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

MM Algorithms for Geometric and Signomial Programming

This paper derives new algorithms for signomial programming, a generalization of geometric programming. The algorithms are based on a generic principle for optimization called the MM algorithm. In this setting, one can apply the geometric-arithmetic mean inequality and a supporting hyperplane inequality to create a surrogate function with parameters separated. Thus, unconstrained signomial programming reduces to a sequence of one-dimensional minimization problems. Simple examples demonstrate that the MM algorithm derived can converge to a boundary point or to one point of a continuum of minimum points. Conditions under which the minimum point is unique or occurs in the interior of parameter space are proved for geometric programming. Convergence to an interior point occurs at a linear rate. Finally, the MM framework easily accommodates equality and inequality constraints of signomial type. For the most important special case, constrained quadratic programming, the MM algorithm involves very simple updates.Comment: 16 pages, 1 figur

Optimization techniques applied to passive measures for in-orbit spacecraft survivability

Spacecraft designers have always been concerned about the effects of meteoroid impacts on mission safety. The engineering solution to this problem has generally been to erect a bumper or shield placed outboard from the spacecraft wall to disrupt/deflect the incoming projectiles. Spacecraft designers have a number of tools at their disposal to aid in the design process. These include hypervelocity impact testing, analytic impact predictors, and hydrodynamic codes. Analytic impact predictors generally provide the best quick-look estimate of design tradeoffs. The most complete way to determine the characteristics of an analytic impact predictor is through optimization of the protective structures design problem formulated with the predictor of interest. Space Station Freedom protective structures design insight is provided through the coupling of design/material requirements, hypervelocity impact phenomenology, meteoroid and space debris environment sensitivities, optimization techniques and operations research strategies, and mission scenarios. Major results are presented

An Analysis of Posynomial MOSFET Models Using Genetic Algorithms and Visualization

MEng thesisAnalog designers are interested in optimization tools which automate the process of circuit sizing. Geometric programming, which uses posynomial models of MOSFET parameters, represents one such tool. Genetic algorithms have been used to evolve posynomial models for geometric programs, with a reasonable mean error when modeling MOSFET parameters. By visualizing MOSFET data using two dimensional plots, this thesis investigates the behavior of various MOSFET small and large signal parameters and consequently proposes a lower bound on the maximum error, which a posynomial cannot improve upon. It then investigates various error metrics which can be used to balance the mean and maximum errors generated by posynomial MOSFET models. Finally, the thesis uses empirical data to verify the existence of the lower bound, and compares the maximum error from various parameters modeled by the genetic algorithm and by monomial fitting. It concludes that posynomial MOSFET models suffer from inherent inaccuracies. Additionally, although genetic algorithms improve on the maximum model error, the improvement, in general, does not vastly surpass results obtained through monomial fitting, which is a less computationally intensive method. Genetic algorithms are hence best used when modeling partially convex MOSFET parameters, such as r0

Modified signomial geometric programming (MSGP) and its applications

A "signomial" is a mathematical function, contains one or more independent variables. Richard J. Duffin and Elmor L. Peterson introduced the term "signomial". Signomial geometric programming (SGP) optimization technique often provides a much better mathematical result of real-world nonlinear optimization problems. In this research paper, we have proposed unconstrained and constrained signomial geometric programming (SGP) problem with positive or negative integral degree of difficulty. Here a modified form of signomial geometric programming (MSGP) has been developed and some theorems have been derived. Finally, these are illustrated by proper examples and applications

Analog circuit optimization using evolutionary algorithms and convex optimization

Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2007.Includes bibliographical references (p. 83-88).In this thesis, we analyze state-of-art techniques for analog circuit sizing and compare them on various metrics. We ascertain that a methodology which improves the accuracy of sizing without increasing the run time or the designer effort is a contribution. We argue that the accuracy of geometric programming can be improved without adversely influencing the run time or increasing the designer's effort. This is facilitated by decomposition of geometric programming modeling into two steps, which decouples accuracy of models and run-time of geometric programming. We design a new algorithm for producing accurate posynomial models for MOS transistor parameters, which is the first step of the decomposition. The new algorithm can generate posynomial models with variable number of terms and real-valued exponents. The algorithm is a hybrid of a genetic algorithm and a convex optimization technique. We study the performance of the algorithm on artificially created benchmark problems. We show that the accuracy of posynomial models of MOS parameters is improved by a considerable amount by using the new algorithm. The new posynomial modeling algorithm can be used in any application of geometric programming and is not limited to MOS parameter modeling. In the last chapter, we discuss various ideas to improve the state-of-art in circuit sizing.by Varun Aggarwal.S.M