Posynomial Models Research Articles

A Variability-Aware Robust Design Methodology for Integrated Circuits by Geometric Programming

Process variations have continuously posed significant challenges to the performance and yield of integrated circuits (ICs). The performance modeling and robust optimization method considering process variations has become an important research task in today’s IC design. Aiming at solving the problems of strong nonlinearity and high-dimensional problems in circuit design, this paper proposes a general robust optimization method for ICs by geometric programming. This method first employs regularization sparse models to model a specific performance metric as a posynomial function in terms of design parameters, in order to reduce parameter space dimensionality and to accurately capture the nonlinear relationship between performance perturbations and process variations. Based on the posynomial performance models, this method further uses an uncertainty set to represent the uncertainties of process variations, and formulates the problem of robust optimization under process variations as a general geometric programming model that can be efficiently solved. Experimental results demonstrate that, the proposed method not only enhances the accuracy and efficiency of circuit performance modeling, but also improves the performance yield significantly compared with traditional circuit design methods.

Log-Sum-Exp Neural Networks and Posynomial Models for Convex and Log-Log-Convex Data.

In this paper, we show that a one-layer feedforward neural network with exponential activation functions in the inner layer and logarithmic activation in the output neuron is a universal approximator of convex functions. Such a network represents a family of scaled log-sum exponential functions, here named log-sum-exp ( LSET ). Under a suitable exponential transformation, the class of LSET functions maps to a family of generalized posynomials GPOST , which we similarly show to be universal approximators for log-log-convex functions. A key feature of an LSET network is that, once it is trained on data, the resulting model is convex in the variables, which makes it readily amenable to efficient design based on convex optimization. Similarly, once a GPOST model is trained on data, it yields a posynomial model that can be efficiently optimized with respect to its variables by using geometric programming (GP). The proposed methodology is illustrated by two numerical examples, in which, first, models are constructed from simulation data of the two physical processes (namely, the level of vibration in a vehicle suspension system, and the peak power generated by the combustion of propane), and then optimization-based design is performed on these models.

Sparse identification of posynomial models

Posynomials are nonnegative combinations of monomials with possibly fractional and both positive and negative exponents. Posynomial models are widely used in various engineering design endeavors, such as circuits, aerospace and structural design, mainly due to the fact that design problems cast in terms of posynomial objectives and constraints can be solved efficiently by means of a convex optimization technique known as geometric programming (GP). However, while quite a vast literature exists on GP-based design, very few contributions can yet be found on the problem of identifying posynomial models from experimental data. Posynomial identification amounts to determining not only the coefficients of the combination, but also the exponents in the monomials, which renders the identification problem hard. In this paper, we propose an approach to the identification of multivariate posynomial models based on the expansion on a given large-scale basis of monomials. The model is then identified by seeking coefficients of the combination that minimize a mixed objective, composed by a term representing the fitting error and a term inducing sparsity in the representation, which results in a problem formulation of the “square-root LASSO” type, with nonnegativity constraints on the variables. We propose to solve the problem via a sequential coordinate-minimization scheme, which is suitable for large-scale implementations. A numerical example is finally presented, dealing with the identification of a posynomial model for a NACA 4412 airfoil.

Sparse Identification of Polynomial and Posynomial Models

Posynomials are nonnegative combinations of monomials with possibly fractional and both positive and negative exponents. Posynomial models are widely used in various engineering design endeavors, such as circuits, aerospace and structural design, mainly due to the fact that design problems cast in terms of posynomial objectives and constraints can be solved efficiently by means of a convex optimization technique known as geometric programming (GP). However, while quite a vast literature exists on GP-based design, very few contributions can yet be found on the problem of identifying posynomial models from experimental data. Posynomial identification amounts to determining not only the coefficients of the combination, but also the exponents in the monomials, which renders the identification problem numerically hard. In this paper, we outline an approach to the identification of both multivariate polynomial and posynomial models, based on the expansion on a given large-scale basis of monomials. The model is then identified by seeking coefficients of the combination that minimize a mixed objective, composed by a term representing the fitting error and a term inducing sparsity in the representation, which result in a problem formulation of the “square-root LASSO” type, with nonnegativity constraints on the variables. We propose to solve the problem via a sequential coordinate-descent scheme, which is suitable for large-scale implementations.

Determining Optimal Demand Rate and Production Decisions: A Geometric Programming Approach

In this paper a nonlinear model is presented to demonstrate the relation between production and marketing departments. By introducing some functions such as pricing cost and market share loss functions it will be tried to show some aspects of market modelling which has not been regarded before. The proposed model will be a constrained signomial geometric programming model. For model solving, after variables’ modifications an iterative technique based on the concept of geometric mean will be introduced to solve the resulting non-standard posynomial model which can be applied to a wide variety of models in non-standard posynomial geometric programming form. At the end a numerical analysis will be presented to accredit the validity of the mentioned model. Keywords—Geometric programming, marketing, nonlinear optimization, production.

