text stringlengths 9 3.55k | source stringlengths 31 280 |
|---|---|
In optics, shot noise describes the fluctuations of the number of photons detected (or simply counted in the abstract) due to their occurrence independent of each other. This is therefore another consequence of discretization, in this case of the energy in the electromagnetic field in terms of photons. In the case of p... | https://en.wikipedia.org/wiki/Poisson_noise |
Of course there are other mechanisms of noise in optical signals which often dwarf the contribution of shot noise. When these are absent, however, optical detection is said to be "photon noise limited" as only the shot noise (also known as "quantum noise" or "photon noise" in this context) remains. Shot noise is easily... | https://en.wikipedia.org/wiki/Poisson_noise |
However the same noise source is present with higher light intensities measured by any photo detector, and is directly measurable when it dominates the noise of the subsequent electronic amplifier. Just as with other forms of shot noise, the fluctuations in a photo-current due to shot noise scale as the square-root of ... | https://en.wikipedia.org/wiki/Poisson_noise |
The shot noise of a coherent optical beam (having no other noise sources) is a fundamental physical phenomenon, reflecting quantum fluctuations in the electromagnetic field. In optical homodyne detection, the shot noise in the photodetector can be attributed to either the zero point fluctuations of the quantised electr... | https://en.wikipedia.org/wiki/Poisson_noise |
In optics, spatial cutoff frequency is a precise way to quantify the smallest object resolvable by an optical system. Due to diffraction at the image plane, all optical systems act as low pass filters with a finite ability to resolve detail. If it were not for the effects of diffraction, a 2" aperture telescope could t... | https://en.wikipedia.org/wiki/Spatial_cutoff_frequency |
Unfortunately, the wave nature of light will never permit this to happen. The spatial cutoff frequency for a perfectly corrected incoherent optical system is given by f o = 1 λ F # c y c l e s / m i l l i m e t e r , {\displaystyle f_{o}={1 \over {\lambda F_{\#}}}\ \ \mathrm {cycles/millimeter} \ ,} where λ {\displayst... | https://en.wikipedia.org/wiki/Spatial_cutoff_frequency |
High-resolution black-and-white film is capable of resolving details on the film as small as 3 micrometers or smaller, thus its cutoff frequency is about 150 cycles/millimeter. So, the telescope's optical resolution is about twice that of high-resolution film, and a crisp, sharp picture would result (provided focus is ... | https://en.wikipedia.org/wiki/Spatial_cutoff_frequency |
The presence of aberrations reduces image contrast and can effectively reduce the system spatial cutoff frequency if the image contrast falls below the ability of the imaging device to discern. The coherent case is given by f o = 1 2 λ F # c y c l e s / m i l l i m e t e r . {\displaystyle f_{o}={1 \over {2\lambda F_{\... | https://en.wikipedia.org/wiki/Spatial_cutoff_frequency |
In optics, spherical aberration (SA) is a type of aberration found in optical systems that have elements with spherical surfaces. Lenses and curved mirrors are prime examples, because this shape is easier to manufacture. Light rays that strike a spherical surface off-centre are refracted or reflected more or less than ... | https://en.wikipedia.org/wiki/Spherical_aberration |
In optics, temporal coherence is measured in an interferometer such as the Michelson interferometer or Mach–Zehnder interferometer. In these devices, a wave is combined with a copy of itself that is delayed by time τ. A detector measures the time-averaged intensity of the light exiting the interferometer. The resulting... | https://en.wikipedia.org/wiki/Temporal_coherence |
In optics, the Abbe sine condition is a condition that must be fulfilled by a lens or other optical system in order for it to produce sharp images of off-axis as well as on-axis objects. It was formulated by Ernst Abbe in the context of microscopes.The Abbe sine condition says that the sine of the object-space angle α ... | https://en.wikipedia.org/wiki/Abbe_sine_condition |
In optics, the Airy disk (or Airy disc) and Airy pattern are descriptions of the best-focused spot of light that a perfect lens with a circular aperture can make, limited by the diffraction of light. The Airy disk is of importance in physics, optics, and astronomy. The diffraction pattern resulting from a uniformly ill... | https://en.wikipedia.org/wiki/Airy_disc |
