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A vast majority of machine learning algorithms train their models and perform inference by solving optimization problems. In order to capture the learning and prediction problems accurately, structural constraints such as sparsity or low rank are frequently imposed or else the objective itself is designed to be a non-convex function. This is especially true of algorithms that operate in high-dimensional spaces or that train non-linear models such as tensor models and deep networks. The freedom to express the learning problem as a non-convex optimization problem gives immense modeling power to the algorithm designer, but often such problems are NP-hard to solve. A popular workaround to this has been to relax non-convex problems to convex ones and use traditional methods to solve the (convex) relaxed optimization problems. However this approach may be lossy and nevertheless presents significant challenges for large scale optimization. On the other hand, direct approaches to non-convex optimization have met with resounding success in several domains and remain the methods of choice for the practitioner, as they frequently outperform relaxation-based techniques - popular heuristics include projected gradient descent and alternating minimization. However, these are often poorly understood in terms of their convergence and other properties. This monograph presents a selection of recent advances that bridge a long-standing gap in our understanding of these heuristics. We hope that an insight into the inner workings of these methods will allow the reader to appreciate the unique marriage of task structure and generative models that allow these heuristic techniques to (provably) succeed. The monograph will lead the reader through several widely used nonconvex optimization techniques, as well as applications thereof. The goal of this monograph is to both, introduce the rich literature in this area, as well as equip the reader with the tools and techniques needed to analyze these simple procedures for non-convex problems.

Nonconvex programming.

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"This is the go-to book for those interested in solving nonconvex problems"--

Nonconvex programming --- Mathematical optimization

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This book deals with the theory and applications of the Reformulation- Linearization/Convexification Technique (RL T) for solving nonconvex optimization problems. A unified treatment of discrete and continuous nonconvex programming problems is presented using this approach. In essence, the bridge between these two types of nonconvexities is made via a polynomial representation of discrete constraints. For example, the binariness on a 0-1 variable x . can be equivalently J expressed as the polynomial constraint x . (1-x . ) = 0. The motivation for this book is J J the role of tight linear/convex programming representations or relaxations in solving such discrete and continuous nonconvex programming problems. The principal thrust is to commence with a model that affords a useful representation and structure, and then to further strengthen this representation through automatic reformulation and constraint generation techniques. As mentioned above, the focal point of this book is the development and application of RL T for use as an automatic reformulation procedure, and also, to generate strong valid inequalities. The RLT operates in two phases. In the Reformulation Phase, certain types of additional implied polynomial constraints, that include the aforementioned constraints in the case of binary variables, are appended to the problem. The resulting problem is subsequently linearized, except that certain convex constraints are sometimes retained in XV particular special cases, in the Linearization/Convexijication Phase. This is done via the definition of suitable new variables to replace each distinct variable-product term. The higher dimensional representation yields a linear (or convex) programming relaxation.

Nonconvex programming --- Programmation non convexe --- Programmation non convexe.

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This book presents a comprehensive description of theory, algorithms and software for solving nonconvex mixed integer nonlinear programs (MINLP). The main focus is on deterministic global optimization methods, which play a very important role in integer linear programming, and are used only recently in MINLP. The presented material consists of two parts. The first part describes basic optimization tools, such as block-separable reformulations, convex and Lagrangian relaxations, decomposition methods and global optimality criteria. Some of these results are presented here for the first time. The second part is devoted to algorithms. Starting with a short overview on existing methods, deformation, rounding, partitioning and Lagrangian heuristics, and a branch-cut-and-price algorithm are presented. The algorithms are implemented as part of an object-oriented library, called LaGO. Numerical results on several mixed integer nonlinear programs are reported to show abilities and limits of the proposed solution methods. The book contains many illustrations and an up-to-date bibliography. Because of the emphasis on practical methods, as well as the introduction into the basic theory, it is accessible to a wide audience and can be used both as a research as well as a graduate text.

Nonconvex programming. --- Nonlinear programming. --- Integer programming. --- Programming (Mathematics) --- Global optimization --- Non-convex programming --- Nonconvex programming --- Nonlinear programming --- Integer programming --- Computer science. --- Mathematics. --- Algorithms. --- Math Applications in Computer Science. --- Applications of Mathematics. --- Computational Science and Engineering. --- Programming Techniques. --- Algorism --- Algebra --- Arithmetic --- Math --- Science --- Informatics --- Foundations --- Computer science—Mathematics. --- Applied mathematics. --- Engineering mathematics. --- Computer mathematics. --- Computer programming. --- Computers --- Electronic computer programming --- Electronic data processing --- Electronic digital computers --- Programming (Electronic computers) --- Coding theory --- Computer mathematics --- Mathematics --- Engineering --- Engineering analysis --- Mathematical analysis --- Programming

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Introduction to Global Optimization Exploiting Space-Filling Curves provides an overview of classical and new results pertaining to the usage of space-filling curves in global optimization. The authors look at a family of derivative-free numerical algorithms applying space-filling curves to reduce the dimensionality of the global optimization problem; along with a number of unconventional ideas, such as adaptive strategies for estimating Lipschitz constant, balancing global and local information to accelerate the search. Convergence conditions of the described algorithms are studied in depth and theoretical considerations are illustrated through numerical examples. This work also contains a code for implementing space-filling curves that can be used for constructing new global optimization algorithms. Basic ideas from this text can be applied to a number of problems including problems with multiextremal and partially defined constraints and non-redundant parallel computations can be organized. Professors, students, researchers, engineers, and other professionals in the fields of pure mathematics, nonlinear sciences studying fractals, operations research, management science, industrial and applied mathematics, computer science, engineering, economics, and the environmental sciences will find this title useful . .

Mathematical optimization --- Nonconvex programming --- Mathematics --- Physical Sciences & Mathematics --- Geometry --- Curves --- Curves on surfaces --- Mathematical models. --- Surfaces, Curves on --- Mathematics. --- Algebraic geometry. --- Computer software. --- Numerical analysis. --- Operations research. --- Management science. --- Manifolds (Mathematics). --- Complex manifolds. --- Manifolds and Cell Complexes (incl. Diff.Topology). --- Operations Research, Management Science. --- Mathematical Software. --- Numerical Analysis. --- Algebraic Geometry. --- Calculus --- Conic sections --- Geometry, Analytic --- Geometry, Differential --- Geometry, Enumerative --- Shapes --- Cell aggregation --- Geometry, algebraic. --- Algebraic geometry --- Mathematical analysis --- Software, Computer --- Computer systems --- Aggregation, Cell --- Cell patterning --- Cell interaction --- Microbial aggregation --- Quantitative business analysis --- Management --- Problem solving --- Operations research --- Statistical decision --- Operational analysis --- Operational research --- Industrial engineering --- Management science --- Research --- System theory --- Analytic spaces --- Manifolds (Mathematics) --- Topology --- Mathematical optimization. --- Nonconvex programming.