{"title":"Alexander M Rubinov","description":null,"products":[{"product_id":"lagrange-type-functions-in-constrained-non-convex-optimization-book-alexander-m-rubinov-9781402076275","title":"Lagrange-type Functions in Constrained Non-Convex Optimization","description":"Lagrange and penalty function methods provide a powerful approach, both as a theoretical tool and a computational vehicle, for the study of constrained optimization problems. However, for a nonconvex constrained optimization problem, the classical Lagrange primal-dual method may fail to find a mini- mum as a zero duality gap is not always guaranteed. A large penalty parameter is, in general, required for classical quadratic penalty functions in order that minima of penalty problems are a good approximation to those of the original constrained optimization problems. It is well-known that penaity functions with too large parameters cause an obstacle for numerical implementation. Thus the question arises how to generalize classical Lagrange and penalty functions, in order to obtain an appropriate scheme for reducing constrained optimiza- tion problems to unconstrained ones that will be suitable for sufficiently broad classes of optimization problems from both the theoretical and computational viewpoints. Some approaches for such a scheme are studied in this book. One of them is as follows: an unconstrained problem is constructed, where the objective function is a convolution of the objective and constraint functions of the original problem. While a linear convolution leads to a classical Lagrange function, different kinds of nonlinear convolutions lead to interesting generalizations. We shall call functions that appear as a convolution of the objective function and the constraint functions, Lagrange-type functions.","brand":"WoB","offers":[{"title":"GB \/ NEW \/ INGRAM","offer_id":52129701626129,"sku":"NLS9781402076275","price":0.0,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0784\/4072\/6801\/files\/9781402076275.jpg?v=1786280342"},{"product_id":"abstract-convexity-and-global-optimization-book-alexander-m-rubinov-9780792363231","title":"Abstract Convexity and Global Optimization","description":"Special tools are required for examining and solving optimization problems. The main tools in the study of local optimization are classical calculus and its modern generalizions which form nonsmooth analysis. The gradient and various kinds of generalized derivatives allow us to ac- complish a local approximation of a given function in a neighbourhood of a given point. This kind of approximation is very useful in the study of local extrema. However, local approximation alone cannot help to solve many problems of global optimization, so there is a clear need to develop special global tools for solving these problems. The simplest and most well-known area of global and simultaneously local optimization is convex programming. The fundamental tool in the study of convex optimization problems is the subgradient, which actu- ally plays both a local and global role. First, a subgradient of a convex function f at a point x carries out a local approximation of f in a neigh- bourhood of x. Second, the subgradient permits the construction of an affine function, which does not exceed f over the entire space and coincides with f at x. This affine function h is called a support func- tion. Since f(y) h(y) for ally, the second role is global. In contrast to a local approximation, the function h will be called a global affine support.","brand":"WoB","offers":[{"title":"- \/ - \/ INTERNAL","offer_id":52414598971665,"sku":null,"price":0.0,"currency_code":"GBP","in_stock":true},{"title":"GB \/ NEW \/ INGRAM","offer_id":52414599758097,"sku":"NLS9780792363231","price":0.0,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0784\/4072\/6801\/files\/9780792363231.jpg?v=1786614662"},{"product_id":"optimization-and-related-topics-book-alexander-m-rubinov-9780792367321","title":"Optimization and Related Topics","description":"This text, comprised predominantly of survey chapters, is a collection of recent results in various fields of theoretical and applied optimization and related topics. It contains survey papers on second order nonsmooth analysis, based on subjects, multiplicative programs and c-programming, optimal algorithms in emergent computation, the extremal principle and its applications, turnpike property for variational problems, asymptotic behavior of random infinite products of some operators, inequalities for Riemann-Stieltjes integral. Other topics covered include nonsmooth analysis and analysis of linear operators and set-valued mappings, numerical methods and generalized penalty functions, applied optimal control problems and Markov decision processes, optimal estimation of signal parameters and the problem of maximal time congestion. Audience: Specialists in optimization, mathematical programming, convex analysis, nonsmoooth analysis, engineers using mathematical tools and optimization technique, specialists in mathematical modelling.","brand":"WoB","offers":[{"title":"- \/ - \/ INTERNAL","offer_id":52425718366481,"sku":null,"price":0.0,"currency_code":"GBP","in_stock":true},{"title":"GB \/ NEW \/ INGRAM","offer_id":52425719054609,"sku":"NLS9780792367321","price":0.0,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0784\/4072\/6801\/files\/9780792367321.jpg?v=1759155623"},{"product_id":"lagrange-type-functions-in-constrained-non-convex-optimization-book-alexander-m-rubinov-9781461348214","title":"Lagrange-type Functions in Constrained Non-Convex Optimization","description":"Lagrange and penalty function methods provide a powerful approach, both