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Theory and Algorithm of LocalRefinementBased Optimization with Application to Device and Interconnect Sizing
 IEEE Transactions on ComputerAided Design of Integrated Circuits and Systems
, 1999
"... In this paper we formulate three classes of optimization problems: the simple, monotonically constrained, and bounded CongHe (CH)programs. We reveal the dominance property under the local refinement (LR) operation for the simple CHprogram, as well as the general dominance property under the pseud ..."
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In this paper we formulate three classes of optimization problems: the simple, monotonically constrained, and bounded CongHe (CH)programs. We reveal the dominance property under the local refinement (LR) operation for the simple CHprogram, as well as the general dominance property under the pseudoLR operation for the monotonically constrained CHprogram and the extendedLR operation for the bounded CHprogram. These properties enable a very efficient polynomialtime algorithm, using different types of LR operations to compute tight lower and upper bounds of the exact solution to any CHprogram. We show that the algorithm is capable of solving many layout optimization problems in deep submicron iterative circuit and/or highperformance multichip module (MCM) and printed circuit board (PCB) designs. In particular, we apply the algorithm to the simultaneous transistor and interconnect sizing problem, and to the global interconnect sizing and spacing problem considering the coupling cap...
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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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.
Optimization Over Symmetric Cones
, 1999
"... We consider the problem of optimizing a linear function over the intersection of an a#ne space and a special class of closed, convex cones, namely the symmetric cones over the reals. This problem subsumes linear programming, convex quadratically constrained quadratic programming, and semidefinite pr ..."
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We consider the problem of optimizing a linear function over the intersection of an a#ne space and a special class of closed, convex cones, namely the symmetric cones over the reals. This problem subsumes linear programming, convex quadratically constrained quadratic programming, and semidefinite programming as special cases. First, we derive some perturbation results for this problem class. Then, we discuss two solution methods: an interior point method capable of delivering highly accurate solutions to problems of modest size, and a first order bundle method which provides solutions of low accuracy, but can handle much larger problems. Finally, we describe an application of semidefinite programming in electronic structure calculations, and give some numerical results on sample problems. vi Contents Dedication iii Acknowledgment iv Abstract vi List of Figures ix List of Tables x List of Symbols and Notations x 1 Conic Optimization Problems 1 1.1 Problem Formulation . . . . . . . ...
Modeling and Optimization of VLSI Interconnects
, 1999
"... As very large scale integrated (VLSI) circuits move into the era of deepsubmicron (DSM) technology and gigahertz frequency, the system performance has increasingly become dominated by the interconnect delay. This dissertation presents five related research topics on interconnect layout optimizati ..."
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As very large scale integrated (VLSI) circuits move into the era of deepsubmicron (DSM) technology and gigahertz frequency, the system performance has increasingly become dominated by the interconnect delay. This dissertation presents five related research topics on interconnect layout optimization, and interconnect extraction and modeling: the multisource wire sizing (MSWS) problem, the simultaneous transistor and interconnect sizing (STIS) problem, the global interconnect sizing and spacing (GISS) problem, the interconnect capacitance extraction problem, and the interconnect inductance extraction problems. Given a routing tree with multiple sources, the MSWS problem determines the optimal widths of the wire segments such that the delay is minimized. We reveal several interesting properties for the optimal MSWS solution, of which the most important is the bundled refinement property. Based on this property, we propose a polynomial time algorithm, which uses iterative bundled refinement operations to compute lower and upper bounds of an optimal solution. Since the algorithm often achieves identical lower and upper bounds in experiments, the optimal solution is obtained simply by the bound computation. Furthermore, this algorithm can be used for singlesource wire sizing problem and runs 100x xxi faster than previous methods. It has replaced previous singlesource wire sizing methods in practice.
Mixed state estimation for a linear gaussian markov model
 in: Proceedings of the IEEE Conference on Decision and Control
"... We consider a discretetime dynamical system with Boolean and continuous states, with the continuous state propagating linearly in the continuous and Boolean state variables, and an additive Gaussian process noise, and where each Boolean state component follows a simple Markov chain. This model, whi ..."
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Cited by 5 (5 self)
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We consider a discretetime dynamical system with Boolean and continuous states, with the continuous state propagating linearly in the continuous and Boolean state variables, and an additive Gaussian process noise, and where each Boolean state component follows a simple Markov chain. This model, which can be considered a hybrid or jumplinear system with very special form, or a standard linear GaussMarkov dynamical system driven by a Boolean Markov process, arises in dynamic fault detection, in which each Boolean state component represents a fault that can occur. We address the problem of estimating the state, given Gaussian noise corrupted linear measurements. Computing the exact maximum a posteriori (MAP) estimate entails solving a mixed integer quadratic program, which is computationally difficult in general, so we propose an approximate MAP scheme, based on a convex relaxation, followed by rounding and (possibly) further local optimization. Our method has a complexity that grows linearly in the time horizon and cubicly with the state dimension, the same as a standard Kalman filter. Numerical experiments suggest that it performs very well in practice. 1
Mixed SemidefiniteQuadraticLinear Programs
, 1998
"... We consider mixed semidenite{quadratic{linear programs. These are linear optimization problems with three kinds of cone constraints, namely: the semidenite cone, the quadratic cone and the nonnegative orthant. We outline a primal{dual path following method to solve these problems and highlight the m ..."
