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.

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Due to the coupled nature of aircraft system design, it is important to consider all of the major subsystems when trying to optimize a configuration. This, however, is easier said than done, particularly because each individual subsystem model can be arbitrarily complex, thus making optimization difficult. By restricting an optimization problem to have a certain mathematical structure, significantly more effective and tractable solution techniques can be used. Geometric programming, an example of one such technique, guarantees finding a globally optimal solution. Although it has been shown that geometric programming can be used to solve some conceptual aircraft design problems, the required formulation can prove too restrictive for certain relationships. Signomial programming is a closely related relaxation of geometric programming that offers enhanced expressiveness, but without the guarantee of global optimality. Despite this, solution methods for signomial programs are disciplined and effective. In the present work, signomial programming models are proposed for optimal preliminary sizing of the vertical tail, fuselage, and landing gear of a commercial aircraft with a tubeand-wing configuration. Signomial programming's relaxed formulation allows it to handle some of the key constraints in tail, fuselage, and landing gear design and therefore a significant improvement in fidelity over geometric programming models is achieved. The models are readily extensible and easily combined with other models, making them effective building blocks for a full aircraft model. A primary contribution of this work is to demonstrate signomial programming as a viable tool for multidisciplinary aircraft design optimization.

Geometric and signomial programming are emerging as promising methods for aircraft design optimization, having both been demonstrated to reliably and quickly find optimal solutions to aircraft design problems. To better understand how they perform compared with more conventional alternatives, this work presents a direct comparison with a general nonlinear programming approach. The crux of geometric programming, and by extension signomial programming, is in the formulation and the logarithmic transformation that makes the problem convex. Starting with the same problem formulation we assess the difference in speed and effectiveness achieved by performing the transformation. Two relatively small aircraft design problems, one a geometric program, the other a signomial program, are solved using the interior point and sequential quadratic programming alogrithms implemented in MATLAB's fmincon function, both with and without performing the log transformation first. Results show that performing the log transformation consistently yields the same optimal solution, independent of initial guess, whereas applying a general nonlinear programming technique directly to the un-transformed problem, at best, takes significantly longer and, at worst, terminates at an infeasible solution. The results also show that the general approach is highly sensitive to the initial guess whereas geometric and signomial programming approaches are not.

The aircraft-induced climate impact has drawn attention in recent years. Aviation operations affect the environment mainly through the release of carbon-dioxide, nitrogen-oxides, and by the formation of contrails. Recent research has shown that altering trajectories can reduce aviation environmental cost by reducing Absolute Global Temperature Change Potential, a climate assessment metric that adapts a linear system for modeling the global temperature response to aviation emissions and contrails. However, these methods will increase fuel consumption that leads to higher fuel costs imposed on airlines. The goal of this work is to identify flights for which the environmental cost of climate impact reduction outweighs the increase in operational cost on an individual aircraft basis. Environmental cost is quantified using the monetary social cost of carbon. The increase in operational cost is considering cost of additional fuel usage only. For this paper, an algorithm has been developed that modifies the trajectories of flights to evaluate the effect of environmental cost and operational cost of flights in the United States National Airspace System. The algorithm identifies flights for which the environmental cost of climate impact can be reduced and modifies their trajectories to achieve maximum environmental net benefit, which is the difference between reduction in environmental cost and additional operational cost. The result shows on a selected day, 16% of the flights among eight major airlines, or 2,043 flights, can achieve environmental net benefit using weather forecast data, resulting in net benefit of around $500,000. The result also suggests that the long-haul flights would be better candidates for cost-efficient climate impact reduction than the short haul flights. The algorithm will help to identify the characteristics of flights that are capable of applying cost-efficient climate impact reduction strategy.

Hyperloop system design is a uniquely coupled problem because it involves the simultaneous design of a complex, high-performance vehicle and its accompanying infrastructure. In the clean-sheet design of this new mode of highspeed mass transportation there is an excellent opportunity for the application of rigorous system optimization techniques. This work presents a system optimization tool, HOPS, that has been adopted as a central component of the Virgin Hyperloop design process. We discuss the choice of objective function, the use of a convex optimization technique called geometric programming, and the level of modeling fidelity that has allowed us to capture the system's many intertwined, and often recursive, design relationships. We also highlight the ways in which the tool has been used. Because organizational confidence in a model is as vital as its technical merit, we close with discussion of the measures taken to build stakeholder trust in HOPS.

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