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Fast and Exact Simultaneous Gate and Wire Sizing by Lagrangian Relaxation
 In Proceedings of the 1998 IEEE/ACM international conference on Computeraided design
, 1997
"... This paper considers simultaneous gate and wire sizing for general VLSI circuits under the Elmore delay model. We present a fast and exact algorithm which can minimize total area subject to maximum delay bound. The algorithm can be easily modified to give exact algorithms for optimizing several othe ..."
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Cited by 90 (8 self)
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This paper considers simultaneous gate and wire sizing for general VLSI circuits under the Elmore delay model. We present a fast and exact algorithm which can minimize total area subject to maximum delay bound. The algorithm can be easily modified to give exact algorithms for optimizing several other objectives (e.g. minimizing maximum delay or minimizing total area subject to arrival time specifications at all inputs and outputs). No previous algorithm for simultaneous gate and wire sizing can guarantee exact solutions for general circuits. Our algorithm is an iterative one with a guarantee on convergence to global optimal solutions. It is based on Lagrangian relaxation and "onegate/wireatatime" local optimizations, and is extremely economical and fast. For example, we can optimize a circuit with 13824 gates and wires in about 13 minutes using under 12 MB memory on an IBM RS/6000 workstation. 1 Introduction Since the invention of integrated circuits almost 40 years ago, gate si...
Optimal Sample Size for Multiple Testing: the Case of Gene Expression Microarrays
 Journal of the American Statistical Association
, 2004
"... We consider the choice of an optimal sample size for multiple comparison problems. The motivating application is the choice of the number of microarray experiments to be carried out when learning about dierential gene expression. However, the approach is valid in any application that involves multip ..."
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Cited by 43 (2 self)
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We consider the choice of an optimal sample size for multiple comparison problems. The motivating application is the choice of the number of microarray experiments to be carried out when learning about dierential gene expression. However, the approach is valid in any application that involves multiple comparison in a large number of hypothesis tests.
An inventorylocation model: Formulation, solution algorithm and computational
, 2002
"... Abstract. We introduce a distribution center (DC) location model that incorporates working inventory and safety stock inventory costs at the distribution centers. In addition, the model incorporates transport costs from the suppliers to the DCs that explicitly reflect economies of scale through the ..."
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Cited by 35 (12 self)
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Abstract. We introduce a distribution center (DC) location model that incorporates working inventory and safety stock inventory costs at the distribution centers. In addition, the model incorporates transport costs from the suppliers to the DCs that explicitly reflect economies of scale through the use of a fixed cost term. The model is formulated as a nonlinear integerprogramming problem. Model properties are outlined. A Lagrangian relaxation solution algorithm is proposed. By exploiting the structure of the problem we can find a loworder polynomial algorithm for the nonlinear integer programming problem that must be solved in solving the Lagrangian relaxation subproblems. A number of heuristics are outlined for finding good feasible solutions. In addition, we describe two variable forcing rules that prove to be very effective at forcing candidate sites into and out of the solution. The algorithms are tested on problems with 88 and 150 retailers. Computation times are consistently below one minute and compare favorably with those of an earlier proposed set partitioning approach for this model (Shen, 2000; Shen, Coullard and Daskin, 2000). Finally, we discuss the sensitivity of the results to changes in key parameters including the fixed cost of placing orders. Significant reductions in these costs might be expected from ecommerce technologies. The model suggests that as these costs decrease it is optimal to locate additional facilities. 1.
Reliability Models for Facility Location: The Expected Failure Cost Case
 Transportation Science
, 2004
"... Classical facility location models like the Pmedian problem (PMP) and the uncapacitated fixedcharge location problem (UFLP) implicitly assume that once constructed, the facilities chosen will always operate as planned. In reality, however, facilities "fail" from time to time due to poor ..."