The disk and rings phenomenon had been known prior to Airy; John Herschel described the appearance of a bright star seen through a telescope under high magnification for an 1828 article on light for the Encyclopedia Metropolitana: ...the star is then seen (in favourable circumstances of tranquil atmosphere, uniform tem... | https://en.wikipedia.org/wiki/Airy_disc |
The most important application of this concept is in cameras, microscopes and telescopes. Due to diffraction, the smallest point to which a lens or mirror can focus a beam of light is the size of the Airy disk. Even if one were able to make a perfect lens, there is still a limit to the resolution of an image created by... | https://en.wikipedia.org/wiki/Airy_disc |
In optics, the Arago spot, Poisson spot, or Fresnel spot is a bright point that appears at the center of a circular object's shadow due to Fresnel diffraction. This spot played an important role in the discovery of the wave nature of light and is a common way to demonstrate that light behaves as a wave. The basic exper... | https://en.wikipedia.org/wiki/Poisson_spot |
The dimensions of the setup must comply with the requirements for Fresnel diffraction. Namely, the Fresnel number must satisfy where d is the diameter of the circular object, ℓ is the distance between the object and the screen, and λ is the wavelength of the source.Finally, the edge of the circular object must be suffi... | https://en.wikipedia.org/wiki/Poisson_spot |
However, with the laser sources available today, it is undemanding to perform an Arago-spot experiment.In astronomy, the Arago spot can also be observed in the strongly defocussed image of a star in a Newtonian telescope. There, the star provides an almost ideal point source at infinity, and the secondary mirror of the... | https://en.wikipedia.org/wiki/Poisson_spot |
In optics, the Ewald–Oseen extinction theorem, sometimes referred to as just the extinction theorem, is a theorem that underlies the common understanding of scattering (as well as refraction, reflection, and diffraction). It is named after Paul Peter Ewald and Carl Wilhelm Oseen, who proved the theorem in crystalline a... | https://en.wikipedia.org/wiki/Ewald–Oseen_extinction_theorem |
In optics, the Fraunhofer diffraction equation is used to model the diffraction of waves when plane waves are incident on a diffracting object, and the diffraction pattern is viewed at a sufficiently long distance (a distance satisfying Fraunhofer condition) from the object (in the far-field region), and also when it i... | https://en.wikipedia.org/wiki/Fraunhofer_diffraction |
In optics, the Fraunhofer diffraction equation is used to model the diffraction of waves when the diffraction pattern is viewed at a long distance from the diffracting object, and also when it is viewed at the focal plane of an imaging lens.The equation was named in honour of Joseph von Fraunhofer although he was not a... | https://en.wikipedia.org/wiki/Fraunhofer_diffraction_equation |
In optics, the Fresnel diffraction equation for near-field diffraction is an approximation of the Kirchhoff–Fresnel diffraction that can be applied to the propagation of waves in the near field. It is used to calculate the diffraction pattern created by waves passing through an aperture or around an object, when viewed... | https://en.wikipedia.org/wiki/Fresnel_Diffraction |
When F << 1 {\displaystyle F<<1} the diffracted wave is considered to be in the Fraunhofer field. However, the validity of the Fresnel diffraction integral is deduced by the approximations derived below. Specifically, the phase terms of third order and higher must be negligible, a condition that may be written as where... | https://en.wikipedia.org/wiki/Fresnel_Diffraction |
In optics, the Hagen–Rubens relation (or Hagen–Rubens formula) is a relation between the coefficient of reflection and the conductivity for materials that are good conductors. The relation states that for solids where the contribution of the dielectric constant to the index of refraction is negligible, the reflection c... | https://en.wikipedia.org/wiki/Hagen–Rubens_relation |
In optics, the Q factor of a resonant cavity is given by Q = 2 π f o E P , {\displaystyle Q={\frac {2\pi f_{o}\,E}{P}},\,} where fo is the resonant frequency, E is the stored energy in the cavity, and P = −dE/dt is the power dissipated. The optical Q is equal to the ratio of the resonant frequency to the bandwidth of t... | https://en.wikipedia.org/wiki/Q_factor |
