Numerical Optimization presents a comprehensive and up-to-date description of the most effective methods in continuous optimization. It responds to the growing interest in optimization in engineering, science, and business by focusing on the methods that are best suited to practical problems. For this new edition the book has been thoroughly updated throughout. There are new chapters on nonlinear interior methods and derivative-free methods for optimization, both of which are used widely in practice and the focus of much current research. Because of the emphasis on practical methods, as well as the extensive illustrations and exercises, the book is accessible to a wide audience. It can be used as a graduate text in engineering, operations research, mathematics, computer science, and business. It also serves as a handbook for researchers and practitioners in the field. The authors have strived to produce a text that is pleasant to read, informative, and rigorous - one that reveals both the beautiful nature of the discipline and its practical side.

Numerical Optimization

We consider the class of iterative shrinkage-thresholding algorithms (ISTA) for solving linear inverse problems arising in signal/image processing. This class of methods, which can be viewed as an extension of the classical gradient algorithm, is attractive due to its simplicity and thus is adequate for solving large-scale problems even with dense matrix data. However, such methods are also known to converge quite slowly. In this paper we present a new fast iterative shrinkage-thresholding algorithm (FISTA) which preserves the computational simplicity of ISTA but with a global rate of convergence which is proven to be significantly better, both theoretically and practically. Initial promising numerical results for wavelet-based image deblurring demonstrate the capabilities of FISTA which is shown to be faster than ISTA by several orders of magnitude.

http://people.rennes.inria.fr/Cedric.Herzet/Cedric.Herzet/Sparse_Seminar/Entrees/2012/11/12_A_Fast_Iterative_Shrinkage-Thresholding_Algorithmfor_Linear_Inverse_Problems_(A._Beck,_M._Teboulle)_files/Breck_2009.pdf

A Fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems

List of Figures. List of Tables. Preface. Foreword. 1. Basic Concepts. 2. Evolutionary Algorithm MOP Approaches. 3. MOEA Test Suites. 4. MOEA Testing and Analysis. 5. MOEA Theory and Issues. 3. MOEA Theoretical Issues. 6. Applications. 7. MOEA Parallelization. 8. Multi-Criteria Decision Making. 9. Special Topics. 10. Epilog. Appendix A: MOEA Classification and Technique Analysis. Appendix B: MOPs in the Literature. Appendix C: Ptrue & PFtrue for Selected Numeric MOPs. Appendix D: Ptrue & PFtrue for Side-Constrained MOPs. Appendix E: MOEA Software Availability. Appendix F: MOEA-Related Information. Index. References.

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Evolutionary algorithms for solving multi-objective problems

/pdf/convex-analysisnoer-san-nojin-zhan-nituite-5dl2rbmoyr.pdf

Convex Analysisの二,三の進展について

In semidefinite programming, one minimizes a linear function subject to the constraint that an affine combination of symmetric matrices is positive semidefinite. Such a constraint is nonlinear and nonsmooth, but convex, so semidefinite programs are convex optimization problems. Semidefinite programming unifies several standard problems (e.g., linear and quadratic programming) and finds many applications in engineering and combinatorial optimization. Although semidefinite programs are much more general than linear programs, they are not much harder to solve. Most interior-point methods for linear programming have been generalized to semidefinite programs. As in linear programming, these methods have polynomial worst-case complexity and perform very well in practice. This paper gives a survey of the theory and applications of semidefinite programs and an introduction to primaldual interior-point methods for their solution.

Semidefinite programming

We present and motivate a new model of the truss topology design problem, where the rigidity of the resulting truss with respect both to given loading scenarios and small "occasional" loads is optimized It is shown that the resulting optimization problem is a semidefinite program We derive and analyze several equivalent reformulations of the problem and present illustrative numerical examples

https://www2.isye.gatech.edu/~nemirovs/SIOPT_RTTD_1997.pdf

Robust Truss Topology Design via Semidefinite Programming

Samenvatting	In this paper we focus on robust linear optimization problems with uncertainty regions defined by o-divergences (for example, chi-squared, Hellinger, Kullback-Leibler). We show how uncertainty regions based on o-divergences arise in a natural way as confidence sets if the uncertain parameters contain elements of a probability vector. Such problems frequently occur in, for example, optimization problems in inventory control or finance that involve terms containing moments of random variables, expected utility, etc. We show that the robust counterpart of a linear optimization problem with o-divergence uncertainty is tractable for most of the choices of o typically considered in the literature. We extend the results to problems that are nonlinear in the optimization variables. Several applications, including an asset pricing example and a numerical multi-item newsvendor example, illustrate the relevance of the proposed approach.

Robust Solutions of Optimization Problems Affected by Uncertain Probabilities

We demonstrate that a conic quadratic problem, is "polynomially reducible" to Linear Programming. We demonstrate this by constructing, for every , an LP program (explicitly given in terms of e and the data of (CQP)) with the following properties:(i)the number dimx + dimu of variables and the number dimp of constraints in (LP) do not exceed (ii)every feasible solutionx to (CQP) can be extended to a feasible solution ( x, u) to (LP); (iii)if ( x, u) is feasible for (LP), thenx satisfies the "e-relaxed" constraints of (CQP), namely,

On Polyhedral Approximations of the Second-Order Cone

Robust Optimization is a rapidly developing methodology for handling optimization problems affected by non-stochastic “uncertain-but- bounded” data perturbations. In this paper, we overview several selected topics in this popular area, specifically, (1) recent extensions of the basic concept of robust counterpart of an optimization problem with uncertain data, (2) tractability of robust counterparts, (3) links between RO and traditional chance constrained settings of problems with stochastic data, and (4) a novel generic application of the RO methodology in Robust Linear Control.

https://www2.isye.gatech.edu/~nemirovs/ismp2006.pdf

Selected topics in robust convex optimization

We propose the use of robust optimization (RO) as a powerful methodology for multiperiod stochastic operations management problems. In particular, we study a two-echelon multiperiod supply chain problem, known as the retailer-supplier flexible commitment (RSFC) problem with uncertain demand that is only known to reside in some uncertainty set. We adopt a min-max criterion, whereby the cost function is minimized against the worst case demand occurrence. To solve the min-max RSFC problem we employ a recent extension of the RO method adapted to dynamic decision problems and known as the affinely adjustable robust counterpart (AARC) methodology. The AARC solution is tested by a large simulation study and found to provide excellent results.

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Aharon Ben-Tal

Papers

Robust Truss Topology Design via Semidefinite Programming

Robust Solutions of Optimization Problems Affected by Uncertain Probabilities

On Polyhedral Approximations of the Second-Order Cone

Selected topics in robust convex optimization

Retailer-Supplier Flexible Commitments Contracts: A Robust Optimization Approach