Mathematical optimizationMathematical optimization (alternatively spelled optimisation) 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.
Combinatorial optimizationCombinatorial optimization is a subfield of mathematical optimization that consists of finding an optimal object from a finite set of objects, where the set of feasible solutions is discrete or can be reduced to a discrete set. Typical combinatorial optimization problems are the travelling salesman problem ("TSP"), the minimum spanning tree problem ("MST"), and the knapsack problem. In many such problems, such as the ones previously mentioned, exhaustive search is not tractable, and so specialized algorithms that quickly rule out large parts of the search space or approximation algorithms must be resorted to instead.
Dynamic programmingDynamic programming is both a mathematical optimization method and an algorithmic paradigm. The method was developed by Richard Bellman in the 1950s and has found applications in numerous fields, from aerospace engineering to economics. In both contexts it refers to simplifying a complicated problem by breaking it down into simpler sub-problems in a recursive manner. While some decision problems cannot be taken apart this way, decisions that span several points in time do often break apart recursively.
SteeringSteering is the control of the direction of locomotion. A conventional automotive steering arrangement allows a driver to control the direction of the vehicle by turning the direction of the front wheels using a hand–operated steering wheel positioned in front of the driver. The steering wheel is attached to a steering column, which is linked to rods, pivots and gears that allow the driver to change the direction of the front wheels. Other arrangements are sometimes found on different types of vehicles; for example, a tiller or rear-wheel steering.
Total variationIn mathematics, the total variation identifies several slightly different concepts, related to the (local or global) structure of the codomain of a function or a measure. For a real-valued continuous function f, defined on an interval [a, b] ⊂ R, its total variation on the interval of definition is a measure of the one-dimensional arclength of the curve with parametric equation x ↦ f(x), for x ∈ [a, b]. Functions whose total variation is finite are called functions of bounded variation.
Convergent boundaryA convergent boundary (also known as a destructive boundary) is an area on Earth where two or more lithospheric plates collide. One plate eventually slides beneath the other, a process known as subduction. The subduction zone can be defined by a plane where many earthquakes occur, called the Wadati–Benioff zone. These collisions happen on scales of millions to tens of millions of years and can lead to volcanism, earthquakes, orogenesis, destruction of lithosphere, and deformation.
Time seriesIn mathematics, a time series is a series of data points indexed (or listed or graphed) in time order. Most commonly, a time series is a sequence taken at successive equally spaced points in time. Thus it is a sequence of discrete-time data. Examples of time series are heights of ocean tides, counts of sunspots, and the daily closing value of the Dow Jones Industrial Average. A time series is very frequently plotted via a run chart (which is a temporal line chart).
Convex optimizationConvex optimization is a subfield of mathematical optimization that studies the problem of minimizing convex functions over convex sets (or, equivalently, maximizing concave functions over convex sets). Many classes of convex optimization problems admit polynomial-time algorithms, whereas mathematical optimization is in general NP-hard.
Steering wheelA steering wheel (also called a driving wheel (UK), a hand wheel, or simply wheel) is a type of steering control in vehicles. Steering wheels are used in most modern land vehicles, including all mass-production automobiles, buses, light and heavy trucks, as well as tractors. The steering wheel is the part of the steering system that is manipulated by the driver; the rest of the steering system responds to such driver inputs.
Leibniz's notationIn calculus, Leibniz's notation, named in honor of the 17th-century German philosopher and mathematician Gottfried Wilhelm Leibniz, uses the symbols dx and dy to represent infinitely small (or infinitesimal) increments of x and y, respectively, just as Δx and Δy represent finite increments of x and y, respectively. Consider y as a function of a variable x, or y = f(x).
Kalman filterFor statistics and control theory, Kalman filtering, also known as linear quadratic estimation (LQE), is an algorithm that uses a series of measurements observed over time, including statistical noise and other inaccuracies, and produces estimates of unknown variables that tend to be more accurate than those based on a single measurement alone, by estimating a joint probability distribution over the variables for each timeframe. The filter is named after Rudolf E. Kálmán, who was one of the primary developers of its theory.
Nintendo Entertainment SystemThe Nintendo Entertainment System (NES) is an 8-bit third-generation home video game console produced by Nintendo. It was first released in Japan in 1983 as the Family Computer (FC), commonly referred to as Famicom. It was redesigned to become the NES, which was released in American test markets on October 18, 1985, and was soon fully launched in North America and other countries. After developing several successful arcade games in the early 1980s such as Donkey Kong (1981), Nintendo planned to create a home video game console.
Power steeringPower steering is a system for reducing a driver's effort to turn a steering wheel of a motor vehicle, by using a power source to assist steering. Hydraulic or electric actuators add controlled energy to the steering mechanism, so the driver can provide less effort to turn the steered wheels when driving at typical speeds, and reduce considerably the physical effort necessary to turn the wheels when a vehicle is stopped or moving slowly. Power steering can also be engineered to provide some artificial feedback of forces acting on the steered wheels.
Constrained optimizationIn mathematical optimization, constrained optimization (in some contexts called constraint optimization) is the process of optimizing an objective function with respect to some variables in the presence of constraints on those variables. The objective function is either a cost function or energy function, which is to be minimized, or a reward function or utility function, which is to be maximized.
Ant colony optimization algorithmsIn computer science and operations research, the ant colony optimization algorithm (ACO) is a probabilistic technique for solving computational problems which can be reduced to finding good paths through graphs. Artificial ants stand for multi-agent methods inspired by the behavior of real ants. The pheromone-based communication of biological ants is often the predominant paradigm used. Combinations of artificial ants and local search algorithms have become a method of choice for numerous optimization tasks involving some sort of graph, e.
DerivativeIn mathematics, the derivative shows the sensitivity of change of a function's output with respect to the input. Derivatives are a fundamental tool of calculus. For example, the derivative of the position of a moving object with respect to time is the object's velocity: this measures how quickly the position of the object changes when time advances. The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point.
Island arcIsland arcs are long chains of active volcanoes with intense seismic activity found along convergent tectonic plate boundaries. Most island arcs originate on oceanic crust and have resulted from the descent of the lithosphere into the mantle along the subduction zone. They are the principal way by which continental growth is achieved. Island arcs can either be active or inactive based on their seismicity and presence of volcanoes. Active arcs are ridges of recent volcanoes with an associated deep seismic zone.
Local search (optimization)In computer science, local search is a heuristic method for solving computationally hard optimization problems. Local search can be used on problems that can be formulated as finding a solution maximizing a criterion among a number of candidate solutions. Local search algorithms move from solution to solution in the space of candidate solutions (the search space) by applying local changes, until a solution deemed optimal is found or a time bound is elapsed.
Calculus of variationsThe calculus of variations (or variational calculus) is a field of mathematical analysis that uses variations, which are small changes in functions and functionals, to find maxima and minima of functionals: mappings from a set of functions to the real numbers. Functionals are often expressed as definite integrals involving functions and their derivatives. Functions that maximize or minimize functionals may be found using the Euler–Lagrange equation of the calculus of variations.
Bounded variationIn mathematical analysis, a function of bounded variation, also known as BV function, is a real-valued function whose total variation is bounded (finite): the graph of a function having this property is well behaved in a precise sense. For a continuous function of a single variable, being of bounded variation means that the distance along the direction of the y-axis, neglecting the contribution of motion along x-axis, traveled by a point moving along the graph has a finite value.