This technique is known as Q-switching. Q factor is of particular importance in plasmonics, where loss is linked to the damping of the surface plasmon resonance. While loss is normally considered a hindrance in the development of plasmonic devices, it is possible to leverage this property to present new enhanced functi... | https://en.wikipedia.org/wiki/Q_factor |
In optics, the complex beam parameter is a complex number that specifies the properties of a Gaussian beam at a particular point z along the axis of the beam. It is usually denoted by q. It can be calculated from the beam's vacuum wavelength λ0, the radius of curvature R of the phase front, the index of refraction n (n... | https://en.wikipedia.org/wiki/Complex_beam_parameter |
In optics, the corpuscular theory of light states that light is made up of small discrete particles called "corpuscles" (little particles) which travel in a straight line with a finite velocity and possess impetus. This was based on an alternate description of atomism of the time period. Isaac Newton laid the foundatio... | https://en.wikipedia.org/wiki/Corpuscular_theory |
In optics, the exit pupil is a virtual aperture in an optical system. Only rays which pass through this virtual aperture can exit the system. The exit pupil is the image of the aperture stop in the optics that follow it. In a telescope or compound microscope, this image is the image of the objective element(s) as produ... | https://en.wikipedia.org/wiki/Exit_pupil |
The size and shape of this disc is crucial to the instrument's performance, because the observer's eye can see light only if it passes through the aperture. The term exit pupil is also sometimes used to refer to the diameter of the virtual aperture. Older literature on optics sometimes refers to the exit pupil as the R... | https://en.wikipedia.org/wiki/Exit_pupil |
In optics, the image of an object is defined as the collection of focus points of light rays coming from the object. A real image is the collection of focus points made by converging rays, while a virtual image is the collection of focus points made by extensions of diverging rays. In other words, a virtual image is fo... | https://en.wikipedia.org/wiki/Virtual_object |
For a (reflecting) mirror, the real image is on the same side of the object while the virtual image is the opposite side to the object. In diagrams of optical systems, virtual rays (forming virtual images) are conventionally represented by dotted lines, to contrast with the solid lines of real rays. Because the rays ne... | https://en.wikipedia.org/wiki/Virtual_object |
In contrast, a real image can be projected on the screen as it is formed by rays that converge on a real location. A real image can be projected onto a diffusely reflecting screen so people can see the image (the image on the screen plays as an object to be imaged by human eyes). A plane mirror forms a virtual image po... | https://en.wikipedia.org/wiki/Virtual_object |
Although the rays of light seem to come from behind the mirror, light from the source only exists in front of the mirror. The image in a plane mirror is not magnified (that is, the image is the same size as the object) and appears to be as far behind the mirror as the object is in front of the mirror. A diverging lens ... | https://en.wikipedia.org/wiki/Virtual_object |
Such an image is reduced in size when compared to the original object. A converging lens (one that is thicker in the middle than at the edges) or a concave mirror is also capable of producing a virtual image if the object is within the focal length. Such an image will be magnified. In contrast, an object placed in fron... | https://en.wikipedia.org/wiki/Virtual_object |
In optics, the inverse Faraday effect is the effect opposite to the Faraday effect. A static magnetization M ( 0 ) {\displaystyle \mathbf {M} (0)} is induced by an external oscillating electrical field with the frequency ω {\displaystyle \omega } , which can be achieved with a high intensity laser pulse for example. Th... | https://en.wikipedia.org/wiki/Inverse_Faraday_effect |
In optics, the nonlinear Schrödinger equation occurs in the Manakov system, a model of wave propagation in fiber optics. The function ψ represents a wave and the nonlinear Schrödinger equation describes the propagation of the wave through a nonlinear medium. The second-order derivative represents the dispersion, while ... | https://en.wikipedia.org/wiki/Nonlinear_Schrödinger_equation |