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Cited by 31 (11 self)
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Classical facility location models like the Pmedian problem (PMP) and the uncapacitated fixedcharge location problem (UFLP) implicitly assume that once constructed, the facilities chosen will always operate as planned. In reality, however, facilities "fail" from time to time due to poor weather, labor actions, changes of ownership, or other factors. Such failures may lead to excessive transportation costs as customers must be served from facilities much farther than their regularly assigned facilities. In this paper, we present models for choosing facility locations to minimize cost while also taking into account the expected transportation cost after failures of facilities. The goal is to choose facility locations that are both inexpensive under traditional objective functions and also reliable. This reliability approach is new in the facility location literature. We formulate reliability models based on both the PMP and the UFLP and present an optimal Lagrangian relaxation algorithm to solve them. We discuss how to use these models to generate a tradeo# curve between the daytoday operating cost and the expected cost taking failures into account, and use these tradeo# curves to demonstrate empirically that substantial improvements in reliability are often possible with minimal increases in operating cost.
Optimal NonUniform WireSizing under the Elmore Delay Model
 in Proc. Int. Conf. on Computer Aided Design
, 1996
"... We consider nonuniform wiresizing for general routing trees under the Elmore delay model. Three minimization objectives are studied: 1) total weighted sinkdelays; 2) total area subject to sinkdelay bounds; and 3) maximum sinkdelay. We first present an algorithm NWSAwd for minimizing total weigh ..."
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Cited by 22 (10 self)
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We consider nonuniform wiresizing for general routing trees under the Elmore delay model. Three minimization objectives are studied: 1) total weighted sinkdelays; 2) total area subject to sinkdelay bounds; and 3) maximum sinkdelay. We first present an algorithm NWSAwd for minimizing total weighted sinkdelays based on iteratively applying the wiresizing formula in [1]. We show that NWSAwd always converges to an optimal wiresizing solution. Based on NWSAwd and the Lagrangian relaxation technique, we obtained two algorithms NWSAdb and NWSAmd which can optimally solve the other two minimization objectives. Experimental results show that our algorithms are efficient both in terms of runtime and storage. For example, NWSAwd, with linear runtime and storage, can solve a 6201wiresegment routingtree problem using about 1.5second runtime and 1.3MB memory on an IBM RS/6000 workstation. 1 Introduction As VLSI technology continues to scale down, interconnect delay has become the d...
On Combinatorial Auction and Lagrangean Relaxation for Distributed Resource Scheduling
 IIE Transactions
, 1998
"... Most existing methods for scheduling are based on centralized or hierarchical decision making using monolithic models. In this study, we investigate a new method based on a distributed and locally autonomous decision structure using the notion of combinatorial auction. In combinatorial auction the b ..."
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Cited by 20 (4 self)
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Most existing methods for scheduling are based on centralized or hierarchical decision making using monolithic models. In this study, we investigate a new method based on a distributed and locally autonomous decision structure using the notion of combinatorial auction. In combinatorial auction the bidders demand a combination of dependent objects with a single bid. We show that not only can we use this auction mechanism to handle complex resource scheduling problems, but there exist strong links between combinatorial auction and Lagrangeanbased decomposition. Exploring some of these properties, we characterize combinatorial auction using auction protocols and payment functions. This study is a #rst step toward developing a distributed scheduling framework that maintains systemwide performance while accommodating local preferences and objectives. We provide some insights to this framework by demonstrating four di#erent versions of the auction mechanism using job shop scheduling proble...
Scheduling Multiple VariableSpeed Machines
 OPERATIONS RESEARCH
, 1994
"... We examine scheduling problems where we control not only the assignment of jobs to machines, but also the time used by the job on the machine. For instance, many tooling machines allow control of the speed at which a job is run. Increasing the speed incurs costs due to machine wear but also incre ..."
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Cited by 17 (0 self)
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We examine scheduling problems where we control not only the assignment of jobs to machines, but also the time used by the job on the machine. For instance, many tooling machines allow control of the speed at which a job is run. Increasing the speed incurs costs due to machine wear but also increases throughput. We discuss some fundamental scheduling problems in this environment and give algorithms for some interesting cases. Some cases are inherently difficult so for these we give heuristics. Our approach illustrates the exploitation of underlying network structure in combinatorial optimization problems. We create heuristics that optimally schedule a large portion of the jobs and then attempt to fit in the remainder. This also gives a method for quickly finding valid inequalities violated by the linear relaxation solution. For the problem of minimizing the sum of makespan and production costs, a rounding heuristic is within a constant factor of optimal. Our heuristics ar...
