functions 641

Abstract: SOSTOOLS v3.00 is the latest release of the freely available MATLAB toolbox for formulating and solving sum of squares (SOS) optimization problems. Such problems arise naturally in the analysis and control of nonlinear dynamical systems, but also in other areas such as combinatorial optimization. Highlights of the new release include the ability to create polynomial matrices and formulate polynomial matrix inequalities, compatibility with MuPAD, the new MATLAB symbolic engine, as well as the multipoly toolbox v2.01. SOSTOOLS v3.00 can interface with five semidefinite programming solvers, and includes ten demonstration examples.

Abstract: This book presents recent advances in the mathematical theory of discrete optimization, particularly those supported by methods from algebraic geometry, commutative algebra, convex and discrete geometry, generating functions, and other tools normally considered outside the standard curriculum in optimization. Algebraic and Geometric Ideas in the Theory of Discrete Optimization offers several research technologies not yet well known among practitioners of discrete optimization, minimizes prerequisites for learning these methods, and provides a transition from linear discrete optimization to nonlinear discrete optimization. Audience: This book can be used as a textbook for advanced undergraduates or beginning graduate students in mathematics, computer science, or operations research or as a tutorial for mathematicians, engineers, and scientists engaged in computation who wish to delve more deeply into how and why algorithms do or do not work. Contents: Part I: Established Tools of Discrete Optimization; Chapter 1: Tools from Linear and Convex Optimization; Chapter 2: Tools from the Geometry of Numbers and Integer Optimization; Part II: Graver Basis Methods; Chapter 3: Graver Bases; Chapter 4: Graver Bases for Block-Structured Integer Programs; Part III: Generating Function Methods; Chapter 5: Introduction to Generating Functions; Chapter 6: Decompositions of Indicator Functions of Polyhedral; Chapter 7: Barvinok s Short Rational Generating Functions; Chapter 8: Global Mixed-Integer Polynomial Optimization via Summation; Chapter 9: Multicriteria Integer Linear Optimization via Integer Projection; Part IV: Grbner Basis Methods; Chapter 10: Computations with Polynomials; Chapter 11: Grbner Bases in Integer Programming; Part V: Nullstellensatz and Positivstellensatz Relaxations; Chapter 12: The Nullstellensatz in Discrete Optimization; Chapter 13: Positivity of Polynomials and Global Optimization; Chapter 14: Epilogue.

Abstract: One of the design objectives in distributed storage system is the minimization of the data traffic during the repair of failed storage nodes. By repairing multiple failures simultaneously and cooperatively rather than successively and independently, further reduction of repair traffic is made possible. A closed-form expression of the optimal tradeoff between the repair traffic and the amount of storage in each node for cooperative repair is given. We show that the points on the tradeoff curve can be achieved by linear cooperative regenerating codes, with an explicit bound on the required finite-field size. The proof relies on a max-flow-min-cut-type theorem from combinatorial optimization for submodular flows. Two families of explicit constructions are given.

Abstract: In this paper, we consider a sequence-dependent disassembly line balancing problem (SDDLBP) with multiple objectives that requires the assignment of disassembly tasks to a set of ordered disassembly workstations while satisfying the disassembly precedence constraints and optimizing the effectiveness of several measures. Since the complexity of SDDLBP increases with the number of parts of the product, an efficient methodology based on artificial bee colony (ABC) is proposed to solve the SDDLBP. ABC is an optimization technique which is inspired by the behavior of honey bees. The performance of the proposed algorithm was tested against six other algorithms. The results show that the proposed ABC algorithm performs well and is superior to the other six algorithms in terms of the objective values performance.

Abstract: This paper reviews the applications of Firefly Algorithm (FA) in various domain of optimization problem. Optimization is a process of determining the best solution to make something as functional and effective as possible by minimizing or maximizing the parameters involved in the problems. Several categories of optimization problem such as discrete, chaotic, multi-objective and many more are addressed by inspiring the behavior of fireflies as mentioned in the literatures. Literatures found that FA was mostly applied by researchers to solve the optimization problems in Computer Science and Engineering domain. Some of them are enhanced or hybridized with other techniques to discover better performance. In addition, literatures found that most of the cases that used FA technique have outperformed compare to other metaheuristic algorithms.