In optics, the numerical aperture (NA) of an optical system is a dimensionless number that characterizes the range of angles over which the system can accept or emit light. By incorporating index of refraction in its definition, NA has the property that it is constant for a beam as it goes from one material to another,... | https://en.wikipedia.org/wiki/Numerical_apertures |
In optics, the optical sine theorem states that the products of the index, height, and sine of the slope angle of a ray in object space and its corresponding ray in image space are equal. That is: n 0 y 0 a 0 = n i y i a i {\displaystyle n_{0}y_{0}a_{0}=n_{i}y_{i}a_{i}} | https://en.wikipedia.org/wiki/Optical_sine_theorem |
In optics, the refractive index (or refraction index) of an optical medium is a dimensionless number that gives the indication of the light bending ability of that medium. The refractive index determines how much the path of light is bent, or refracted, when entering a material. This is described by Snell's law of refr... | https://en.wikipedia.org/wiki/Complex_refractive_index |
This implies that vacuum has a refractive index of 1, and assumes that the frequency (f = v/λ) of the wave is not affected by the refractive index. The refractive index may vary with wavelength. This causes white light to split into constituent colors when refracted. | https://en.wikipedia.org/wiki/Complex_refractive_index |
This is called dispersion. This effect can be observed in prisms and rainbows, and as chromatic aberration in lenses. Light propagation in absorbing materials can be described using a complex-valued refractive index. | https://en.wikipedia.org/wiki/Complex_refractive_index |
The imaginary part then handles the attenuation, while the real part accounts for refraction. For most materials the refractive index changes with wavelength by several percent across the visible spectrum. Nevertheless, refractive indices for materials are commonly reported using a single value for n, typically measure... | https://en.wikipedia.org/wiki/Complex_refractive_index |
The concept of refractive index applies across the full electromagnetic spectrum, from X-rays to radio waves. It can also be applied to wave phenomena such as sound. In this case, the speed of sound is used instead of that of light, and a reference medium other than vacuum must be chosen.For lenses (such as eye glasses... | https://en.wikipedia.org/wiki/Complex_refractive_index |
In optics, the small-angle approximations form the basis of the paraxial approximation. | https://en.wikipedia.org/wiki/Small-angle_formula |
In optics, the surface vertices are the points where each optical surface crosses the optical axis. They are important primarily because they are the physically measurable parameters for the position of the optical elements, and so the positions of the cardinal points must be known with respect to the vertices to descr... | https://en.wikipedia.org/wiki/Surface_vertex |
In optics, the term soliton is used to refer to any optical field that does not change during propagation because of a delicate balance between nonlinear and dispersive effects in the medium. There are two main kinds of solitons: spatial solitons: the nonlinear effect can balance the dispersion. The electromagnetic fie... | https://en.wikipedia.org/wiki/Spatial_solitons |
In optics, there are several methods of filtering selected wavelengths from a source or to a detector. They rely on scattering or destructive interference. | https://en.wikipedia.org/wiki/Band-reject_filter |
In optics, tilt is a deviation in the direction a beam of light propagates. | https://en.wikipedia.org/wiki/Tilt_(optics) |
In optics, transmission is the property of a substance to permit the passage of light, with some or none of the incident light being absorbed in the process. If some light is absorbed by the substance, then the transmitted light will be a combination of the wavelengths of the light that was transmitted and not absorbed... | https://en.wikipedia.org/wiki/Transmission_coefficient_(physics) |
The transmission coefficient is a measure of how much of an electromagnetic wave (light) passes through a surface or an optical element. Transmission coefficients can be calculated for either the amplitude or the intensity of the wave. Either is calculated by taking the ratio of the value after the surface or element t... | https://en.wikipedia.org/wiki/Transmission_coefficient_(physics) |