Nonconvex Gate Delay Modeling and Delay Optimization

Convex delay models like the Elmore model, the related Logical Effort model, posynomial, and generalized posynomial models have always been favored by researchers, as convexity has a priori guarantees of global optimum solutions. The accuracy of the model may be sacrificed in this quest to generate convex delay models. In this paper, we investigate the use of signomial delay modeling for area/delay optimization. We present a procedure to automatically generate signomial gate delay models by nonlinear least squares fitting. As opposed to posynomial models, signomial models achieve better fits to SPICE generated data. However, signomials are not convex in general. Nevertheless, we show via duality arguments that we obtain near optimum (within 1%) solutions. Our optimization considers beta-ratio constraints, minimum and maximum size constraints for n- and p-transistors, rise/fall delays, and edge rates. The gate sizes for the fastest delay solution for a 44000-cell design, using the IBM 130-nm process, can be achieved in about 16 min of CPU time on a PC, and the area-delay tradeoff curve for 21 points can be generated in about 2 h of CPU time. To the best of our knowledge, this is the first report of using a true signomial delay model and its application to optimum gate sizing. In addition, we give performance details for the automatic data fitting for an 11-function library of static CMOS gates.

Generalized Simulation-Based Posynomial Model Generation for Analog Integrated Circuits

This paper presents a new method to automatically generate posynomial symbolic expressions for the performance characteristics of analog integrated circuits. Both the coefficient set as well as the exponent set of the posynomial expression, for some performance as a function of the design variables, are determined based on performance data extracted from SPICE simulation results with device-level accuracy. Techniques from design of experiments (DOE) are used to generate an optimal set of sample points to fit the models. We will prove that the optimization problem formulated for this problem typically corresponds to a non-convex problem, but has no 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 can be used to compose a geometric program that fully describes the analog circuit sizing problem. The automatic generation avoids the time-consuming nature of hand-crafted analytic model generation. Experimental results illustrate the capabilities of the presented modeling technique.

Optimal design of protein production plants with time and size factor process models.

In this work we propose an optimization model for the design of a biotechnological multiproduct batch plant. A first level of detail posynomial model is constructed for each unit, as well as decisions regarding the structural optimization of the plant. A particular feature of this model is that it contains composite units in which semicontinuous items operate on the material contained by batch items. This occurs in the purification steps, in particular with the microfilters operating between retentate and permeate vessels, and with the homogenizer and ultrafilters operating on the material contained in a batch holding vessel. Also, the unit models rely on batch operating time expressions that depend on both the batch size and the size of semicontinuous items. The model takes into account all of the available options to increase the efficiency of the batch plant design: unit duplication in-phase and out-of-phase and intermediate storage tanks. The resulting mathematical model for the minimization of the plant capital cost is a mixed integer non-linear program (MINLP), which is solved to global optimality with an implementation of the outer approximation/ equality relaxation/ augmented penalty (OA/ER/AP) method. A plant that produces four recombinant proteins in eight processing stages is used to illustrate the proposed approach. An interesting feature of this example is that it represents an attempt to standardize a plant for the production of both therapeutic and nontherapeutic proteins; the model applied is generic and can thus be applied to any such modular plant. Results indicate that the best solution in terms of minimal capital cost contains no units in parallel and with intermediate storage tank allocation.

Optimal design of macrocells for low power and high speed

The problem of designing individual macrocells for a library for low power and high speed is addressed here, and a new technique for optimization using posynomial approximating functions is devised. In the design of each macrocell, optimality in design is critical and highly accurate techniques for measuring the performance are required during optimization. This paper presents methods for accurately estimating the worst-case contribution of the power and delay of a cell to a circuit. The program uses circuit-level simulation to calculate the power dissipation and delay of the cell with the highest accuracy. A rationale for using arbitrary degree posynomial modeling functions for area, delay, and power modeling is presented. The problem is then formulated as a convex programming problem, and a rigorous optimization technique is used to arrive at the optimal macrocell.

Posynomial modeling of batch plants: A procedure to include process decision variables

A general procedure for the generation of posynomial models of batch plants is presented. Posynomial models describe process stages with size factors and cycle times, allowing the synthesis and sizing of batch plants through well-established literature developments. Starting from modular process design equations, proper algebraic manipulation defines size factors and cycle times which are functions of process decision variables. A performance index optimization on those variables, with a decoupling level of intermediate storage, provides a posynomial model for the process. A citric acid fermentative process is modeled as an example.

OPTIMAL SELECTION OF STEAM TURBINE EXHAUST ANNULUS AND CONDENSER SIZES BY GEOMETRIC PROGRAMMING

Abstract In this paper, a posynomial programming model is developed for the optimal selection of steam turbine exhaust annulus and condenser sizes. After formulating the general model, a numerical example is presented and sensitivity analysis is considered. A simple formula expressing the optimal cost of the condensing system as a function of selected parameters is obtained.