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About 6,905 papers for “Geometric programming”

Geometric Programming

Elmor L. Peterson, Gerhard Plenert · Wiley Encyclopedia of Electrical and Electronics Engineering · 1999

Abstract The sections in this article are An Elementary Example: the Optimal Design of a Power Line Generalized Polynomials Traditional Calculus and Numerical Approaches The Geometric Programming Approach Unconstrained Posynomial Minimization VIA Geometric Programming: The General Case An Important Example: Entropy Optimization in Information Theory, Thermodynamics, and Statistical Mechanics

Cited by 14

Geometric Programming

Jayant Rajgopal · Wiley Encyclopedia of Operations Research and Management Science · 2011

Abstract Geometric programming (GP) constitutes a special class of nonlinear optimization problems involving objective and constraint functions that are stated in the form of generalized polynomials. While the field is only about 50 years old, it has been studied extensively. This article reviews the important theoretical and computational aspects of GP and also discusses extensions to the original form that have been developed over the years.

Cited by 21

Geometric Programming

Engineering Optimization · 2009

This chapter contains sections titled: Introduction Posynomial Unconstrained Minimization Problem Solution of an Unconstrained Geometric Programming Program Using Differential Calculus Solution of an Unconstrained Geometric Programming Problem Using Arithmetic–Geometric Inequality Primal–Dual Relationship and Sufficiency Conditions in the Unconstrained Case Constrained Minimization Solution of a Constrained Geometric Programming Problem Primal and Dual Programs in the Case of Less-Than Inequalities Geometric Programming with Mixed Inequality Constraints Complementary Geometric Programming Applications of Geometric Programming References and Bibliography Review Questions Problems

Cited by 9

Geometric Programming.

Paul Willig, R. J. Duffin, Elmor L. Peterson et al. · American Mathematical Monthly · 1970

Cited by 0

Geometric Programming

Elmor L. Peterson · SIAM Review · 1976

Contrary to popular belief, geometric programming is not just a special technique for studying the very important class of posynomial (optimization) problems. It is really a very general mathematical theory that is especially useful for studying a large class of separable problems. Its practical efficacy is due mainly to the fact that many important (seemingly inseparable) problems can actually be formulated as separable geometric programming problems, by fully exploiting their linear algebraic structure. Some examples are: nonlinear network flow problems (both single-commodity and multicommodity), discrete optimal control problems with linear dynamics, optimal location problems of the generalized Fermat type, ($l_P $ constrained) $l_P $ regression problems, chemical equilibrium problems, ordinary programming problems, (quadratically constrained) quadratic programming problems, and general algebraic programming problems. The theory of geometric programming includes (i) very strong existence, uniqueness, and characterization theorems, (ii) useful parametric and economic analyses, (iii) illuminating decomposition principles, and (iv) powerful numerical solution techniques. In addition to indicating the true scope of geometric programming and its potential impact on many fields of application, this paper can serve either as a modern introduction to the subject (including an enlightening review of the original posynomial case), or as an up-tate handbook of the most fundamental aspects of the subject.

Cited by 124

Geometric Programming

R. J. Duffin, Elmor L. Peterson, Clarence Zener · IEEE Transactions on Systems Man and Cybernetics · 1974

Cited by 180

Geometric programming

Alberto Paoluzzi, Valerio Pascucci, Michele Vicentino · ACM Transactions on Graphics · 1995

This article presents a functional programming approach to geometric design with embedded polyhedral complexes. Its main goals are to show the expressive power of the language as well as its usefulness for geometric design. The language, named PLASM (the Programming LAnguage for Solid Modeling), introduces a very high level approach to “constructive” or “generative” modeling. Geometrical objects are generated by evaluating some suitable language expressions. Because generating expressions can be easily combined, the language also extends the standard variational geometry approach by supporting classes of geometric objects with varying topology and shape. The design language PLASM can be roughly considered as a geometry-oriented extension of a subset of the functional language FL. The language takes a dimension-independent approach to geometry representation and algorithms. In particular it implements an algebraic calculus over embedded polyhedra of any dimension. The generated objects are always geometrically consistent because the validity of geometry is guaranteed at a syntactical level. Such an approach allows one to use a representation scheme which is weaker than those usually adopted in solid modelers, thus encompassing a broader geometric domain, which contains solids, surfaces, and wire-frames, as well as higher-dimensional objects.