Abstract: Most swarm intelligence algorithms were devised for continuous optimization problems. However, they have been adapted for discrete optimization as well with applications in different domains. This survey aims at providing an updated review of research of swarm intelligence algorithms for discrete optimization problems, comprising combinatorial or binary. The biological inspiration that motivated the creation of each swarm algorithm is introduced, and later, the discretization and encoding methods are used to adapt each algorithm for discrete problems. Methods are compared for different classes of problems and a critical analysis is provided, pointing to future trends.

Abstract: The single-row facility layout problem (SRFLP) is an NP-hard combinatorial optimization problem that is concerned with the arrangement of n departments of given lengths on a line so as to minimize the weighted sum of the distances between department pairs. (SRFLP) is the one-dimensional version of the facility layout problem that seeks to arrange rectangular departments so as to minimize the overall interaction cost. This paper compares the different modelling approaches for (SRFLP) and applies a recent SDP approach for general quadratic ordering problems from Hungerlander and Rendl to (SRFLP). In particular, we report optimal solutions for several (SRFLP) instances from the literature with up to 42 departments that remained unsolved so far. Secondly we significantly reduce the best known gaps and running times for large instances with up to 110 departments.

Abstract: Purpose – The purpose of this paper is to introduce sequence‐dependent disassembly line balancing problem (SDDLBP) to the literature and propose an efficient metaheuristic solution methodology to this NP‐complete problem.Design/methodology/approach – This manuscript utilizes a well‐proven metaheuristics solution methodology, namely, ant colony optimization, to address the problem.Findings – Since SDDLBP is NP‐complete, finding an optimal balance becomes computationally prohibitive due to exponential growth of the solution space with the increase in the number of parts. The proposed methodology is very fast, generates (near) optimal solutions, preserves precedence requirements and is easy to implement.Practical implications – Since development of cost effective and profitable disassembly systems is an important issue in end‐of‐life product treatment, every step towards improving disassembly line balancing brings us closer to cost savings and compelling practicality.Originality/value – This paper introduces...

Abstract: The Distance Geometry Problem in three dimensions consists in finding an embedding in $${\mathbb{R}^3}$$ of a given nonnegatively weighted simple undirected graph such that edge weights are equal to the corresponding Euclidean distances in the embedding. This is a continuous search problem that can be discretized under some assumptions on the minimum degree of the vertices. In this paper we discuss the case where we consider the full-atom representation of the protein backbone and some of the edge weights are subject to uncertainty within a given nonnegative interval. We show that a discretization is still possible and propose the iBP algorithm to solve the problem. The approach is validated by some computational experiments on a set of artificially generated instances.

Abstract: We develop exact algorithms for multi-objective integer programming (MIP) problems. The algorithms iteratively generate nondominated points and exclude the regions that are dominated by the previously-generated nondominated points. One algorithm generates new points by solving models with additional binary variables and constraints. The other algorithm employs a search procedure and solves a number of models to find the next point avoiding any additional binary variables. Both algorithms guarantee to find all nondominated points for any MIP problem. We test the performance of the algorithms on randomly-generated instances of the multi-objective knapsack, multi-objective shortest path and multi-objective spanning tree problems. The computational results show that the algorithms work well.

Abstract: This paper shows how the Bayesian network paradigm can be used in order to solve combinatorial optimization problems. To do it some methods of structure learning from data and simulation of Bayesian networks are inserted inside Estimation of Distribution Algorithms (EDA). EDA are a new tool for evolutionary computation in which populations of individuals are created by estimation and simulation of the joint probability distribution of the selected individuals. We propose new approaches to EDA for combinatorial optimization based on the theory of probabilistic graphical models. Experimental results are also presented.

Abstract: This paper investigates the cost requirement for a space debris collecting mission aimed at removing heavy debris from low Earth orbits. The problem mixes combinatorial optimization to select the debris among a list of candidates and functional optimization to define the orbital manoeuvres. The solving methodology proceeds in two steps: Firstly, a specific transfer strategy with impulsive manoeuvres is defined so that the problem becomes of finite dimension; secondly the problem is linearized around an initial reference solution. A Branch and Bound algorithm is then applied iteratively to optimize simultaneously the debris selection and the orbital manoeuvres, yielding a new reference solution. The optimal solutions found are close to the initial guess despite a very complicated design space. The method is exemplified on a representative application case.