The Stochastic Location Model with Risk Pooling
 EUROPEAN JOURNAL OF OPERATIONAL RESEARCH
, 2007
"... In this paper, we present a stochastic version of the Location Model with Risk Pooling (LMRP) that optimizes location, inventory, and allocation decisions under random parameters described by discrete scenarios. The goal of our model (called the stochastic LMRP, or SLMRP) is to find solutions that m ..."
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Cited by 17 (5 self)
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In this paper, we present a stochastic version of the Location Model with Risk Pooling (LMRP) that optimizes location, inventory, and allocation decisions under random parameters described by discrete scenarios. The goal of our model (called the stochastic LMRP, or SLMRP) is to find solutions that minimize the expected total cost (including location, transportation, and inventory costs) of the system across all scenarios. The location model explicitly handles the economies of scale and riskpooling effects that result from consolidating inventory sites. The SLMRP framework can also be used to solve multicommodity and multiperiod problems. We present a Lagrangianrelaxation–based exact algorithm for the SLMRP. The Lagrangian subproblem is a nonlinear integer program, but it can be solved by a loworder polynomial algorithm. We discuss simple variablefixing routines that can drastically reduce the size of the problem. We present quantitative and qualitative computational results on problems with up to 150 nodes and 9 scenarios, describing both algorithm performance and solution behavior as key parameters change.
Conditional Subgradient Optimization  Theory and Applications
 European Journal of Operational Research
, 1996
"... We generalize the subgradient optimization method for nondifferentiable convex programming to utilize conditional subgradients. Firstly, we derive the new method and establish its convergence by generalizing convergence results for traditional subgradient optimization. Secondly, we consider a partic ..."
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Cited by 15 (4 self)
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We generalize the subgradient optimization method for nondifferentiable convex programming to utilize conditional subgradients. Firstly, we derive the new method and establish its convergence by generalizing convergence results for traditional subgradient optimization. Secondly, we consider a particular choice of conditional subgradients, obtained by projections, which leads to an easily implementable modification of traditional subgradient optimization schemes. To evaluate the subgradient projection method we consider its use in three applications: uncapacitated facility location, twoperson zerosum matrix games, and multicommodity network flows. Computational experiments show that the subgradient projection method performs better than traditional subgradient optimization; in some cases the difference is considerable. These results suggest that our simple modification may improve subgradient optimization schemes significantly. This finding is important as such schemes are very popula...
TOMLAB  A General Purpose, Open MATLAB Environment for Research and Teaching in Optimization
, 1998
"... TOMLAB is a general purpose, open and integrated MATLAB environment for research and teaching in optimization on UNIX and PC systems. The motivation for TOMLAB is to simplify research on practical optimization problems, giving easy access to all types of solvers; at the same time having full acce ..."
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Cited by 14 (13 self)
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TOMLAB is a general purpose, open and integrated MATLAB environment for research and teaching in optimization on UNIX and PC systems. The motivation for TOMLAB is to simplify research on practical optimization problems, giving easy access to all types of solvers; at the same time having full access to the power of MATLAB. By using a simple, but general input format, combined with the ability in MATLAB to evaluate string expressions, it is possible to run internal TOMLAB solvers, MATLAB Optimization Toolbox and commercial solvers written in FORTRAN or C/C++ using MEXfile interfaces. Currently MEXfile interfaces have been developed for MINOS, NPSOL, NPOPT, NLSSOL, LPOPT, QPOPT and LSSOL. TOMLAB may either be used totally parameter driven or menu driven. The basic principles will be discussed. The menu system makes it suitable for teaching. Many standard test problems are included. More test problems are easily added. There are many example and demonstration files. Iterati...