Linear approximation

About: Linear approximation is a research topic. Over the lifetime, 3901 publications have been published within this topic receiving 74764 citations.

A logarithmic barrier cutting plane method for convex programming

Abstract: The paper presents a logarithmic barrier cutting plane algorithm for convex (possibly non-smooth, semi-infinite) programming. Most cutting plane methods, like that of Kelley, and Cheney and Goldstein, solve a linear approximation (localization) of the problem and then generate an additional cut to remove the linear program's optimal point. Other methods, like the “central cutting” plane methods of Elzinga-Moore and Goffin-Vial, calculate a center of the linear approximation and then adjust the level of the objective, or separate the current center from the feasible set. In contrast to these existing techniques, we develop a method which does not solve the linear relaxations to optimality, but rather stays in the interior of the feasible set. The iterates follow the central path of a linear relaxation, until the current iterate either leaves the feasible set or is too close to the boundary. When this occurs, a new cut is generated and the algorithm iterates. We use the tools developed by den Hertog, Roos and Terlaky to analyze the effect of adding and deleting constraints in long-step logarithmic barrier methods for linear programming. Finally, implementation issues and computational results are presented. The test problems come from the class of numerically difficult convex geometric and semi-infinite programming problems.

A fast tour design method using non-tangent v-infinity leveraging transfer

Abstract: The announced missions to the Saturn and Jupiter systems renewed the space community interest in simple design methods for gravity assist tours at planetary moons A key element in such trajectories are the V-Infinity Leveraging Transfers (VILT) which link simple impulsive maneuvers with two consecutive gravity assists at the same moon VILTs typically include a tangent impulsive maneuver close to an apse location, yielding to a desired change in the excess velocity relative to the moon In this paper we study the VILT solution space and derive a linear approximation which greatly simplifies the computation of the transfers, and is amenable to broad global searches Using this approximation, Tisserand graphs, and heuristic optimization procedure we introduce a fast design method for multiple-VILT tours We use this method to design a trajectory from a highly eccentric orbit around Saturn to a 200-km science orbit at Enceladus The trajectory is then recomputed removing the linear approximation, showing a Δv change of <4% The trajectory is 27 years long and comprises 52 gravity assists at Titan, Rhea, Dione, Tethys, and Enceladus, and several deterministic maneuvers Total Δv is only 445 m/s, including the Enceladus orbit insertion, almost 10 times better then the 39 km/s of the Enceladus orbit insertion from the Titan–Enceladus Hohmann transfer The new method and demonstrated results enable a new class of missions that tour and ultimately orbit small mass moons Such missions were previously considered infeasible due to flight time and Δv constraints

Inference for best linear approximations to set identified functions

Abstract: This paper provides inference methods for best linear approximations to functions which are known to lie within a band. It extends the partial identification literature by allowing the upper and lower functions defining the band to be any functions, including ones carrying an index, which can be estimated parametrically or non-parametrically. The identification region of the parameters of the best linear approximation is characterized via its support function, and limit theory is developed for the latter. We prove that the support function approximately converges to a Gaussian process and establish validity of the Bayesian bootstrap. The paper nests as special cases the canonical examples in the literature: mean regression with interval valued outcome data and interval valued regressor data. Because the bounds may carry an index, the paper covers problems beyond mean regression; the framework is extremely versatile. Applications include quantile and distribution regression with interval valued data, sample selection problems, as well as mean, quantile, and distribution treatment effects. Moreover, the framework can account for the availability of instruments. An application is carried out, studying female labor force participation along the lines of Mulligan and Rubinstein (2008).