In optics, two non-Lambertian sources that emit beamed energy can interact in a way that causes a shift in the spectral lines. It is analogous to a pair of tuning forks with similar frequencies (pitches), connected together mechanically with a sounding board; there is a strong coupling that results in the resonant freq... | https://en.wikipedia.org/wiki/Wolf_effect |
It can produce either redshifts or blueshifts, depending on the observer's point of view, but is redshifted when the observer is head-on.For two sources interacting while separated by a vacuum, the Wolf effect cannot produce shifts greater than the linewidth of the source spectral line, since it is a position-dependent... | https://en.wikipedia.org/wiki/Wolf_effect |
In optics, various autocorrelation functions can be experimentally realized. The field autocorrelation may be used to calculate the spectrum of a source of light, while the intensity autocorrelation and the interferometric autocorrelation are commonly used to estimate the duration of ultrashort pulses produced by model... | https://en.wikipedia.org/wiki/Optical_autocorrelation |
In the following examples, the autocorrelation signal is generated by the nonlinear process of second-harmonic generation (SHG). Other techniques based on two-photon absorption may also be used in autocorrelation measurements, as well as higher-order nonlinear optical processes such as third-harmonic generation, in whi... | https://en.wikipedia.org/wiki/Optical_autocorrelation |
In optics, vergence is the angle formed by rays of light that are not perfectly parallel to one another. Rays that move closer to the optical axis as they propagate are said to be converging, while rays that move away from the axis are diverging. These imaginary rays are always perpendicular to the wavefront of the lig... | https://en.wikipedia.org/wiki/Vergence_(optics) |
Beyond that focal point, the rays diverge. Conversely, a concave lens or convex mirror will cause parallel rays to diverge. | https://en.wikipedia.org/wiki/Vergence_(optics) |
Light does not actually consist of imaginary rays and light sources are not single-point sources, thus vergence is typically limited to simple ray modeling of optical systems. In a real system, the vergence is a product of the diameter of a light source, its distance from the optics, and the curvature of the optical su... | https://en.wikipedia.org/wiki/Vergence_(optics) |
Likewise, a decrease in curvature decreases vergence, resulting in a longer focal length and an increase in image or spot diameter. This reciprocal relationship between vergence, focal length, and waist diameter are constant throughout an optical system, and is referred to as the optical invariant. A beam that is expan... | https://en.wikipedia.org/wiki/Vergence_(optics) |
In optics, wall-plug efficiency or radiant efficiency is the energy conversion efficiency with which the system converts electrical power into optical power. It is defined as the ratio of the radiant flux (i.e., the total optical output power) to the input electrical power.In laser systems, this efficiency includes los... | https://en.wikipedia.org/wiki/Wall-plug_efficiency |
In optics, when a beam of light shines at an angle through two stacked transparent plates of different materials of different refractive indexes, it may reflect off three surfaces: the top, middle, and bottom surfaces of the two plates. The number of different beam paths that have k reflections, for k > 1, is the k {\d... | https://en.wikipedia.org/wiki/Fibonacci_Number |
Since the conversion factor 1.609344 for miles to kilometers is close to the golden ratio, the decomposition of distance in miles into a sum of Fibonacci numbers becomes nearly the kilometer sum when the Fibonacci numbers are replaced by their successors. This method amounts to a radix 2 number register in golden ratio... | https://en.wikipedia.org/wiki/Fibonacci_Number |
The measured values of voltages and currents in the infinite resistor chain circuit (also called the resistor ladder or infinite series-parallel circuit) follow the Fibonacci sequence. The intermediate results of adding the alternating series and parallel resistances yields fractions composed of consecutive Fibonacci n... | https://en.wikipedia.org/wiki/Fibonacci_Number |
Brasch et al. 2012 show how a generalized Fibonacci sequence also can be connected to the field of economics. In particular, it is shown how a generalized Fibonacci sequence enters the control function of finite-horizon dynamic optimisation problems with one state and one control variable. The procedure is illustrated ... | https://en.wikipedia.org/wiki/Fibonacci_Number |