Cited by 55Open access

Geometric programming method.

Lifeng Wang (121568), Hu Liang (1776022), Yuxin Jian (19513517) et al. · Figshare · 2024

<div><p>This paper focuses on the cooperative driving challenges of connected and automated vehicles (CAVs) at single-lane roundabouts. First, a geometric path planning method is proposed for CAVs navigating a single-lane roundabout. Based on this method, a vehicle roundabout model is established. Four potential traffic scenarios for CAVs are established, and the optimal arrival times at conflict points are analyzed. By correlating the optimal arrival times at conflict points with the optimal entry times into the roundabout, the multi-vehicle coordination problem in complex intersections is simplified to a speed control issue during entry. Utilizing the principles of optimal control and Pontryagin minimization, two speed optimization strategies are proposed. Finally, MATLAB is employed for simulation analysis. The results indicate that the control strategy proposed in this paper enables the system to clearly identify potential conflicts between vehicles and implement an optimal control strategy, ensuring that vehicles can navigate the roundabout efficiently in terms of time and fuel without collisions. Additionally, the minimum time interval is established at 0.2 seconds to completely prevent vehicle collisions. In this study, the fusion problem involving two vehicles at a single conflict point is further expanded to encompass multiple vehicles at multiple conflict points. Thus, the efficient scheduling of multiple vehicles in single-lane roundabouts is realized.</p></div>

Cited by 0Open access

Conic Geometric Programming

Venkat Chandrasekaran, Parikshit Shah · arXiv (Cornell University) · 2013

We introduce and study conic geometric programs (CGPs), which are convex optimization problems that unify geometric programs (GPs) and conic optimization problems such as semidefinite programs (SDPs). A CGP consists of a linear objective function that is to be minimized subject to affine constraints, convex conic constraints, and upper bound constraints on sums of exponential and affine functions. The conic constraints are the central feature of conic programs such as SDPs, while upper bounds on combined exponential/affine functions are generalizations of the types of constraints found in GPs. The dual of a CGP involves the maximization of the negative relative entropy between two nonnegative vectors jointly, subject to affine and conic constraints on the two vectors. Although CGPs contain GPs and SDPs as special instances, computing global optima of CGPs is not much harder than solving GPs and SDPs. More broadly, the CGP framework facilitates a range of new applications that fall outside the scope of SDPs and GPs. Specifically, we demonstrate the utility of CGPs in providing solutions to problems such as permanent maximization, hitting-time estimation in dynamical systems, the computation of the capacity of channels transmitting quantum information, and robust optimization formulations of GPs.

Cited by 1Open access

Applied Geometric Programming

R. A. Cuninghame‐Green, Charles S. Beightler, Don T. Phillips · Operational Research Quarterly (1970-1977) · 1977

Cited by 318

GEOMETRIC PROGRAMMING ALGORITHM.

G Westley · 1971

The Documentation is given for an effective solution procedure for the basic geometric programming problem.The report includes a numerical test evaluation for the procedure.

Cited by 1Open access

Conic Geometric Programming

Venkat Ch, Parikshit Shahw · 2013

We introduce and study conic geometric programs (CGPs), which are convex optimiza-tion problems that unify geometric programs (GPs) and conic optimization problems such as semidefinite programs (SDPs). A CGP consists of a linear objective function that is to be minimized subject to affine constraints, convex conic constraints, and upper bound constraints on sums of exponential and affine functions. The conic constraints are the central feature of conic programs such as SDPs, while upper bounds on combined exponential/affine functions are generalizations of the types of constraints found in GPs. The dual of a CGP involves the maximization of the negative relative entropy between two nonnegative vectors jointly, subject to affine and conic constraints on the two vectors. Although CGPs contain GPs and SDPs as special instances, computing global optima of CGPs is not much harder than solving GPs and SDPs. More broadly, the CGP framework facilitates a range of new applications that fall out-side the scope of SDPs and GPs. Specifically, we demonstrate the utility of CGPs in providing solutions to problems such as permanent maximization, hitting-time estimation in dynamical systems, the computation of the capacity of channels transmitting quantum information, and robust optimization formulations of GPs.

Cited by 0