as a theoretical tool and a computational vehicle, for the study of constrained optimization problems. However, for a nonconvex constrained optimization problem, the classical Lagrange primal-dual method may fail to find a mini- mum as a zero duality gap is not always guaranteed. A large penalty parameter is, in general, required for classical quadratic penalty functions in order that minima of penalty problems are a good approximation to those of the original constrained optimization problems. It is well-known that penaity functions with too large parameters cause an obstacle for numerical implementation. Thus the question arises how to generalize classical Lagrange and penalty functions, in order to obtain an appropriate scheme for reducing constrained optimiza- tion problems to unconstrained ones that will be suitable for sufficiently broad classes of optimization problems from both the theoretical and computational viewpoints. Some approaches for such a scheme are studied in this book. One of them is as follows: an unconstrained problem is constructed, where the objective function is a convolution of the objective and constraint functions of the original problem. While a linear convolution leads to a classical Lagrange function, different kinds of nonlinear convolutions lead to interesting generalizations. We shall call functions that appear as a convolution of the objective function and the constraint functions, Lagrange-type functions.","brand":"WoB","offers":[{"title":"- \/ - \/ INTERNAL","offer_id":52453318983953,"sku":null,"price":0.0,"currency_code":"GBP","in_stock":true},{"title":"GB \/ NEW \/ INGRAM","offer_id":52453319573777,"sku":"NLS9781461348214","price":0.0,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0784\/4072\/6801\/files\/9781461348214.jpg?v=1786442222"},{"product_id":"optimization-and-related-topics-book-alexander-m-rubinov-9781441948441","title":"Optimization and Related Topics","description":"This volume contains, in part, a selection of papers presented at the sixth Australian Optimization Day Miniconference (Ballarat, 16 July 1999), and the Special Sessions on Nonlinear Dynamics and Optimization and Operations Re- search - Methods and Applications, which were held in Melbourne, July 11-15 1999 as a part of the Joint Meeting of the American Mathematical Society and Australian Mathematical Society. The editors have strived to present both con- tributed papers and survey style papers as a more interesting mix for readers. Some participants from the meetings mentioned above have responded to this approach by preparing survey and 'semi-survey' papers, based on presented lectures. Contributed paper, which contain new and interesting results, are also included. The fields of the presented papers are very large as demonstrated by the following selection of key words from selected papers in this volume: - optimal control, stochastic optimal control, MATLAB, economic models, implicit constraints, Bellman principle, Markov process, decision-making under uncertainty, risk aversion, dynamic programming, optimal value function. - emergent computation, complexity, traveling salesman problem, signal estimation, neural networks, time congestion, teletraffic. - gap functions, nonsmooth variational inequalities, derivative-free algo- rithm, Newton's method. - auxiliary function, generalized penalty function, modified Lagrange func- tion. - convexity, quasiconvexity, abstract convexity.","brand":"WoB","offers":[{"title":"- \/ - \/ INTERNAL","offer_id":52476076589329,"sku":null,"price":0.0,"currency_code":"GBP","in_stock":true},{"title":"GB \/ NEW \/ INGRAM","offer_id":52476077965585,"sku":"NLS9781441948441","price":0.0,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0784\/4072\/6801\/files\/9781441948441.jpg?v=1789043657"},{"product_id":"abstract-convexity-and-global-optimization-book-alexander-m-rubinov-9781441948311","title":"Abstract Convexity and Global Optimization","description":"Special tools are required for examining and solving optimization problems. The main tools in the study of local optimization are classical calculus and its modern generalizions which form nonsmooth analysis. The gradient and various kinds of generalized derivatives allow us to ac- complish a local approximation of a given function in a neighbourhood of a given point. This kind of approximation is very useful in the study of local extrema. However, local approximation alone cannot help to solve many problems of global optimization, so there is a clear need to develop special global tools for solving these problems. The simplest and most well-known area of global and simultaneously local optimization is convex programming. The fundamental tool in the study of convex optimization problems is the subgradient, which actu- ally plays both a local and global role. First, a subgradient of a convex function f at a point x carries out a local approximation of f in a neigh- bourhood of x. Second, the subgradient permits the construction of an affine function, which does not exceed f over the entire space and coincides with f at x. This affine function h is called a support func- tion. Since f(y) h(y) for ally, the second role is global. In contrast to a local approximation, the function h will be called a global affine support.","brand":"WoB","offers":[{"title":"GB \/ NEW \/ INGRAM","offer_id":52481078952209,"sku":"NLS9781441948311","price":0.0,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0784\/4072\/6801\/files\/9781441948311.jpg?v=1786442603"}],"url":"https:\/\/www.worldofbooks.com\/en-gb\/collections\/author-books-by-alexander-m-rubinov.oembed","provider":"World of Books ","version":"1.0","type":"link"}