Mario Merz included the Fibonacci sequence in some of his artworks beginning in 1970. Joseph Schillinger (1895–1943) developed a system of composition which uses Fibonacci intervals in some of its melodies; he viewed these as the musical counterpart to the elaborate harmony evident within nature. See also Golden ratio ... | https://en.wikipedia.org/wiki/Fibonacci_Number |
In optimal LS SPCA (USPCA, uncorrelated SPCA) the orthogonality constraints require that the cardinality of the solutions is not smaller than the order of the component, These constraints may also create numerical problems when computing components of order larger than two. Correlated SPCA (CSPCA, correlated SPCA) is a... | https://en.wikipedia.org/wiki/Sparse_PCA |
The computation of the USPCA and CSPCA solutions is demanding when the data matrix is large. With Projection SPCA (PSPCA) approximate CSPCA solutions can computed much more efficiently by simply projecting the current first PC onto subsets of the variables. This means that the solutions can be computed with efficient l... | https://en.wikipedia.org/wiki/Sparse_PCA |
In optimal control problems, it is sometimes the case that a control is restricted to be between a lower and an upper bound. If the optimal control switches from one extreme to the other (i.e., is strictly never in between the bounds), then that control is referred to as a bang-bang solution. Bang–bang controls frequen... | https://en.wikipedia.org/wiki/Bang-bang_control |
For example, if it is desired for a car starting at rest to arrive at a certain position ahead of the car in the shortest possible time, the solution is to apply maximum acceleration until the unique switching point, and then apply maximum braking to come to rest exactly at the desired position. A familiar everyday exa... | https://en.wikipedia.org/wiki/Bang-bang_control |
In optimal control theory, a control is a variable chosen by the controller or agent to manipulate state variables, similar to an actual control valve. Unlike the state variable, it does not have a predetermined equation of motion. The goal of optimal control theory is to find some sequence of controls (within an admis... | https://en.wikipedia.org/wiki/Control_(optimal_control_theory) |
In optimal control theory, a transversality condition is a boundary condition for the terminal values of the costate variables. They are one of the necessary conditions for optimality infinite-horizon optimal control problems without an endpoint constraint on the state variables. | https://en.wikipedia.org/wiki/Transversality_condition |
In optimal control theory, the Lagrange multipliers are interpreted as costate variables, and Lagrange multipliers are reformulated as the minimization of the Hamiltonian, in Pontryagin's minimum principle. | https://en.wikipedia.org/wiki/Lagrange_multiplier_method |
In optimal control theory, the concept of shadow price is reformulated as costate equations, and one solves the problem by minimization of the associated Hamiltonian via Pontryagin's minimum principle. | https://en.wikipedia.org/wiki/Shadow_price |
In optimal control theory, the evolution of n state variables through time depends at any time on their own values and on the values of k control variables. With linear evolution, matrices of coefficients appear in the state equation (equation of evolution). In some problems the values of the parameters in these matric... | https://en.wikipedia.org/wiki/Random_matrices |
In optimal control, problems of singular control are problems that are difficult to solve because a straightforward application of Pontryagin's minimum principle fails to yield a complete solution. Only a few such problems have been solved, such as Merton's portfolio problem in financial economics or trajectory optimiz... | https://en.wikipedia.org/wiki/Singular_control |
The most common difficulty in applying Pontryagin's principle arises when the Hamiltonian depends linearly on the control u {\displaystyle u} , i.e., is of the form: H ( u ) = ϕ ( x , λ , t ) u + ⋯ {\displaystyle H(u)=\phi (x,\lambda ,t)u+\cdots } and the control is restricted to being between an upper and a lower boun... | https://en.wikipedia.org/wiki/Singular_control |
{\displaystyle u(t)={\begin{cases}b,&\phi (x,\lambda ,t)<0\\?,&\phi (x,\lambda ,t)=0\\a,&\phi (x,\lambda ,t)>0.\end{cases}}} If ϕ {\displaystyle \phi } is positive at some times, negative at others and is only zero instantaneously, then the solution is straightforward and is a bang-bang control that switches from b {\d... | https://en.wikipedia.org/wiki/Singular_control |
