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Operations Research
Published in Paul L. Goethals, Natalie M. Scala, Daniel T. Bennett, Mathematics in Cyber Research, 2022
Paul L. Goethals, Natalie M. Scala, Nathaniel D. Bastian
Mathematical optimization is a sub-field of operations research concerned with the theory, solution methods, and modeling techniques for finding the extrema of functions on sets defined by linear and/or nonlinear constraints (equalities and inequalities) in a finite-dimensional vector space. In more practical terms, mathematical optimization is concerned with finding a best decision, according to some criterion, out of a (typically much) larger set of decisions. The respective mathematical model will involve decision variables (one for every elementary decision, often called a solution), constraints (limitations on some combination of decision variables), and an objective function that measures the quality of the solution. Mathematical optimization has been widely accepted because of its ability to model important and complex decision problems across many application domains, including cyber.
The Role of Optimization
Published in Louis Theodore, R. Ryan Dupont, Water Resource Management Issues, 2019
Louis Theodore, R. Ryan Dupont
The optimization process has been described by Aris as “getting the best you can out of a given situation” (Aris 1964). Problems amenable to solutions by mathematical optimization techniques generally have one or more independent variables whose values yield one or more viable solutions. In addition to the optimization definition presented by Aris, one might also offer the following generic definition for engineers: “Optimization is concerned with determining the “best” solution to a given problem” (Theodore 2011). Alternately, a dictionary would offer the following definition: “to make the most of… develop or realize to the utmost extent… often the most efficient or optimum use of.” Mathematical optimization techniques are also used for guiding the problem solver to the choice of variables that maximizes a “goodness” measure (e.g., profit) or that minimizes a “badness” measure (e.g., cost).
Introduction to Design of Composite Structures
Published in Robert M. Jones, Mechanics of Composite Materials, 2018
The term nonlinear in nonlinear programming does not refer to a material or geometric nonlinearity but instead refers to the nonlinearity in the mathematical optimization problem itself. The first step in the optimization process involves answering questions such as: what is the buckling response, what is the vibration response, what is the deflection response, and what is the stress response? Requirements usually exist for every one of those response variables. Putting those response characteristics and constraints together leads to an equation set that is inherently nonlinear, irrespective of whether the material properties themselves are linear or nonlinear, and that nonlinear equation set is where the term nonlinear programming comes from.
A grey production planning model on a ready-mixed concrete plant
Published in Engineering Optimization, 2020
Erdal Aydemir, Gokhan Yılmaz, Kenan Oguzhan Oruc
Linear programming (LP) models, which are special cases of mathematical optimization modelling, are most widely used as a decision tool in the quantitative analysis of industrial, theoretical and practical problems. An LP problem may be defined as the problem of maximizing or minimizing with a linear objective function subject to linear equations. The classical LP model form is generated with fixed decision parameters using crisp values. To model real-life problems, many of the decision parameters must be dealt with in an uncertainty (inexactness) form. This form, which builds from the real-life model according to the measures of the system value using high-tech sensors and data-processing systems, consists of inaccuracy, simplification of physical models, variations of the parameters of the system, computational errors, etc.
Abstract convergence theorem for quasi-convex optimization problems with applications
Published in Optimization, 2019
Carisa Kwok Wai Yu, Yaohua Hu, Xiaoqi Yang, Siu Kai Choy
Mathematical optimization provides a unified framework for a wide variety of important problems in many disciplines and application fields. Convex optimization plays a central role in mathematical optimization; however, it is too restrictive for many real-life problems encountered in economics, finance and management science. Quasi-convex optimization usually provides a much more accurate representation of realities than convex optimization does, while it still inherits some nice properties of convex optimization. This leads to a significant increase of studies in quasi-convex optimization; see [1–4] and references therein. However, the development of numerical algorithms for quasi-convex optimization, in particular for large-scale problems, is still in its infancy. Hence, there is a great demand for developing efficient numerical algorithms for solving large-scale quasi-convex optimization problems.
Optimization methods applied to stormwater management problems: a review
Published in Urban Water Journal, 2018
Shadab Shishegar, Sophie Duchesne, Geneviève Pelletier
Where x is a set of decision variables, f(x) is the function that defines the objective of the problem, gi(x) represents all the functions that together with bi, the boundaries, and S, as the set constraints on x, determine the problem constraints. It should be noted that a problem can have more than one objective function. In this case, besides f(x) other functions will be added to the objective function part of the model. The goal of a mathematical optimization model is to minimize or maximize the objective function(s) while satisfying all the constraints.