Nonlinear programming

About: Nonlinear programming is a research topic. Over the lifetime, 19486 publications have been published within this topic receiving 656602 citations. The topic is also known as: non-linear programming & NLP.

Optimization Strategies for the Vulnerability Analysis of the Electric Power Grid

Abstract: Identifying small groups of lines, whose removal would cause a severe blackout, is critical for the secure operation of the electric power grid. We show how power grid vulnerability analysis can be studied as a bilevel mixed integer nonlinear programming problem. Our analysis reveals a special structure in the formulation that can be exploited to avoid nonlinearity and approximate the original problem as a pure combinatorial problem. The key new observation behind our analysis is the correspondence between the Jacobian matrix (a representation of the feasibility boundary of the equations that describe the flow of power in the network) and the Laplacian matrix in spectral graph theory (a representation of the graph of the power grid). The reduced combinatorial problem is known as the network inhibition problem, for which we present a mixed integer linear programming formulation. Our experiments on benchmark power grids show that the reduced combinatorial model provides an accurate approximation, to enable vulnerability analyses of real-sized problems with more than 16,520 power lines.

Spacecraft Trajectory Optimization

Abstract: 1. The problem of spacecraft trajectory optimization Bruce A. Conway 2. Primer vector theory and application John E. Prussing 3. Spacecraft trajectory optimization using direct transcription and nonlinear programming Stephen W. Paris and Bruce A. Conway 4. Elements of a software system for spacecraft trajectory optimization Cesar Ocampo 5. Low-thrust trajectory optimization using orbital averaging and control parameterization Craig A. Kluever 6. Analytic representation of optimal low-thrust transfer in circular orbit Jean A. Kechichian 7. Global optimization and space pruning for spacecraft trajectory design Dario Izzo 8. Incremental techniques for global space trajectory design Massimiliano Vasile and Matteo Ceriotti 9. Optimal low-thrust trajectories using stable manifolds Christopher Martin and Bruce A. Conway.

The ESA NLP Solver WORHP

Abstract: We Optimize Really Huge Problems (WORHP) is a solver for large-scale, sparse, nonlinear optimization problems with millions of variables and constraints. Convexity is not required, but some smoothness and regularity assumptions are necessary for the underlying theory and the algorithms based on it. WORHP has been designed from its core foundations as a sparse sequential quadratic programming (SQP) / interior-point (IP) method; it includes efficient routines for computing sparse derivatives by applying graph-coloring methods to finite differences, structure-preserving sparse named after Broyden, Fletcher, Goldfarb and Shanno (BFGS) update techniques for Hessian approximations, and sparse linear algebra. Furthermore it is based on reverse communication, which offers an unprecedented level of interaction between user and nonlinear programming (NLP) solver. It was chosen by ESA as the European NLP solver on the basis of its high robustness and its application-driven design and development philosophy. Two large-scale optimization problems from space applications that demonstrate the robustness of the solver complement the cursory description of general NLP methods and some WORHP implementation details.

A Mathematical Programming Approach for the Solution of the Railway Yield Management Problem

Abstract: Railway passenger transportation plays a fundamental role in Europe, particularly in view of the growing number of trains offering valuable services such as high speed travel, high comfort, etc. Hence, it is advantageous to submit seat inventories to a Yield Management system to get the maximum revenue. We consider a deterministic linear programming model and a probabilistic nonlinear programming model for the network problem with non-nested seat allocation. A first comparative analysis of the computational results obtained by the two models, both in terms of the overall expected revenue and in terms of CPU time, is carried out. Furthermore, we describe a new nonlinear algorithm for the solution of the probabilistic nonlinear programming model that exploits the structure of the optimization problem. The numerical results obtained on a set of real data show that, for this class of problems, this algorithm is more efficient than other standard algorithms for nonlinear programming problems.

Simplicial decomposition in nonlinear programming algorithms

Abstract: Simplicial decomposition is a special version of the Dantzig—Wolfe decomposition principle, based on Caratheodory's theorem. The associated class of algorithms has the following features and advantages: The master and the subprogram are constructed without dual variables; the methods remain therefore well-defined for non-concave objective functions, and pseudo-concavity suffices for convergence to global maxima. The subprogram produces affinely independent sets of feasible generator points defining simplices, which the master program keeps minimal by dropping redundant generator points and finding maximizers in the relative interiors of the resulting subsimplices. The use of parallel subspaces allows the direct application of any unrestricted optimization method in the master program; thus the best unconstrained procedure for any type of objective function can be used to find constrained maximizers for it.