Data fitting with geometric-programming-compatible softmax functions

Abstract

Motivated by practical applications in engineering, this article considers the problem of approximating a set of data with a function that is compatible with geometric programming (GP). Starting with well-established methods for fitting max-affine functions, it is shown that improved fits can be obtained using an extended function class based on the softmax of a set of affine functions. The softmax is generalized in two steps, with the most expressive function class using an implicit representation that allows fitting algorithms to locally tune softness. Each of the proposed function classes is directly compatible with the posynomial constraint forms in geometric programming. Maxmonomial fitting and posynomial fitting are shown to correspond to fitting special cases of the proposed implicit softmax function class. The fitting problem is formulated as a nonlinear least squares regression, solved locally using a Levenberg-Marquardt algorithm. Practical implementation considerations are discussed. The article concludes with numerical examples from aerospace engineering and electrical engineering.

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

“…Several tools have been proposed in the literature to fit data via convex or log-log-convex functions (see Section II-B for a definition of log-log convexity). Some remarkable examples are, for instance, [8], where an efficient least-squares partition algorithm is proposed to fit data through max-affine functions; [9], where a similar method has been proposed to fit maxmonomial functions; [10], where a technique based on fitting the data through implicit softmax-affine functions has been proposed; and [11], [12], where methods to fit data through posynomial models have been proposed.…”

“…, the function f T given in (3) approximatesf as T tends to zero, see [10]. This deformation is familiar in tropical geometry under the name of "Maslov dequantization," [26], and it is a key ingredient of Viro's patchworking method, [24].…”

“…Since c k x α (k) is log-log-convex, then by Proposition 2 each function in the POS class is log-log-convex. Posynomials are of great interest in practical applications since, under a loglog transform, they become convex functions [10], [28]. More precisely, by letting q .…”

Turbofan Engine Sizing and Tradeoff Analysis via Signomial Programming

“…The parabolic drag model, with the induced drag parameter K, is used to model induced drag. GPfit [18,16] was used to develop a GP compatible fit to Xfoil [9] drag data for an NC130 airfoil [10] at a Reynolds number of 20 million. The fit is plotted in Figure C- …”

Efficient Aircraft Multidisciplinary Design Optimization and Sensitivity Analysis via Signomial Programming

“…Other techniques to make constraints GP compatible include using variable transformations and Taylor approximations. Variable transformations are used in the tail cone model from [5] and the stagnation relations in Appendix B. Taylor approximations are used in the GP-compatible Breguet range equation presented in [2] and the structural model in Appendix A. Additionally, empirical relations can be used to formulate constraints using the methods described in [14]. An example of GP-compatible fits can be found in [5], where XFOIL [15] data is used to generate a posynomial inequality constraint for TASOPT C-series airfoil parasitic drag coefficient as a function of wing thickness, lift coefficient, Reynolds number, and Mach number.…”