Resizing: Predicting a New Geometric Programming Optimum as Coefficients Vary

1984 ◽  
Vol 106 (1) ◽  
pp. 11-16
Author(s):  
D. J. Wilde
Keyword(s):  
Design Problem ◽  
Geometric Programming ◽  
Optimal Solution ◽  
Matrix Equations ◽  
Dual Variable ◽  
Method Of Least Squares ◽  
Satisfactory Solution ◽  
Geometric Program ◽  
The Matrix ◽  
Dual Variables

Imagine that an optimal solution is available for a constrained geometric program, and suppose one wishes a satisfactory solution for greatly different values of some of the coefficients. An estimate can be constructed by using the values of the dual variables for the old optimum in the invariance conditions for the new problem. Although these are inconsistent except at the precise optimum, a unique primal solution can easily be generated from them by the method of least squares with individual equations weighted by the value of the corresponding dual variable. The matrix equations for these linear operations are derived and applied to a well-known merchant fleet design problem. The predictions are remarkably accurate—1.4 percent error for a 100 percent coefficient change.

Entropy Maximization and Geometric Programming

1977 ◽  
Vol 9 (4) ◽  
pp. 419-427 ◽  
Author(s):  
J J Dinkel ◽  
G A Kochenberger ◽  
S-N Wong
Keyword(s):  
Sensitivity Analysis ◽  
Convex Function ◽  
Geometric Programming ◽  
Optimal Solution ◽  
Entropy Model ◽  
Entropy Maximization ◽  
Geometric Programs ◽  
Geometric Program

This paper shows the equivalence of entropy-maximization models to geometric programs. As a result we derive a dual geometric program which consists of the minimization of an unconstrained convex function. We develop the necessary duality equivalencies between the two dual programs and show the computational attractiveness of our approach. We also develop some characterizations of the optimal solution of the entropy model which have important implications with regard to postoptimal or sensitivity analysis.

scholarly journals Improving ADMMs for solving doubly nonnegative programs through dual factorization

4OR ◽  
2020 ◽  
Author(s):  
Martina Cerulli ◽  
Marianna De Santis ◽  
Elisabeth Gaar ◽  
Angelika Wiegele
Keyword(s):  
Objective Function ◽  
Large Scale ◽  
Objective Function Value ◽  
Optimal Solution ◽  
Stable Set ◽  
Dual Variable ◽  
Post Processing ◽  
Semidefinite Programs ◽  
The Matrix ◽  
The Impact

Abstract Alternating direction methods of multipliers (ADMMs) are popular approaches to handle large scale semidefinite programs that gained attention during the past decade. In this paper, we focus on solving doubly nonnegative programs (DNN), which are semidefinite programs where the elements of the matrix variable are constrained to be nonnegative. Starting from two algorithms already proposed in the literature on conic programming, we introduce two new ADMMs by employing a factorization of the dual variable. It is well known that first order methods are not suitable to compute high precision optimal solutions, however an optimal solution of moderate precision often suffices to get high quality lower bounds on the primal optimal objective function value. We present methods to obtain such bounds by either perturbing the dual objective function value or by constructing a dual feasible solution from a dual approximate optimal solution. Both procedures can be used as a post-processing phase in our ADMMs. Numerical results for DNNs that are relaxations of the stable set problem are presented. They show the impact of using the factorization of the dual variable in order to improve the progress towards the optimal solution within an iteration of the ADMM. This decreases the number of iterations as well as the CPU time to solve the DNN to a given precision. The experiments also demonstrate that within a computationally cheap post-processing, we can compute bounds that are close to the optimal value even if the DNN was solved to moderate precision only. This makes ADMMs applicable also within a branch-and-bound algorithm.

scholarly journals A New Iterative Algorithm for Solving a Class of Matrix Nearness Problem

