Algorithms for the continuous nonlinear resource allocation problem---new implementations and numerical studies

Patriksson (2008) provided a then up-to-date survey on the continuous,separable, differentiable and convex resource allocation problem with a single resource constraint. Since the publication of that paper the interest in the problem has grown: several new applications have arisen where the problem at hand constitutes a subproblem, and several new algorithms have been developed for its efficient solution. This paper therefore serves three purposes. First, it provides an up-to-date extension of the survey of the literature of the field, complementing the survey in Patriksson (2008) with more then 20 books and articles. Second, it contributes improvements of some of these algorithms, in particular with an improvement of the pegging (that is, variable fixing) process in the relaxation algorithm, and an improved means to evaluate subsolutions. Third, it numerically evaluates several relaxation (primal) and breakpoint (dual) algorithms, incorporating a variety of pegging strategies, as well as a quasi-Newton method. Our conclusion is that our modification of the relaxation algorithm performs the best. At least for problem sizes up to 30 million variables the practical time complexity for the breakpoint and relaxation algorithms is linear

OPTIMASS: A Package for the Minimization of Kinematic Mass Functions with Constraints

Reconstructed mass variables, such as $M_2$, $M_{2C}$, $M_T^\star$, and $M_{T2}^W$, play an essential role in searches for new physics at hadron colliders. The calculation of these variables generally involves constrained minimization in a large parameter space, which is numerically challenging. We provide a C++ code, OPTIMASS, which interfaces with the MINUIT library to perform this constrained minimization using the Augmented Lagrangian Method. The code can be applied to arbitrarily general event topologies and thus allows the user to significantly extend the existing set of kinematic variables. We describe this code and its physics motivation, and demonstrate its use in the analysis of the fully leptonic decay of pair-produced top quarks using the $M_2$ variables.Comment: 39 pages, 12 figures, (1) minor revision in section 3, (2) figure added in section 4.3, (3) reference added and (4) matched with published versio

Structural optimization of large structural systems by optimality criteria methods

The fundamental concepts of the optimality criteria method of structural optimization are presented. The effect of the separability properties of the objective and constraint functions on the optimality criteria expressions is emphasized. The single constraint case is treated first, followed by the multiple constraint case with a more complex evaluation of the Lagrange multipliers. Examples illustrate the efficiency of the method