Mathematical optimization or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. It is generally divided into two subfields: discrete optimization and continuous optimization. Optimization problems arise in all quantitative disciplines from computer science and engineering to operations research and economics, and the development of solution methods has been of interest in mathematics for centuries.
In mathematical optimization and decision theory, a loss function or cost function is a function that maps an event or values of one or more variables onto a real number intuitively representing some "cost" associated with the event. An optimization problem seeks to minimize a loss function. An objective function is either a loss function or its opposite, in which case it is to be maximized. The loss function could include terms from several levels of the hierarchy.
Multi-disciplinary design optimization (MDO) is a field of engineering that uses optimization methods to solve design problems incorporating a number of disciplines. It is also known as multidisciplinary system design optimization (MSDO), and multidisciplinary design analysis and optimization (MDAO).
Sensitivity analysis is the study of how the uncertainty in the output of a mathematical model or system can be divided and allocated to different sources of uncertainty in its inputs. A related practice is uncertainty analysis, which has a greater focus on uncertainty quantification and propagation of uncertainty; ideally, uncertainty and sensitivity analysis should be run in tandem.
In the design of experiments, optimal experimental designs are a class of experimental designs that are optimal with respect to some statistical criterion. The creation of this field of statistics has been credited to Danish statistician Kirstine Smith.
In mathematical optimization, a trust region is the subset of the region of the objective function that is approximated using a model function. If an adequate model of the objective function is found within the trust region, then the region is expanded; conversely, if the approximation is poor, then the region is contracted.
A surrogate model is an engineering method used when an outcome of interest cannot be easily measured or computed, so an approximate mathematical model of the outcome is used instead. Most engineering design problems require experiments and/or simulations to evaluate design objective and constraint functions as a function of design variables. For example, in order to find the optimal airfoil shape for an aircraft wing, an engineer simulates the airflow around the wing for different shape variables. For many real-world problems, however, a single simulation can take many minutes, hours, or even days to complete. As a result, routine tasks such as design optimization, design space exploration, sensitivity analysis and "what-if" analysis become impossible since they require thousands or even millions of simulation evaluations.
Stochastic approximation methods are a family of iterative methods typically used for root-finding problems or for optimization problems. The recursive update rules of stochastic approximation methods can be used, among other things, for solving linear systems when the collected data is corrupted by noise, or for approximating extreme values of functions which cannot be computed directly, but only estimated via noisy observations.
Probabilistic design is a discipline within engineering design. It deals primarily with the consideration and minimization of the effects of random variability upon the performance of an engineering system during the design phase. Typically, these effects studied and optimized are related to quality and reliability. It differs from the classical approach to design by assuming a small probability of failure instead of using the safety factor. Probabilistic design is used in a variety of different applications to assess the likelihood of failure. Disciplines which extensively use probabilistic design principles include product design, quality control, systems engineering, machine design, civil engineering and manufacturing.
Stochastic control or stochastic optimal control is a sub field of control theory that deals with the existence of uncertainty either in observations or in the noise that drives the evolution of the system. The system designer assumes, in a Bayesian probability-driven fashion, that random noise with known probability distribution affects the evolution and observation of the state variables. Stochastic control aims to design the time path of the controlled variables that performs the desired control task with minimum cost, somehow defined, despite the presence of this noise. The context may be either discrete time or continuous time.
IOSO is a multiobjective, multidimensional nonlinear optimization technology.
SimulationX is a CAE software application running on Microsoft Windows for the physical simulation of technical systems. It is developed and sold by ESI Group.
OptiY is a design environment software that provides modern optimization strategies and state of the art probabilistic algorithms for uncertainty, reliability, robustness, sensitivity analysis, data-mining and meta-modeling.
Red Cedar Technology is a software development and engineering services company. Red Cedar Technology was founded by Michigan State University professors Ron Averill and Erik Goodman in 1999. The headquarters is located in East Lansing, Michigan, near MSU's campus. Red Cedar Technology develops and distributes the HEEDS Professional suite of design optimization software. HEEDS is based on spin-out technology from Michigan State University. On June 30, 2013 Red Cedar Technology was acquired by CD-adapco. CD-adapco was acquired in 2016 by Siemens Digital Industries Software.
SmartDO is a multidisciplinary design optimization software, based on the Direct Global Search technology developed and marketed by FEA-Opt Technology. SmartDO specialized in the CAE-Based optimization, such as CAE, FEA, CAD, CFD and automatic control, with application on various physics phenomena. It is both GUI and scripting driven, allowed to be integrated with almost any kind of CAD/CAE and in-house codes.
pSeven is a design space exploration (DSE) software platform that was developed by pSeven SAS that features design, simulation, and analysis capabilities and assists in design decisions. It provides integration with third-party CAD and CAE software tools; multi-objective and robust optimization algorithms; data analysis, and uncertainty quantification tools.
Simulation-based optimization integrates optimization techniques into simulation modeling and analysis. Because of the complexity of the simulation, the objective function may become difficult and expensive to evaluate. Usually, the underlying simulation model is stochastic, so that the objective function must be estimated using statistical estimation techniques.
Probabilistic numerics is an active field of study at the intersection of applied mathematics, statistics, and machine learning centering on the concept of uncertainty in computation. In probabilistic numerics, tasks in numerical analysis such as finding numerical solutions for integration, linear algebra, optimization and simulation and differential equations are seen as problems of statistical, probabilistic, or Bayesian inference.
(Stochastic) variance reduction is an algorithmic approach to minimizing functions that can be decomposed into finite sums. By exploiting the finite sum structure, variance reduction techniques are able to achieve convergence rates that are impossible to achieve with methods that treat the objective as an infinite sum, as in the classical Stochastic approximation setting.
