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34
FenchelLagrange versus Geometric Duality in Convex Optimization
"... We present a new duality theory in order to treat convex optimization problems and we prove that the geometric duality used by C.H. Scott and T.R. Jefferson in different papers during the last quarter of century is a special case of it. Moreover, weaker sufficient conditions in order to achieve st ..."
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We present a new duality theory in order to treat convex optimization problems and we prove that the geometric duality used by C.H. Scott and T.R. Jefferson in different papers during the last quarter of century is a special case of it. Moreover, weaker sufficient conditions in order to achieve strong duality are considered and optimality conditions are derived. Next we apply our approach to some problems considered by Scott and Jefferson, determining their duals. We give weaker sufficient conditions in order to achieve strong duality and the corresponding optimality conditions. Finally, posynomial geometric programming is viewed also as a particular case of the duality approach we present.
A Polynomial Time Optimal Algorithm for Simultaneous Buffer and Wire Sizing
"... An interconnect joining a source and a sink is divided into fixedlength uniformwidth wire segments, and some adjacent segments have buffers in between. The problem we considered is to simultaneously size the buffers and the segments so that the Elmore delay from the source to the sink is minimized ..."
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An interconnect joining a source and a sink is divided into fixedlength uniformwidth wire segments, and some adjacent segments have buffers in between. The problem we considered is to simultaneously size the buffers and the segments so that the Elmore delay from the source to the sink is minimized. Previously, no polynomial time algorithm for the problem has been reported in literature. In this paper, we present a polynomial time algorithm SBWS for the simultaneous buffer and wire sizing problem. SBWS is an iterative algorithm with guaranteed convergence to the optimal solution. It runs in quadratic time and uses constant memory for computation. Also, experimental results show that SBWS is extremely efcient in practice. For example, for an interconnect of 10000 segments and buffers, the CPU time is only 0.127 second.
Generalized Posynomial Performance Modeling
 IN DATE ’03
, 2003
"... This paper presents a new method to automatically generate posynomial symbolic expressions for the performance characteristics of analog integrated circuits. The coefficient set as well as the exponent set of the posynomial expression are determined based on SPICE simulation data with devicelevel a ..."
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This paper presents a new method to automatically generate posynomial symbolic expressions for the performance characteristics of analog integrated circuits. The coefficient set as well as the exponent set of the posynomial expression are determined based on SPICE simulation data with devicelevel accuracy. We will prove that this problem corresponds to solving a nonconvex optimization problem without local minima. The presented method is capable of generating posynomial performance expressions for both linear and nonlinear circuits and circuit characteristics. This approach allows to automatically generate an accurate sizing model that composes a geometric program that fully describes the analog circuit sizing problem. The automatic generation avoids the timeconsuming nature of handcrafted analytic model generation. Experimental results illustrate the capabilities and effectiveness of the presented modeling technique.
A Fitting Approach to Generate Symbolic Expressions for Linear and Nonlinear Analog Circuit Performance Characteristics
 In Proceedings Design Automation and Test in Europe Conference
, 2002
"... This paper presents a novel method to automatically generate symbolic expressions for both linear and nonlinear circuit characteristics using a templatebased fitting of numerical, simulated data. The aim of the method is to generate convex, interpretable expressions. The posynomiality of the genera ..."
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Cited by 4 (1 self)
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This paper presents a novel method to automatically generate symbolic expressions for both linear and nonlinear circuit characteristics using a templatebased fitting of numerical, simulated data. The aim of the method is to generate convex, interpretable expressions. The posynomiality of the generated expressions enables the use of efficient geometric programming techniques when using these expressions for circuit sizing and optimization. Attention is paid to estimating the relative `goodnessoffit' of the generated expressions. Experimental results illustrate the capabilities of the approach.
Using underapproximations for sparse nonnegative matrix factorization
 Pattern Recognition
, 2010
"... Nonnegative Matrix Factorization (NMF) has gathered a lot of attention in the last decade and has been successfully applied in numerous applications. It consists in the factorization of a nonnegative matrix by the product of two lowrank nonnegative matrices: M ≈ V W. In this paper, we attempt to so ..."
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Nonnegative Matrix Factorization (NMF) has gathered a lot of attention in the last decade and has been successfully applied in numerous applications. It consists in the factorization of a nonnegative matrix by the product of two lowrank nonnegative matrices: M ≈ V W. In this paper, we attempt to solve NMF problems in a recursive way. In order to do that, we introduce a new variant called Nonnegative Matrix Underapproximation (NMU) by adding the upper bound constraint V W ≤ M. Besides enabling a recursive procedure for NMF, these inequalities make NMU particularly wellsuited to achieve a sparse representation, improving the partbased decomposition. Although NMU is NPhard (which we prove using its equivalence with the maximum edge biclique problem in bipartite graphs), we present two approaches to solve it: a method based on convex reformulations and a method based on Lagrangian relaxation. Finally, we provide some encouraging numerical results for image processing applications.
Automated design of operational transconductance amplifiers using reversed geometric programming
 In Proceedings of the 41th IEEE/ACM Design Automation Conference
, 2004
"... We present a method for designing operational amplifiers using reversed geometric programming, which is an extension of geometric programming that allows both convex and nonconvex constraints. Adding a limited set of nonconvex constraints can improve the accuracy of convex equationbased optimizati ..."