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We consider mixed semidenite{quadratic{linear programs. These are linear optimization problems with three kinds of cone constraints, namely: the semidenite cone, the quadratic cone and the nonnegative orthant. We outline a primal{dual path following method to solve these problems and highlight the main features of SDPpack, a Matlab package which solves such programs. We give some examples where such mixed programs arise, and provide numerical results on benchmark problems. 1 Introduction We consider the following mixed semidenite{quadratic{linear program (SQLP): max b T y (1) s:t: F (k) 0 + P m i=1 y i F (k) i 0; k = 1; : : : ; L (2) k(c (k) A (k) ) T yk (k) (g (k) ) T y; k = 1; : : : ; M (3) (A (0) ) T y c (0) (4) where y 2 R m and F (k) i 2 S nk ; i = 0; : : : ; m; k = 1; : : : ; L A (k) 2 R mpk ; c (k) 2 R pk ; g (k) 2 R m ; (k) 2 R; k = 1; : : : ; M A (0) 2 R mp0 ; c (0) 2 R p0 : The rst set of constraints (2)...
1 Interconnect Matching Design Rule Inferring and Optimization through Correlation Extraction
"... Abstract — New backend design for manufacturability rules have brought guarantee rules for interconnect matching. These rules indicate a certain capacitance matching guarantee given spacing between interconnects and interconnect area. Yet, the number of these rules is so few that they are of limite ..."
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Abstract — New backend design for manufacturability rules have brought guarantee rules for interconnect matching. These rules indicate a certain capacitance matching guarantee given spacing between interconnects and interconnect area. Yet, the number of these rules is so few that they are of limited value in circuit or interconnect optimization. A method to infer additional guarantees from the provided guarantees is necessary so that optimization can be optimal. In this paper, we target two problems. First, we present a methodology to infer additional matching guarantees through extracting correlation information from the given limited set of matching guarantees in the design manual. In order to achieve this, we propose a multifunction variant of multivariate NewtonRaphson method to extract parameters of the proposed dimension and distancebased process correlation model for interconnects. We propose to use the extracted correlation information to infer a continuum of matching rules through simulation with proposed modifications to the standard capacitance extraction procedure. Secondly, we show how to directly incorporate the inferred interconnect matching guarantees for accurate interconnect optimization in a flexible geometric programming construction. We show how much resource savings are possible through inferring of new matching rules. Applying the inferred mismatch guarantees allows a geometric programmingbased Htree optimization to reduce the clock tree resources 27 % on average and up to 56%. Index Terms — design guarantee inferring, correlation extraction, interconnect matching, Htree optimization I.
A Fast Hybrid Algorithm for LargeScale ℓ1Regularized Logistic Regression
"... ℓ1regularized logistic regression, also known as sparse logistic regression, is widely used in machine learning, computer vision, data mining, bioinformatics and neural signal processing. The use of ℓ1 regularization attributes attractive properties to the classifier, such as feature selection, rob ..."
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ℓ1regularized logistic regression, also known as sparse logistic regression, is widely used in machine learning, computer vision, data mining, bioinformatics and neural signal processing. The use of ℓ1 regularization attributes attractive properties to the classifier, such as feature selection, robustness to noise, and as a result, classifier generality in the context of supervised learning. When a sparse logistic regression problem has largescale data in high dimensions, it is computationally expensive to minimize the nondifferentiable ℓ1norm in the objective function. Motivated by recent work (Koh et al., 2007; Hale et al., 2008), we propose a novel hybrid algorithm based on combining two types of optimization iterations: one being very fast and memory friendly while the other being slower but more accurate. Called hybrid iterative shrinkage (HIS), the resulting algorithm is comprised of a fixed point continuation phase and an interior point phase. The first phase is based completely on memory efficient operations such as matrixvector multiplications, while the second phase is based on a truncated Newton’s method. Furthermore, we show that various optimization techniques, including line search and continuation, can significantly accelerate convergence. The algorithm has global convergence at a geometric rate (a Qlinear rate in optimization terminology).
Contents lists available at ScienceDirect Signal Processing
"... journal homepage: www.elsevier.com/locate/sigpro ..."
WeC02.5 Mixed State Estimation for a Linear Gaussian Markov Model
"... Abstract — We consider a discretetime dynamical system with Boolean and continuous states, with the continuous state propagating linearly in the continuous and Boolean state variables, and an additive Gaussian process noise, and where each Boolean state component follows a simple Markov chain. This ..."
Abstract
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Abstract — We consider a discretetime dynamical system with Boolean and continuous states, with the continuous state propagating linearly in the continuous and Boolean state variables, and an additive Gaussian process noise, and where each Boolean state component follows a simple Markov chain. This model, which can be considered a hybrid or jumplinear system with very special form, or a standard linear GaussMarkov dynamical system driven by a Boolean Markov process, arises in dynamic fault detection, in which each Boolean state component represents a fault that can occur. We address the problem of estimating the state, given Gaussian noise corrupted linear measurements. Computing the exact maximum a posteriori (MAP) estimate entails solving a mixed integer quadratic program, which is computationally difficult in general, so we propose an approximate MAP scheme, based on a convex relaxation, followed by rounding and (possibly) further local optimization. Our method has a complexity that grows linearly in the time horizon and cubicly with the state dimension, the same as a standard Kalman filter. Numerical experiments suggest that it performs very well in practice. I.