Topic
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.
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TL;DR: Integer programming and network flow-based lower-bounding methods that can solve moderately large instances of the WTA problem optimally and obtain almost optimal solutions of fairly large instances within a few seconds are suggested.
Abstract: The weapon-target assignment (WTA) problem is a fundamental problem arising in defense-related applications of operations research. This problem consists of optimally assigning n weapons to m targets so that the total expected survival value of the targets after all the engagements is minimal. The WTA problem can be formulated as a nonlinear integer programming problem and is known to be NP-complete. No exact methods exist for the WTA problem that can solve even small-size problems (for example, with 20 weapons and 20 targets). Although several heuristic methods have been proposed to solve the WTA problem, due to the absence of exact methods, no estimates are available on the quality of solutions produced by such heuristics. In this paper, we suggest integer programming and network flow-based lower-bounding methods that we obtain using a branch-and-bound algorithm for the WTA problem. We also propose a network flow-based construction heuristic and a very large-scale neighborhood (VLSN) search algorithm. We present computational results of our algorithms, which indicate that we can solve moderately large instances (up to 80 weapons and 80 targets) of the WTA problem optimally and obtain almost optimal solutions of fairly large instances (up to 200 weapons and 200 targets) within a few seconds.
200 citations
TL;DR: A trust region algorithm for equality constrained optimization is proposed that employs a differentiable exact penalty function that under certain conditions global convergence and local superlinear convergence results are proved.
Abstract: A trust region algorithm for equality constrained optimization is proposed that employs a differentiable exact penalty function. Under certain conditions global convergence and local superlinear convergence results are proved.
200 citations
TL;DR: The method is based on combining, modifying and extending the nonsmooth optimization work of Wolfe, Leraarechal, Feuer, Poljak and Merrill and can be thought of as a generalized reset conjugate gradient algorithm.
Abstract: We present an implementable algorithm for solving constrained optmization problems defined by functions that are not everywhere differenliable. The method is based on combining, modifying and extending the nonsmooth optimization work of Wolfe, Leraarechal, Feuer, Poljak and Merrill. It can be thought of as a generalized reset conjugate gradient algorithm. We also introduce the class of weakly upper semismooth functions. These functions are locally Lipschitz and have a semicontinuous relationship between their generalized gradient sets and their directional derivatives. The algorithm is shown to converge to stationary points of the optimization problem if the objective and constraint functions are weakly upper semismooth. Such points are optimal points if the problem functions are also semiconvex and a constraint qualification is satisfied. Under stronger convexity assumptions, bounds on the deviation from optimally of the algorithm iterates are given.
199 citations
TL;DR: In this paper a portfolio selection model which is based on Markowitz's portfolio selection problem including three of the most important limitations is considered and the results can lead Markowitz’s model to a more practical one.
Abstract: Heuristic algorithms strengthen researchers to solve more complex and combinatorial problems in a reasonable time. Markowitz's Mean-Variance portfolio selection model is one of those aforesaid problems. Actually, Markowitz's model is a nonlinear (quadratic) programming problem which has been solved by a variety of heuristic and non-heuristic techniques. In this paper a portfolio selection model which is based on Markowitz's portfolio selection problem including three of the most important limitations is considered. The results can lead Markowitz's model to a more practical one. Minimum transaction lots, cardinality constraints (both of which have been presented before in other researches) and market (sector) capitalization (which is proposed in this research for the first time as a constraint for Markowitz model), are considered in extended model. No study has ever proposed and solved this expanded model. To solve this mixed-integer nonlinear programming (NP-Hard), a corresponding genetic algorithm (GA) is utilized. Computational study is performed in two main parts; first, verifying and validating proposed GA and second, studying the applicability of presented model using large scale problems.
199 citations
TL;DR: Some properties of regularized and penalized nonlinear programming formulations of mathematical programs with equilibrium constraints (MPECs) are described, and estimates are obtained for the distance of these solutions to the MPEC solution under various assumptions.
Abstract: Some properties of regularized and penalized nonlinear programming formulations of mathematical programs with equilibrium constraints (MPECs) are described. The focus is on the properties of these formulations near a local solution of the MPEC at which strong stationarity and a second-order sufficient condition are satisfied. In the regularized formulations, the complementarity condition is replaced by a constraint involving a positive parameter that can be decreased to zero. In the penalized formulation, the complementarity constraint appears as a penalty term in the objective. The existence and uniqueness of solutions for these formulations are investigated, and estimates are obtained for the distance of these solutions to the MPEC solution under various assumptions.
199 citations