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We present a method for designing operational amplifiers using reversed geometric programming, which is an extension of geometric programming that allows both convex and nonconvex constraints. Adding a limited set of nonconvex constraints can improve the accuracy of convex equationbased optimization, without compromising global optimality. These constraints allow increased accuracy for critical modeling equations, such as the relationship between gm and IDS. To demonstrate the design methodology, a foldedcascode amplifier is designed in a 0.18 µm technology for varying speed requirements and is compared with simulations and designs obtained from geometric programming. Categories and Subject Descriptors:
Timing modeling and optimization under the transmission line model
 IEEE Transactions on Very Large Scale Integration Systems
, 2004
"... Abstract—As the operating frequency increases to gigahertz and the rise time of a signal is less than or comparable to the timeofflight delay of a wire, it is necessary to consider the transmission line behavior for delay computation. We present in this paper, an analytical formula for the delay c ..."
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Abstract—As the operating frequency increases to gigahertz and the rise time of a signal is less than or comparable to the timeofflight delay of a wire, it is necessary to consider the transmission line behavior for delay computation. We present in this paper, an analytical formula for the delay computation under the transmission line model. Extensive simulations with SPICE show the high fidelity of the formula. Compared with previous works, our model leads to smaller average errors in delay estimation. Based on this formula, we show the property that the minimum delay for a transmission line with reflection occurs when the number of round trips is minimized (i.e., equals one). Besides, we show that the delay of a circuit path is a posynomial function in wire and buffer sizes, implying that a local optimum is equal to the global optimum. Thus, we can apply any efficient search algorithm such as the wellknown gradient search procedure to compute the globally optimal solution. Experimental results show that simultaneous wire and buffer sizing is very effective for performance optimization under the transmission line model. Index Terms—Buffer sizing, delay model, inductance, interconnect, performance optimization, transmission line, wire sizing.
Global injectivity and multiple equilibria in uni and bimolecular reaction networks
, 2012
"... Abstract. Dynamical system models of complex biochemical reaction networks are highdimensional, nonlinear, and contain many unknown parameters. The capacity for multiple equilibria in such systems plays a key role in important biochemical processes. Examples show that there is a very delicate relat ..."
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Abstract. Dynamical system models of complex biochemical reaction networks are highdimensional, nonlinear, and contain many unknown parameters. The capacity for multiple equilibria in such systems plays a key role in important biochemical processes. Examples show that there is a very delicate relationship between the structure of a reaction network and its capacity to give rise to several positive equilibria. In this paper we focus on networks of reactions governed by massaction kinetics. As is almost always the case in practice, we assume that no reaction involves the collision of three or more molecules at the same place and time, which implies that the associated massaction differential equations contain only linear and quadratic terms. We describe a general injectivity criterion for quadratic functions of several variables, and relate this criterion to a network’s capacity for multiple equilibria. In order to take advantage of this criterion we look for explicit general conditions that imply nonvanishing of polynomial functions on the positive orthant. In particular, we investigate in detail the case of polynomials with only one negative monomial, and we fully characterize the case of affinely independent exponents. We describe several examples, including an example that shows how these methods may be used for designing multistable chemical systems in synthetic biology. 1. Introduction. A
Robustness of Posynomial Geometric Programming Optima
 Mathematical Programming
, 1999
"... This paper develops a simple bounding procedure for the optimal value of a posynomial geometric programming (GP) problem when some of the coefficients for terms in the problem's objective function are estimated with error. The bound may be computed even before the problem is solved and it is shown ..."
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This paper develops a simple bounding procedure for the optimal value of a posynomial geometric programming (GP) problem when some of the coefficients for terms in the problem's objective function are estimated with error. The bound may be computed even before the problem is solved and it is shown analytically that the optimum value is very insensitive to errors in the coefficients; for example, a 20% error could cause the optimum to be wrong by no more than 1.67%. Key Words: Geometric Programming, Posynomials, Sensitivity Analysis *Corresponding Author Address: Department of Industrial Engineering 1048 Benedum Hall University of Pittsburgh Pittsburgh, PA 15261 email: rajgopal@engrng.pitt.edu fax: (412) 6249831 1 Introduction Geometric Programming (GP) is a technique for solving certain classes of algebraic nonlinear optimization problems. Since its original development by Duffin, Peterson and Zener (1967) at the Westinghouse R & D Center, it has been studied extensively and...
A generalization of Pólya’s theorem to signomials with rational exponents, in preparation. See draft at www.math.lsu.edu/∼preprints
"... Pólya proved that if a real, homogeneous polynomial is positive on the nonnegative orthant (except at the origin), then it is the quotient of two homogeneous polynomials with no negative coefficients. We generalize this from polynomials to signomials with arbitrary rational exponents; we also show t ..."
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Pólya proved that if a real, homogeneous polynomial is positive on the nonnegative orthant (except at the origin), then it is the quotient of two homogeneous polynomials with no negative coefficients. We generalize this from polynomials to signomials with arbitrary rational exponents; we also show that Pólya’s theorem does not generalize to arbitrary signomials (i.e., with irrational (real) exponents). 1