An Unconstrained Convex Programming Dual Approach to a Class of Linearly-constrained Entropy Maximization Problem with a Quadric Cost and Its Applications to Transportation Planning Problems PDF Download

Author: S. C. Fang
Publisher:
ISBN:
Category : Convex programming
Languages : en
Pages : 46

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Author:
Publisher:
ISBN:
Category : Electronic journals
Languages : en
Pages : 428

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Author: University of California, Berkeley. Institute of Transportation Studies
Publisher:
ISBN:
Category : Transportation
Languages : en
Pages : 28

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Author: Patrick Brockett
Publisher:
ISBN:
Category : Decision making
Languages : en
Pages : 11

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This document formulates the generalized constrained maximum entropy problem often used in a decision making context as an extended dual convex programming problem. The dual problem is then presented. In this dual setting the primal Lagrange multipliers are precisely the dual variables, and are easily calculated directly by virtue of the simple structure of the dual problem. An example involving the selection of best equipment for an oil spill is presented as an illustration. The authors contrast their solution with those given by previous authors. (Author).

Author: Mung Chiang
Publisher: Now Publishers Inc
ISBN: 9781933019093
Category : Computers
Languages : en
Pages : 172

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Recently Geometric Programming has been applied to study a variety of problems in the analysis and design of communication systems from information theory and queuing theory to signal processing and network protocols. Geometric Programming for Communication Systems begins its comprehensive treatment of the subject by providing an in-depth tutorial on the theory, algorithms, and modeling methods of Geometric Programming. It then gives a systematic survey of the applications of Geometric Programming to the study of communication systems. It collects in one place various published results in this area, which are currently scattered in several books and many research papers, as well as to date unpublished results. Geometric Programming for Communication Systems is intended for researchers and students who wish to have a comprehensive starting point for understanding the theory and applications of geometric programming in communication systems.

Author:
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ISBN:
Category : Mathematics
Languages : en
Pages : 958

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Author: Dimitri Bertsekas
Publisher: Athena Scientific
ISBN: 1886529450
Category : Mathematics
Languages : en
Pages : 560

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A uniquely pedagogical, insightful, and rigorous treatment of the analytical/geometrical foundations of optimization. The book provides a comprehensive development of convexity theory, and its rich applications in optimization, including duality, minimax/saddle point theory, Lagrange multipliers, and Lagrangian relaxation/nondifferentiable optimization. It is an excellent supplement to several of our books: Convex Optimization Theory (Athena Scientific, 2009), Convex Optimization Algorithms (Athena Scientific, 2015), Nonlinear Programming (Athena Scientific, 2016), Network Optimization (Athena Scientific, 1998), and Introduction to Linear Optimization (Athena Scientific, 1997). Aside from a thorough account of convex analysis and optimization, the book aims to restructure the theory of the subject, by introducing several novel unifying lines of analysis, including: 1) A unified development of minimax theory and constrained optimization duality as special cases of duality between two simple geometrical problems. 2) A unified development of conditions for existence of solutions of convex optimization problems, conditions for the minimax equality to hold, and conditions for the absence of a duality gap in constrained optimization. 3) A unification of the major constraint qualifications allowing the use of Lagrange multipliers for nonconvex constrained optimization, using the notion of constraint pseudonormality and an enhanced form of the Fritz John necessary optimality conditions. Among its features the book: a) Develops rigorously and comprehensively the theory of convex sets and functions, in the classical tradition of Fenchel and Rockafellar b) Provides a geometric, highly visual treatment of convex and nonconvex optimization problems, including existence of solutions, optimality conditions, Lagrange multipliers, and duality c) Includes an insightful and comprehensive presentation of minimax theory and zero sum games, and its connection with duality d) Describes dual optimization, the associated computational methods, including the novel incremental subgradient methods, and applications in linear, quadratic, and integer programming e) Contains many examples, illustrations, and exercises with complete solutions (about 200 pages) posted at the publisher's web site http://www.athenasc.com/convexity.html

Author: Aharon Ben-Tal
Publisher: SIAM
ISBN: 9780898718829
Category : Technology & Engineering
Languages : en
Pages : 504

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Here 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.