2012 ◽  
Vol 2012 ◽  
pp. 1-6
Author(s):  
Xuefeng Duan ◽  
Chunmei Li
Keyword(s):  
Iterative Algorithm ◽  
Projection Algorithm ◽  
Frobenius Norm ◽  
Matrix Equations ◽  
Von Neumann ◽  
Numerical Examples ◽  
Matrix Nearness Problem ◽  
Least Frobenius Norm Solution ◽  
Alternating Projection ◽  
The Matrix

Based on the alternating projection algorithm, which was proposed by Von Neumann to treat the problem of finding the projection of a given point onto the intersection of two closed subspaces, we propose a new iterative algorithm to solve the matrix nearness problem associated with the matrix equations AXB=E, CXD=F, which arises frequently in experimental design. If we choose the initial iterative matrix X0=0, the least Frobenius norm solution of these matrix equations is obtained. Numerical examples show that the new algorithm is feasible and effective.

scholarly journals An Iterative Algorithm for the Generalized Reflexive Solutions of the Generalized Coupled Sylvester Matrix Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-28 ◽  
Author(s):  
Feng Yin ◽  
Guang-Xin Huang
Keyword(s):  
Iterative Algorithm ◽  
Special Kind ◽  
Frobenius Norm ◽  
Matrix Equations ◽  
Sylvester Matrix ◽  
Special Cases ◽  
The Matrix ◽  
Solution Pair ◽  
Sylvester Matrix Equations ◽  
Matrix Pair

An iterative algorithm is constructed to solve the generalized coupled Sylvester matrix equations(AXB-CYD,EXF-GYH)=(M,N), which includes Sylvester and Lyapunov matrix equations as special cases, over generalized reflexive matricesXandY. When the matrix equations are consistent, for any initial generalized reflexive matrix pair[X1,Y1], the generalized reflexive solutions can be obtained by the iterative algorithm within finite iterative steps in the absence of round-off errors, and the least Frobenius norm generalized reflexive solutions can be obtained by choosing a special kind of initial matrix pair. The unique optimal approximation generalized reflexive solution pair[X̂,Ŷ]to a given matrix pair[X0,Y0]in Frobenius norm can be derived by finding the least-norm generalized reflexive solution pair[X̃*,Ỹ*]of a new corresponding generalized coupled Sylvester matrix equation pair(AX̃B-CỸD,EX̃F-GỸH)=(M̃,Ñ), whereM̃=M-AX0B+CY0D,Ñ=N-EX0F+GY0H. Several numerical examples are given to show the effectiveness of the presented iterative algorithm.

scholarly journals A note on the Ritz method with an application to overtone stellar pulsation theory

1968 ◽  
Vol 8 (2) ◽  
pp. 275-286 ◽  
Author(s):  
A. L. Andrew
Keyword(s):  
Numerical Solution ◽  
Eigenvalue Problems ◽  
Ritz Method ◽  
Linear Operators ◽  
Matrix Equations ◽  
Infinite Dimensional ◽  
Stellar Pulsation ◽  
Finite Matrix ◽  
The Matrix ◽  
Matrix Eigenvalue Problems

The Ritz method reduces eigenvalue problems involving linear operators on infinite dimensional spaces to finite matrix eigenvalue problems. This paper shows that for a certain class of linear operators it is possible to choose the coordinate functions so that numerical solution of the matrix equations is considerably simplified, especially when the matrices are large. The method is applied to the problem of overtone pulsations of stars.

scholarly journals Numerical Solution of Duffing Equation by Using an Improved Taylor Matrix Method

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Berna Bülbül ◽  
Mehmet Sezer
Keyword(s):  
Matrix Method ◽  
Numerical Approach ◽  
Duffing Equation ◽  
Matrix Equations ◽  
Algebraic Equations ◽  
Solution Function ◽  
Numerical Examples ◽  
Collocation Points ◽  
Nonlinear Algebraic Equations ◽  
The Matrix

We have suggested a numerical approach, which is based on an improved Taylor matrix method, for solving Duffing differential equations. The method is based on the approximation by the truncated Taylor series about center zero. Duffing equation and conditions are transformed into the matrix equations, which corresponds to a system of nonlinear algebraic equations with the unknown coefficients, via collocation points. Combining these matrix equations and then solving the system yield the unknown coefficients of the solution function. Numerical examples are included to demonstrate the validity and the applicability of the technique. The results show the efficiency and the accuracy of the present work. Also, the method can be easily applied to engineering and science problems.

