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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Book•
06 Jun 2005TL;DR: This text provides both an in-depth tutorial on the theory, algorithms, and modeling methods of GP, and a comprehensive survey on the applications of GP to the study of communication systems.
Abstract: Geometric Programming (GP) is a class of nonlinear optimization with many useful theoretical and computational properties. Over the last few years, GP has been used to solve a variety of problems in the analysis and design of communication systems in several 'layers' in the communication network architecture, including information theory problems, signal processing algorithms, basic queuing system optimization, many network resource allocation problems such as power control and congestion control, and cross-layer design. We also start to understand why, in addition to how, GP can be applied to a surprisingly wide range of problems in communication systems. These applications have in turn spurred new research activities on GP, especially generalizations of GP formulations and development of distributed algorithms to solve GP in a network. This text provides both an in-depth tutorial on the theory, algorithms, and modeling methods of GP, and a comprehensive survey on the applications of GP to the study of communication systems.
510 citations
TL;DR: In this paper, it was shown that if a finite solution to the problem exists, only one linear programming problem must be solved, and this is because the denominator cannot have two different signs in the feasible region except in ways which are not of practical importance.
Abstract: Charnes and Cooper [1] showed that a linear programming problem with a linear fractional objective function could be solved by solving at most two ordinary linear programming problems. In addition, they showed that where it is known a priori that the denominator of the objective function has a unique sign in the feasible region, only one problem need be solved. In the present note it is shown that if a finite solution to the problem exists, only one linear programming problem must be solved. This is because the denominator cannot have two different signs in the feasible region, except in ways which are not of practical importance.
507 citations
TL;DR: The improved genetic algorithm (GA) formulation for pipe network optimization has been developed and found a solution for the New York tunriels problem which is the lowest-cost feasible discrete size solution yet presented in the literature.
Abstract: An improved genetic algorithm (GA) formulation for pipe network optimization has been developed. The new GA uses variable power scaling of the fitness function. The exponent introduced into the fitness function is increased in magnitude as the GA computer run proceeds. In addition to the more commonly used bitwise mutation operator, an adjacency or creeping mutation operator is introduced. Finally, Gray codes rather than binary codes are used to represent the set of decision variables which make up i the pipe network design. Results are presented comparing the performance of the traditional or simple GA formulation and the improved GA formulation for the New York City tunnels problem. The case study results indicate the improved GA performs significantly better than the simple GA. In addition, the improved GA performs better than previously used traditional optimization methods such as linear, dynamic, and nonlinear programming methods and an enumerative search method. The improved GA found a solution for the New York tunriels problem which is the lowest-cost feasible discrete size solution yet presented in the literature.
507 citations
Book•
01 Jan 1991
TL;DR: In this paper, the Lovasz Extensions of Submodular Functions are extended to include nonlinear weight functions and linear weight functions with continuous variables, and a Decomposition Algorithm is proposed.
Abstract: Introduction. 1. Mathematical Preliminaries. Submodular Systems and Base Polyhedra. 2. From Matroids to Submodular Systems. Matroids. Polymatroids. Submodular Systems. 3. Submodular Systems and Base Polyhedra. Fundamental Operations on Submodular Systems. Greedy Algorithm. Structures of Base Polyhedra. Intersecting- and Crossing-Submodular Functions. Related Polyhedra. Submodular Systems of Network Type. Neoflows. 4. The Intersection Problem. The Intersection Theorem. The Discrete Separation Theorem. The Common Base Problem. 5. Neoflows. The Equivalence of the Neoflow Problems. Feasibility for Submodular Flows. Optimality for Submodular Flows. Algorithms for Neoflows. Matroid Optimization. Submodular Analysis. 6. Submodular Functions and Convexity. Conjugate Functions and a Fenchel-Type Min-Max Theorem for Submodular and Supermodular Functions. Subgradients of Submodular Functions. The Lovasz Extensions of Submodular Functions. 7. Submodular Programs. Submodular Programs - Unconstrained Optimization. Submodular Programs - Constrained Optimization. Nonlinear Optimization with Submodular Constraints. 8. Separable Convex Optimization. Optimality Conditions. A Decomposition Algorithm. Discrete Optimization. 9. The Lexicographically Optimal Base Problem. Nonlinear Weight Functions. Linear Weight Functions. 10. The Weighted Max-Min and Min-Max Problems. Continuous Variables. Discrete Variables. 11. The Fair Resource Allocation Problem. Continuous Variables. Discrete Variables. 12. The Neoflow Problem with a Separable Convex Cost Function. References. Index.
505 citations
TL;DR: This work reviews some of the key developments in the modern era of interior-point methods, including comments on both the complexity theory and practical algorithms for linear programming, semi-definite programming, monotone linear complementarity, and convex programming over sets that can be characterized by self-concordant barrier functions.
505 citations