(One approach would be to repeatedly differentiate ∂ H / ∂ u {\displaystyle \partial H/\partial u} with respect to time until the control u again explicitly appears, though this is not guaranteed to happen eventually. One can then set that expression to zero and solve for u. This amounts to saying that between t 1 {\di... | https://en.wikipedia.org/wiki/Singular_control |
In optimal control, the situation is more complicated because of the possibility of a singular solution. The generalized Legendre–Clebsch condition, also known as convexity, is a sufficient condition for local optimality such that when the linear sensitivity of the Hamiltonian to changes in u is zero, i.e., ∂ H ∂ u = 0... | https://en.wikipedia.org/wiki/Legendre–Clebsch_condition |
In optimal transport, a branch of mathematics, polar factorization of vector fields is a basic result due to Brenier (1987), with antecedents of Knott-Smith (1984) and Rachev (1985), that generalizes many existing results among which are the polar decomposition of real matrices, and the rearrangement of real-valued fun... | https://en.wikipedia.org/wiki/Polar_factorization_theorem |
In optimization and other branches of mathematics, and in search algorithms (a topic in computer science), a candidate solution is a member of the set of possible solutions in the feasible region of a given problem. A candidate solution does not have to be a likely or reasonable solution to the problem—it is simply in ... | https://en.wikipedia.org/wiki/Feasible_solution |
The space of all candidate solutions, before any feasible points have been excluded, is called the feasible region, feasible set, search space, or solution space. This is the set of all possible solutions that satisfy the problem's constraints. Constraint satisfaction is the process of finding a point in the feasible s... | https://en.wikipedia.org/wiki/Feasible_solution |
In optimization problems in applied mathematics, the duality gap is the difference between the primal and dual solutions. If d ∗ {\displaystyle d^{*}} is the optimal dual value and p ∗ {\displaystyle p^{*}} is the optimal primal value then the duality gap is equal to p ∗ − d ∗ {\displaystyle p^{*}-d^{*}} . This value i... | https://en.wikipedia.org/wiki/Duality_gap |
Otherwise the gap is strictly positive and weak duality holds.In general given two dual pairs separated locally convex spaces ( X , X ∗ ) {\displaystyle \left(X,X^{*}\right)} and ( Y , Y ∗ ) {\displaystyle \left(Y,Y^{*}\right)} . Then given the function f: X → R ∪ { + ∞ } {\displaystyle f:X\to \mathbb {R} \cup \{+\inft... | https://en.wikipedia.org/wiki/Duality_gap |
Then let F: X × Y → R ∪ { + ∞ } {\displaystyle F:X\times Y\to \mathbb {R} \cup \{+\infty \}} be a perturbation function such that F ( x , 0 ) = f ( x ) {\displaystyle F(x,0)=f(x)} . The duality gap is the difference given by inf x ∈ X − sup y ∗ ∈ Y ∗ {\displaystyle \inf _{x\in X}-\sup _{y^{*}\in Y^{*}}} where F ∗ {\d... | https://en.wikipedia.org/wiki/Duality_gap |
In optimization theory, maximum flow problems involve finding a feasible flow through a flow network that obtains the maximum possible flow rate. The maximum flow problem can be seen as a special case of more complex network flow problems, such as the circulation problem. The maximum value of an s-t flow (i.e., flow fr... | https://en.wikipedia.org/wiki/Integral_flow_theorem |
In optimization theory, semi-infinite programming (SIP) is an optimization problem with a finite number of variables and an infinite number of constraints, or an infinite number of variables and a finite number of constraints. In the former case the constraints are typically parameterized. | https://en.wikipedia.org/wiki/Semi-infinite_programming |
In optimization theory, the duality principle states that optimization problems may be viewed from either of two perspectives, the primal problem or the dual problem. In general given two dual pairs separated locally convex spaces ( X , X ∗ ) {\displaystyle \left(X,X^{*}\right)} and ( Y , Y ∗ ) . {\displaystyle \left(Y... | https://en.wikipedia.org/wiki/Convex_analysis |
{\displaystyle \inf _{x\in X}f(x).} If there are constraint conditions, these can be built into the function f {\displaystyle f} by letting f = f + I c o n s t r a i n t s {\displaystyle f=f+I_{\mathrm {constraints} }} where I {\displaystyle I} is the indicator function. Then let F: X × Y → {\displaystyle F:X\times Y\... | https://en.wikipedia.org/wiki/Convex_analysis |
