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Geometric, Linear, and Dynamic Programming and Other Methods for Optimization
Published in Yogesh Jaluria, Design and Optimization of Thermal Systems, 2019
Linear programming is an important optimization technique that has been applied to a wide range of problems, particularly to those in mathematics, economics, industrial engineering, power transmission, and material flow. This method is applicable if the objective functions as well as the constraints are linear functions of the independent variables. The constraints may be equalities or inequalities. Since its first appearance about 70 years ago, linear programming has found increasing use due to the need to model, manage, and optimize large systems such as those concerned with manufacturing, transportation, energy, and telecommunications (Hadley, 1962; Murtagh, 1981; Dantzig, 1998; Dantzig and Thapa, 2003; Gass, 2010; Karloff, 2006; Vanderbei, 2014). A large number of efficient optimization algorithms for linear programming have been developed and are available commercially as well as in the public domain. For instance, Matlab toolboxes have software that can be easily employed to solve linear programming problems for system or process optimization.
Devising and Synthesis of Mems and Nems
Published in Sergey Edward Lyshevski, Mems and Nems, 2018
It is evident that the user needs to integrate the inequality constraints in order to solve the specific practical problems in the synthesis of MEMS and NEMS. In fact, the importance of linear programming derives by its straightforward applications and by the existence of well-developed generalpurpose techniques and computationally-efficient software for finding optimal solutions. Simplex methods, introduced 50 years ago, use the basic solutions computed by fixing the variables at their bounds to reduce the constraints Ax=b to a square system in order to solve it for unique values of the remaining variables. The basic solutions give extreme boundary points of the feasible region defined by Ax = b, x ≥ 0. Therefore, the simplex method is based on moving from one point to another along the edges of the boundary. In contrast, barrier (interior-point) methods utilize points within the interior of the feasible region. The integer linear programming requires that some or all variables are integers. Widely used general-purpose techniques for solving integer linear programming use the solutions to a series of linear programming problems to manage the search for integer solutions and to prove the optimality.
State Estimation
Published in Leonard L. Grigsby, Power System Stability and Control, 2017
Jason G. Lindquist, Danny Julian
Another solution method that addresses the state estimation problem is linear programming. Linear programming is an optimization technique that serves to minimize a linear objective function subject to a set of constraints: min{c¯Tx¯}s.t.Ax¯=b¯x¯≥0
Optimizing the Implementation of Small Modular Reactors into Ontario’s Future Energy Mix
Published in Nuclear Technology, 2023
C. Colterjohn, S. Nagasaki, Y. Fujii
A subset of mathematical programming, linear programming is one of the simplest methods of optimization. Given a set of decision variables and subject to constraints defined by linear equalities and/or inequalities, a linear objective function may be optimized to find its maximum or minimum values. A feasibility region is first created by the function, whose boundaries are defined by the linear constraints, and each point along which represents a possible solution corresponding to varying values for the decision variables. In most practical situations, the number of decision variables can grow very quickly depending on the system that is being modeled. The Simplex method is an iterative process for solving linear programming problems, where the values of the decision variables are sequentially tested until an optimal solution is found. This method involves writing the objective function in the form
Objective Functions and Infrastructure for Optimal Placement of Electrical Vehicle Charging Station: A Comprehensive Survey
Published in IETE Journal of Research, 2021
Mohammad Suhail, Iram Akhtar, Sheeraz Kirmani
Linear programming is an optimization technique for a system of linear constraints and a linear objective function. An objective function defines the quantity to be optimized, and the goal of linear programming is to find the values of the variables that maximize or minimize the objective function [46]. This technique can be used to minimize the total costs by taking into account the transportation costs, land cost and other cost as constraints. This technique can also be used to maximize the energy efficiency which is an important parameter while determining the optimal location of a charging station. In this method, at least 2 linear inequality equations are formed based on the constraints and then maximum and minimum conditions are checked in all equations and a graphical representation gives the best operating area for the operation of the charging station.
MOS Amplifier Design Methodology for Optimum Performance
Published in IETE Journal of Research, 2020
Abir J. Mondal, Paromita Bhattacharjee, Pinaki Chakraborty, Bidyut K. Bhattacharyya
Optimization is a process of providing the best possible solution for a problem with regard to a given situation. It includes maximizing or minimizing an objective function relative to some constraints, which usually extends over a range of available choices. Depending on the nature of objective function and constraints, mathematical optimization problem can be distinguished into many categories. Broadly, it can be categorized into linear programming and NLP optimization problem. With respects to only constraint equations, optimization problem can be unconstrained, linearly constrained or nonlinearly constrained. Linear programming has a strict restriction on both of its objective function and constraint equations to be in linear form. Many methods have been developed to solve linear programming problems. One of the most popular is the simplex method invented in 1947 by Dantzig, an American mathematical scientist, for solving the linear programming problems that arose in U.S. Air Force planning problems [11]. Nonlinear programming is another technique to solve an optimization problem, where an objective function f has to be minimized or maximized subject to some constraint equations gi. It is worthy to mention that either of the objective function or constraint equation can be nonlinear. Therefore, analog circuit problems having linear and nonlinear equations in terms of voltage, current or any design parameter can readily be optimized using NLP.