Author: Patrick Brockett
Publisher:
ISBN:
Category :
Languages : en
Pages : 11

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Author: Mehdi Karimi
Publisher:
ISBN:
Category : Convex functions
Languages : en
Pages : 139

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This thesis studies the theory and implementation of infeasible-start primal-dual interior-point methods for convex optimization problems. Convex optimization has applications in many fields of engineering and science such as data analysis, control theory, signal processing, relaxation and randomization, and robust optimization. In addition to strong and elegant theories, the potential for creating efficient and robust software has made convex optimization very popular. Primal-dual algorithms have yielded efficient solvers for convex optimization problems in conic form over symmetric cones (linear-programming (LP), second-order cone programming (SOCP), and semidefinite programing (SDP)). However, many other highly demanded convex optimization problems lack comparable solvers. To close this gap, we have introduced a general optimization setup, called \emph{Domain-Driven}, that covers many interesting classes of optimization. Domain-Driven means our techniques are directly applied to the given ``good" formulation without a forced reformulation in a conic form. Moreover, this approach also naturally handles the cone constraints and hence the conic form. A problem is in the Domain-Driven setup if it can be formulated as minimizing a linear function over a convex set, where the convex set is equipped with an efficient self-concordant barrier with an easy-to-evaluate Legendre-Fenchel conjugate. We show how general this setup is by providing several interesting classes of examples. LP, SOCP, and SDP are covered by the Domain-Driven setup. More generally, consider all convex cones with the property that both the cone and its dual admit efficiently computable self-concordant barriers. Then, our Domain-Driven setup can handle any conic optimization problem formulated using direct sums of these cones and their duals. Then, we show how to construct interesting convex sets as the direct sum of the epigraphs of univariate convex functions. This construction, as a special case, contains problems such as geometric programming, $p$-norm optimization, and entropy programming, the solutions of which are in great demand in engineering and science. Another interesting class of convex sets that (optimization over it) is contained in the Domain-Driven setup is the generalized epigraph of a matrix norm. This, as a special case, allows us to minimize the nuclear norm over a linear subspace that has applications in machine learning and big data. Domain-Driven setup contains the combination of all the above problems; for example, we can have a problem with LP and SDP constraints, combined with ones defined by univariate convex functions or the epigraph of a matrix norm. We review the literature on infeasible-start algorithms and discuss the pros and cons of different methods to show where our algorithms stand among them. This thesis contains a chapter about several properties for self-concordant functions. Since we are dealing with general convex sets, many of these properties are used frequently in the design and analysis of our algorithms. We introduce a notion of duality gap for the Domain-Driven setup that reduces to the conventional duality gap if the problem is a conic optimization problem, and prove some general results. Then, to solve our problems, we construct infeasible-start primal-dual central paths. A critical part in achieving the current best iteration complexity bounds is designing algorithms that follow the path efficiently. The algorithms we design are predictor-corrector algorithms. Determining the status of a general convex optimization problem (as being unbounded, infeasible, having optimal solutions, etc.) is much more complicated than that of LP. We classify the possible status (seven possibilities) for our problem as: solvable, strictly primal-dual feasible, strictly and strongly primal infeasible, strictly and strongly primal unbounded, and ill-conditioned. We discuss the certificates our algorithms return (heavily relying on duality) for each of these cases and analyze the number of iterations required to return such certificates. For infeasibility and unboundedness, we define a weak and a strict detector. We prove that our algorithms return these certificates (solve the problem) in polynomial time, with the current best theoretical complexity bounds. The complexity results are new for the infeasible-start models used. The different patterns that can be detected by our algorithms and the iteration complexity bounds for them are comparable to the current best results available for infeasible-start conic optimization, which to the best of our knowledge is the work of Nesterov-Todd-Ye (1999). In the applications, computation, and software front, based on our algorithms, we created a Matlab-based code, called DDS, that solves a large class of problems including LP, SOCP, SDP, quadratically-constrained quadratic programming (QCQP), geometric programming, entropy programming, and more can be added. Even though the code is not finalized, this chapter shows a glimpse of possibilities. The generality of the code lets us solve problems that CVX (a modeling system for convex optimization) does not even recognize as convex. The DDS code accepts constraints representing the epigraph of a matrix norm, which, as we mentioned, covers minimizing the nuclear norm over a linear subspace. For acceptable classes of convex optimization problems, we explain the format of the input. We give the formula for computing the gradient and Hessian of the corresponding self-concordant barriers and their Legendre-Fenchel conjugates, and discuss the methods we use to compute them efficiently and robustly. We present several numerical results of applying the DDS code to our constructed examples and also problems from well-known libraries such as the DIMACS library of mixed semidefinite-quadratic-linear programs. We also discuss different numerical challenges and our approaches for removing them.