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.

/pdf/evolutionary-algorithms-for-solving-multi-objective-problems-2txxk4mqci.pdf

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

This is a book devoted to well-structured and thus efficiently solvable convex optimization problems, with emphasis on conic quadratic and semidefinite programming. The authors present the basic theory underlying these problems as well as their numerous applications in engineering, including synthesis of filters, Lyapunov stability analysis, and structural design. The authors also discuss the complexity issues and provide an overview of the basic theory of state-of-the-art polynomial time interior point methods for linear, conic quadratic, and semidefinite programming. The book's focus on well-structured convex problems in conic form allows for unified theoretical and algorithmical treatment of a wide spectrum of important optimization problems arising in applications.

Lectures on modern convex optimization: analysis, algorithms, and engineering applications

We study convex optimization problems for which the data is not specified exactly and it is only known to belong to a given uncertainty set U, yet the constraints must hold for all possible values of the data from U. The ensuing optimization problem is called robust optimization. In this paper we lay the foundation of robust convex optimization. In the main part of the paper we show that if U is an ellipsoidal uncertainty set, then for some of the most important generic convex optimization problems (linear programming, quadratically constrained programming, semidefinite programming and others) the corresponding robust convex program is either exactly, or approximately, a tractable problem which lends itself to efficientalgorithms such as polynomial time interior point methods.

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

Robust Convex Optimization

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

Robust solutions of uncertain linear programs

Optimal solutions of Linear Programming problems may become severely infeasible if the nominal data is slightly perturbed. We demonstrate this phenomenon by studying 90 LPs from the well-known NETLIB collection. We then apply the Robust Optimization methodology (Ben-Tal and Nemirovski [1–3]; El Ghaoui et al. [5, 6]) to produce “robust” solutions of the above LPs which are in a sense immuned against uncertainty. Surprisingly, for the NETLIB problems these robust solutions nearly lose nothing in optimality.

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

Robust solutions of Linear Programming problems contaminated with uncertain data

We consider linear programs with uncertain parameters, lying in some prescribed uncertainty set, where part of the variables must be determined before the realization of the uncertain parameters (‘‘non-adjustable variables’’), while the other part are variables that can be chosen after the realization (‘‘adjustable variables’’). We extend the Robust Optimization methodology ([1, 3-6, 9, 13, 14]) to this situation by introducing the Adjustable Robust Counterpart (ARC) associated with an LP of the above structure. Often the ARC is significantly less conservative than the usual Robust Counterpart (RC), however, in most cases the ARC is computationally intractable (NP-hard). This difficulty is addressed by restricting the adjustable variables to be affine functions of the uncertain data. The ensuing Affinely Adjustable Robust Counterpart (AARC) problem is then shown to be, in certain important cases, equivalent to a tractable optimization problem (typically an LP or a Semidefinite problem), and in other cases, having a tight approximation which is tractable. The AARC approach is illustrated by applying it to a multi-stage inventory management problem.

https://www2.isye.gatech.edu/~nemirovs/MP_Elana_2004.pdf

Aharon Ben-Tal

Papers

Lectures on modern convex optimization: analysis, algorithms, and engineering applications

Robust Convex Optimization

Robust solutions of uncertain linear programs

Robust solutions of Linear Programming problems contaminated with uncertain data

Adjustable robust solutions of uncertain linear programs