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Handbook of semidefinite programming
"... Semidefinite programming (or SDP) has been one of the most exciting and active research areas in optimization during the 1990s. It has attracted researchers with very diverse backgrounds, including experts in convex programming, linear algebra, numerical optimization, combinatorial optimization, con ..."
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Cited by 89 (3 self)
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Semidefinite programming (or SDP) has been one of the most exciting and active research areas in optimization during the 1990s. It has attracted researchers with very diverse backgrounds, including experts in convex programming, linear algebra, numerical optimization, combinatorial optimization, control theory, and statistics. This tremendous research activity was spurred by the discovery of important applications in combinatorial optimization and control theory, the development of efficient interiorpoint algorithms for solving SDP problems, and the depth and elegance of the underlying optimization theory. This book includes nineteen chapters on the theory, algorithms, and applications of semidefinite programming. Written by the leading experts on the subject, it offers an advanced and broad overview of the current state of the field. The coverage is somewhat less comprehensive, and the overall level more advanced, than we had planned at the start of the project. In order to finish the book in a timely fashion, we have had to abandon hopes for separate chapters on some important topics (such as a discussion of SDP algorithms in the
Optimization Of Multiclass Queueing Networks with Changeover Times via the Achievable Region Approach: Part I, The Singlestation Case
, 1999
"... ..."
The achievable region approach to the optimal control of stochastic systems
 JOURNAL OF THE ROYAL STATISTICAL SOCIETY
, 1998
"... The achievable region approach seeks solutions to stochastic optimisation problems by: (i) characterising the space of all possible performances (the achievable region) of the system of interest, and (ii) optimising the overall systemwide performance objective over this space. This is radically di ..."
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Cited by 14 (4 self)
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The achievable region approach seeks solutions to stochastic optimisation problems by: (i) characterising the space of all possible performances (the achievable region) of the system of interest, and (ii) optimising the overall systemwide performance objective over this space. This is radically different from conventional formulations based on dynamic programming. The approach is explained with reference to a simple twoclass queueing system. Powerful new methodologies due to the authors and coworkers are deployed to analyse a general multiclass queueing system with parallel servers and then to develop an approach to optimal load distribution across a network of interconnected stations. Finally, the approach is used for the first time to analyse a class of intensity control problems.
Moment Problems and Semidefinite Optimization
 WORKING PAPER, SLOAN SCHOOL OF MANAGEMENT, MIT
, 2000
"... Problems involving moments of random variables arise naturally in many areas of mathematics, economics, and operations research. How dowe obtain optimal bounds on the probability that a random variable belongs in a set, given some of its moments? How dowe price financial derivatives without assuming ..."
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Cited by 12 (0 self)
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Problems involving moments of random variables arise naturally in many areas of mathematics, economics, and operations research. How dowe obtain optimal bounds on the probability that a random variable belongs in a set, given some of its moments? How dowe price financial derivatives without assuming any model for the underlying price dynamics, given only moments of the price of the underlying asset? How do we obtain stronger relaxations for stochastic optimization problems exploiting the knowledge that the decision variables are moments of random variables? Can we generate near optimal solutions for a discrete optimization problem from a semidefinite relaxation by interpreting an optimal solution of the relaxation as a covariance matrix? In this paper, we demonstrate that convex, and in particular semidefinite, optimization methods lead to interesting and often unexpected answers to these questions.
Applications of Semidefinite Programming
, 1998
"... A wide variety of nonlinear convex optimization problems can be cast as problems involving linear matrix inequalities (LMIs), and hence efficiently solved using recently developed interiorpoint methods. In this paper, we will consider two classes of optimization problems with LMI constraints: ffl ..."
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Cited by 9 (0 self)
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A wide variety of nonlinear convex optimization problems can be cast as problems involving linear matrix inequalities (LMIs), and hence efficiently solved using recently developed interiorpoint methods. In this paper, we will consider two classes of optimization problems with LMI constraints: ffl The semidefinite programming problem, i.e., the problem of minimizing a linear function subject to a linear matrix inequality. Semidefinite programming is an important numerical tool for analysis and synthesis in systems and control theory. It has also been recognized in combinatorial optimization as a valuable technique for obtaining bounds on the solution of NPhard problems.