{\displaystyle F(x,0)=f(x).} The dual problem with respect to the chosen perturbation function is given by sup y ∗ ∈ Y ∗ − F ∗ ( 0 , y ∗ ) {\displaystyle \sup _{y^{*}\in Y^{*}}-F^{*}\left(0,y^{*}\right)} where F ∗ {\displaystyle F^{*}} is the convex conjugate in both variables of F . {\displaystyle F.} | https://en.wikipedia.org/wiki/Convex_analysis |
The duality gap is the difference of the right and left hand sides of the inequality sup y ∗ ∈ Y ∗ − F ∗ ( 0 , y ∗ ) ≤ inf x ∈ X F ( x , 0 ) . {\displaystyle \sup _{y^{*}\in Y^{*}}-F^{*}\left(0,y^{*}\right)\leq \inf _{x\in X}F(x,0).} This principle is the same as weak duality. If the two sides are equal to each other, ... | https://en.wikipedia.org/wiki/Convex_analysis |
In optimization, a descent direction is a vector p ∈ R n {\displaystyle \mathbf {p} \in \mathbb {R} ^{n}} that points towards a local minimum x ∗ {\displaystyle \mathbf {x} ^{*}} of an objective function f: R n → R {\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} } . Computing x ∗ {\displaystyle \mathbf {x} ^{*}} by an... | https://en.wikipedia.org/wiki/Descent_direction |
Numerous methods exist to compute descent directions, all with differing merits, such as gradient descent or the conjugate gradient method. More generally, if P {\displaystyle P} is a positive definite matrix, then p k = − P ∇ f ( x k ) {\displaystyle p_{k}=-P\nabla f(x_{k})} is a descent direction at x k {\displaystyl... | https://en.wikipedia.org/wiki/Descent_direction |
In optimization, a gradient method is an algorithm to solve problems of the form min x ∈ R n f ( x ) {\displaystyle \min _{x\in \mathbb {R} ^{n}}\;f(x)} with the search directions defined by the gradient of the function at the current point. Examples of gradient methods are the gradient descent and the conjugate gradie... | https://en.wikipedia.org/wiki/Gradient_method |
In optimization, a self-concordant function is a function f: R → R {\displaystyle f:\mathbb {R} \rightarrow \mathbb {R} } for which | f ‴ ( x ) | ≤ 2 f ″ ( x ) 3 / 2 {\displaystyle |f'''(x)|\leq 2f''(x)^{3/2}} or, equivalently, a function f: R → R {\displaystyle f:\mathbb {R} \rightarrow \mathbb {R} } that, wherever f ... | https://en.wikipedia.org/wiki/Self-concordant_function |
In optimization, the line search strategy is one of two basic iterative approaches to find a local minimum x ∗ {\displaystyle \mathbf {x} ^{*}} of an objective function f: R n → R {\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} } . The other approach is trust region. The line search approach first finds a descent dire... | https://en.wikipedia.org/wiki/Line_search |
In option pricing, a jump-diffusion model is a form of mixture model, mixing a jump process and a diffusion process. Jump-diffusion models have been introduced by Robert C. Merton as an extension of jump models. Due to their computational tractability, the special case of a basic affine jump diffusion is popular for so... | https://en.wikipedia.org/wiki/Jump-diffusion_model |
In options trading, a bear spread is a bearish, vertical spread options strategy that can be used when the options trader is moderately bearish on the underlying security. Because of put–call parity, a bear spread can be constructed using either put options or call options. If constructed using calls, it is a bear call... | https://en.wikipedia.org/wiki/Bear_spread |
In options trading, a box spread is a combination of positions that has a certain (i.e., riskless) payoff, considered to be simply "delta neutral interest rate position". For example, a bull spread constructed from calls (e.g., long a 50 call, short a 60 call) combined with a bear spread constructed from puts (e.g., lo... | https://en.wikipedia.org/wiki/Box_spread_(options) |
An alternate name is "alligator spread," derived from the large number of trades required to open and close them "eating" one's profit via commission fees. Box spreads are usually only opened with European options, whose exercise is not allowed until the option's expiration. Most other styles of options, such as Americ... | https://en.wikipedia.org/wiki/Box_spread_(options) |
In options trading, a bull spread is a bullish, vertical spread options strategy that is designed to profit from a moderate rise in the price of the underlying security. Because of put–call parity, a bull spread can be constructed using either put options or call options. If constructed using calls, it is a bull call s... | https://en.wikipedia.org/wiki/Bull_spread |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.