On Solutions of Inverse Problem for Hermitian Generalized Hamiltonian Matrices

2013 ◽  
Vol 860-863 ◽  
pp. 2727-2731
Author(s):  
Kai Fu Liang ◽  
Ming Jun Li ◽  
Ze Lin Zhu
Keyword(s):  
Inverse Problem ◽  
Inverse Problems ◽  
Design Automation ◽  
Sufficient Conditions ◽  
Matrix Equations ◽  
Complex Matrix ◽  
Linear Quadratic ◽  
Real Representation ◽  
The Matrix ◽  
Necessary And Sufficient

Hamiltonian matrices have many applications to design automation and autocontrol, in particular in the linear-quadratic autocontrol problem. This paper studies the inverse problems of generalized Hamiltonian matrices for matrix equations. By real representation of complex matrix, we give the necessary and sufficient conditions for the existence of a Hermitian generalized Hamiltonian solutions to the matrix equations, and then derive the representation of the general solutions.

scholarly journals Regional Logistics Network Design in Mitigating Truck Flow-Caused Congestion Problems

2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Mi Gan ◽  
Xinyuan Li ◽  
Fadong Zhang ◽  
Zhenggang He
Keyword(s):  
Network Design ◽  
Traffic Congestion ◽  
Design Problem ◽  
Transportation Network ◽  
Optimal Solution ◽  
Service Level ◽  
Initial Distribution ◽  
Urban Traffic ◽  
Logistics Network ◽  
Regional Logistics

Truck flow plays a vital role in urban traffic congestion and has a significant influence on cities. In this study, we develop a novel model for solving regional logistics network (RLN) design problems considering the traffic status of the background transportation network. The models determine not only the facility location, initial distribution planning, roadway construction, and expansion decisions but also offer an optimal solution to the logistics network service level and truck-type selections. We first analyze the relationship between the urban transportation network and the RLN design problem using real truck data and traffic flow status in a typical city. Then, we develop the uncover degree function (UDF), which reflects the service degree of the RLN and formulates based on an impedance function. Subsequently, the integrated logistics network design models are proposed. We model the RLN design problem as a minimal cost problem and design double-layer Lagrangian relaxation heuristics algorithms to solve the model problems. Through experiments with data from the six-node problem and Sioux-Falls network, the effectiveness of the models and algorithms is verified. This study contributes to the planning of regional logistics networks while mitigating traffic congestion caused by truck flow.

Haptic Wrists: An Alternative Design Strategy Based on User Perception

2009 ◽  
Vol 9 (1) ◽  
Author(s):  
Javier Martín ◽  
Joan Savall ◽  
Iñaki Díaz ◽  
Josune Hernantes ◽  
Diego Borro
Keyword(s):  
Collision Detection ◽  
Design Problem ◽  
Force Feedback ◽  
Mechanical Design ◽  
Design Strategy ◽  
User Perception ◽  
Satisfactory Solution ◽  
Alternative Design ◽  
Three Degree Of Freedom ◽  
Mechanical Design Problem

A new three degree-of-freedom (3DOF) torque feedback wrist is being developed to be added to an existing 3DOF force feedback haptic device. It is difficult to find a satisfactory solution to the mechanical design problem, mainly because of the required large rotational workspace and severe weight constraints. This work proposes an alternative design strategy based on user perception, which allows simplification of the mechanics. The proposed approach consists of substituting the last rotational DOF of the wrist with a pseudohaptic DOF. Thanks to specially designed visuotactile cues, the pseudohaptic DOF is integrated with the active DOF into the same device, being able to generate free motion and collision detection perception to the user. This approach provides for simpler kinematics, lightweight designs, lower inertias, and less friction, which are key advantages for the inclusion of torque feedback into force feedback devices.

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