Robustness and the Internet: Theoretical Foundations
, 2002
"... While control and communications theory have played a crucial role throughout in designing aspects of the Internet, a unified and integrated theory of the Internet as a whole has only recently become a practical and achievable research objective. Dramatic progress has been made recently in analytica ..."
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Cited by 5 (1 self)
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While control and communications theory have played a crucial role throughout in designing aspects of the Internet, a unified and integrated theory of the Internet as a whole has only recently become a practical and achievable research objective. Dramatic progress has been made recently in analytical results that provide for the first time a nascent but promising foundation for a rigorous and coherent mathematical theory underpinning Internet technology. This new theory addresses directly the performance and robustness of both the “horizontal ” decentralized and asynchronous nature of control in TCP/IP as well as the “vertical ” separation into the layers of the TCP/IP protocol stack from application down to the link layer. These results generalize notions of source and channel coding from information theory as well as decentralized versions of robust control. The new theoretical insights gained about the Internet also combine with our understanding of its origins and evolution to provide a rich source of ideas about complex systems in general. Most surprisingly, our deepening understanding from genomics and molecular biology has revealed that at the network and protocol level, cells and organisms are strikingly similar to technological networks, despite having completely different material substrates, evolution, and development/construction. 1
Connections Between SemiInfinite and Semidefinite Programming
"... We consider convex optimization problems with linear matrix inequality (LMI) constraints, i.e., constraints of the form F (x) =F0+x1F1+ +xmFm 0; (1.1) where the matrices Fi = F T ..."
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Cited by 4 (1 self)
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We consider convex optimization problems with linear matrix inequality (LMI) constraints, i.e., constraints of the form F (x) =F0+x1F1+ +xmFm 0; (1.1) where the matrices Fi = F T
Linear Optimization over a Polymatroid with Side Constraints – Scheduling Queues and Minimizing Submodular Functions
, 2007
"... Two seemingly unrelated problems, scheduling a multiclass queueing system and minimizing a submodular function, share a rather deep connection via the polymatroid that is characterized by a submodular set function on the one hand and represents the performance polytope of the queueing system on the ..."
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Two seemingly unrelated problems, scheduling a multiclass queueing system and minimizing a submodular function, share a rather deep connection via the polymatroid that is characterized by a submodular set function on the one hand and represents the performance polytope of the queueing system on the other hand. We first develop what we call a grouping algorithm that solves the queueing scheduling problem under side constraints, with a computational effort of O(n 3), n being the number of job classes. The algorithm organizes the job classes into groups, and identifies the optimal policy to be a priority rule across the groups and a randomized rule within each group (to enforce the side constraints). We then apply the grouping algorithm to the submodular function minimization, mapping the latter to a queueing scheduling problem with side constraints. We show the minimizing subset can be identified by applying the grouping algorithm n times. Hence, this results in a fully combinatorial algorithm that minimizes a submodular function with an effort of O(n 4). 1
Bounds on linear PDEs . . .
, 2006
"... Using recent progress on moment problems, and their connections with semidefinite optimization, we present in this paper a new methodology based on semidefinite optimization, to obtain a hierarchy of upper and lower bounds on linear functionals defined on solutions of linear partial differential equ ..."
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Using recent progress on moment problems, and their connections with semidefinite optimization, we present in this paper a new methodology based on semidefinite optimization, to obtain a hierarchy of upper and lower bounds on linear functionals defined on solutions of linear partial differential equations. We apply the proposed method to examples of PDEs in one and two dimensions, with very encouraging results. We pay particular attention to a PDE with oblique derivative conditions, commonly arising in queueing theory. We also provide computational evidence that the semidefinite constraints are critically important in improving the quality of the bounds, that is, without them the bounds are weak.