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.

Entropic Proximal Mappings with Applications to Nonlinear Programming

Abstract: We introduce a family of new transforms based on imitating the proximal mapping of Moreau and the associated Moreau-Yosida proximal approximation of a function. The transforms are constructed in terms of the AÂ†-divergence functional a generalization of the relative entropy and of Bregman's measure of distance. An analogue of Moreau's theorem associated with these entropy-like distances is proved. We show that the resulting Entropic Proximal Maps share properties similar to the proximal mapping and provide a fairly general framework for constructing approximation and smoothing schemes for optimization problems. Applications of the results to the construction of generalized augmented Lagrangians for nonlinear programs and the minimax problem are presented.

Recent advances in trust region algorithms

Abstract: Trust region methods are a class of numerical methods for optimization. Unlike line search type methods where a line search is carried out in each iteration, trust region methods compute a trial step by solving a trust region subproblem where a model function is minimized within a trust region. Due to the trust region constraint, nonconvex models can be used in trust region subproblems, and trust region algorithms can be applied to nonconvex and ill-conditioned problems. Normally it is easier to establish the global convergence of a trust region algorithm than that of its line search counterpart. In the paper, we review recent results on trust region methods for unconstrained optimization, constrained optimization, nonlinear equations and nonlinear least squares, nonsmooth optimization and optimization without derivatives. Results on trust region subproblems and regularization methods are also discussed.

Bacteria Foraging-Based Solution to Optimize Both Real Power Loss and Voltage Stability Limit

Abstract: Optimal location and control of a unified power flow controller (UPFC) along with transformer taps are tuned with a view to simultaneously optimize the real power losses and voltage stability limit (VSL) of a mesh power network. This issue is formulated as a nonlinear equality and inequality constrained optimization problem with an objective function incorporating both the real power loss and VSL. A new evolutionary algorithm known as bacteria foraging is applied for solving the multiobjective multivariable problem, with the UPFC location, its series injected voltage, and the transformer tap positions as the variables. For a single objective of only real power loss, the same problem is also solved with interior point successive linearization program (IPSLP) technique using the LINPROG command of MATLAB. A comparison between the two suggests the superiority of the proposed algorithm. A cost effectiveness analysis of UPFC installation vis-agrave-vis loss reduction is carried out to establish the benefit of investment in a UPFC

Solving Nonconvex Optimal Control Problems by Convex Optimization

Abstract: Motivated by aerospace applications, this paper presents a methodology to use second-order cone programming to solve nonconvex optimal control problems The nonconvexity arises from the presence of concave state inequality constraints and nonlinear terminal equality constraints The development relies on a solution paradigm, in which the concave inequality constraints are approximated by successive linearization Analysis is performed to establish the guaranteed satisfaction of the original inequality constraints, the existence of the successive solutions, and the equivalence of the solution of the original problem to the converged successive solution These results lead to a rigorous proof of the convergence of the successive solutions under appropriate conditions as well as nonconservativeness of the converged solution The nonlinear equality constraints are treated in a two-step procedure in which the constraints are first approximated by first-order expansions, then compensated by second-order correct

The nonlinear programming method of Wilson, Han, and Powell with an augmented Lagrangian type line search function

Abstract: The paper represents an outcome of an extensive comparative study of nonlinear optimization algorithms. This study indicates that quadratic approximation methods which are characterized by solving a sequence of quadratic subproblems recursively, belong to the most efficient and reliable nonlinear programming algorithms available at present. The purpose of this paper is to analyse the theoretical convergence properties and to investigate the numerical performance in more detail. In Part 1, the exactL 1-penalty function of Han and Powell is replaced by a differentiable augmented Lagrange function for the line search computation to the able to prove the global convergence and to show that the steplength one is chosen in the neighbourhood of a solution. In Part 2, the quadratic subproblem is exchanged by a linear least squares problem to improve the efficiency, and to test the dependence of the performance from different solution methods for the quadratic or least squares subproblems.