An approximation algorithm efficiently returns a feasible solution with a proved performance guarantee relative to the optimum. The guarantee must specify whether the problem is a maximization or minimization problem.

An outcome is Pareto efficient when no feasible alternative weakly improves every agent's utility and strictly improves at least one. For strict preference orders over distinct alternatives, an outcome unanimously ranked below another is inefficient. Onto strategyproof social choice functions on an unrestricted domain are Pareto efficient: rank-raising monotonicity and unanimity rule out selecting a unanimously dominated alternative.

Variational analysis studies optimization, perturbation stability and generalized differentiation through functions, sets and set-valued mappings. Convex analysis is a foundational special case; limiting normal constructions also handle nonconvex geometry.

A parametric optimization problem has objective or constraints depending on input data. Its optimizer set is a solution map of a parametric optimization problem. Sensitivity analysis asks how this map changes as the data vary.

A minimization problem seeks a feasible point with smallest objective value. Its infimum can be finite without being attained, or equal to $-\infty$ when unbounded below. Negating the objective converts it to a maximization problem.

A maximization problem seeks a feasible point with largest objective value. Its supremum can fail to be attained or be infinite. Negating the objective gives a minimization problem. A global extremum of an optimization Lagrangian can certify a feasible optimum by the Lagrangian sufficiency theorem.

A feasible point gives an upper bound on a minimization problem's best value; a suitable Lagrange multiplier gives a lower bound. Lagrangian duality searches for the strongest bound obtainable this way. Matching bounds certify a global minimum.

Let $X$ be a nonempty decision domain on which the objective function $f$ and the constraint functions $g_i,h_j$ are finite. Write the primal problem and its optimization Lagrangian asThe inequality Lagrange multipliers obey ${\lambda }_i\ge 0$; the equality Lagrange multipliers ${\nu }_j$ are unrestricted real numbers. Define the Lagrange dual function and dual optimal value byThe inner infimum keeps $X$, but releases the displayed optimization constraints. It can be $-\infty$, giving an uninformative lower bound. A different domain changes the bound.^[ref]

Here is the entire weak duality argument. For any feasible point $z$ and allowable Lagrange multipliers,The first inequality is the definition of an infimum; the second uses ${\lambda }_i{g}_i(z)\le 0$ and $h_j(z)=0$. Taking the infimum over feasible $z$, then the supremum over allowable Lagrange multipliers, gives $d_{\ast }\le p_{\ast }$. No convexity is used.

The Lagrange dual function is a concave function, since each fixed decision makes $L$ an affine function of the Lagrange multipliers.^[ref] Explicitly, write $y=(\lambda ,\nu )$ and choose two points where $q$ is finite. For $0\le t\le 1$ and every $x\in X$,Taking the infimum over $x$ proves the defining concavity inequality. Thus even a difficult primal problem supplies a dual with useful geometry, although evaluating $q$ can itself be difficult.

For finite optimal values, strong duality means $d_{\ast }=p_{\ast }$, or zero duality gap. It need not imply dual attainment. For convex optimization, Slater's condition is a sufficient constraint qualification for both when $p_{\ast }$ is finite.^[ref] The Strong Lagrangian property requires an actual finite Lagrange multiplier achieving equality. Convex perturbation duality interprets such a bound as a supporting slope of the perturbed value function.

At an interior local extremum of a differentiable function, its gradient is zero: restricting the function to every line through the point shows that all directional derivatives vanish. This is only a necessary condition in general. For a strictly concave function, an interior stationary point is the unique global maximizer on its convex domain. Constrained or boundary optima instead need conditions adapted to their feasible directions.

The density of individuals at age $a$ evolves by transport with unit ageing speed and loss rate $\mu (a)$. New individuals enter at age zero through the integral birth boundary. Exponential separated profiles lead to the Euler-Lotka equation.

In collisional ionization, an energetic collision frees a bound electron. The process consumes kinetic energy and changes the electron and ion populations controlling plasma cooling. In a thermal gas its rate competes with recombination and depends strongly on the ionization threshold relative to temperature.

A free electron can be captured by an ion while a photon carries away binding and kinetic energy. This radiative recombination process contributes free-bound radiation to plasma cooling and balances collisional ionization in a collisional-equilibrium plasma.

Photoionization removes a bound electron when an atom or ion absorbs a sufficiently energetic photon. The threshold is its ionization energy. The excess photon energy becomes kinetic energy of the products; subsequent collisions can heat the gas or cause secondary ionizations. A photoionization rate and a photoionization equilibrium are distinct concepts: equilibrium also requires specified recombination and other reaction rates.

The periodic Hamiltonian chain $H=\frac{1}{2}\sum p_i^2+\sum e^{q_i-q_{i+1}}$, with cyclic indices, is a Liouville integrable system. Exponentials of neighbouring coordinate differences lead to a Lax pair and conserved characteristic coefficients.

An auxiliary function used in Hirota's bilinear method to encode a solution of an integrable partial differential equation. Products and ratios of tau functions replace nonlinear field variables; their differential equations become bilinear. In a Sine-Gordon multisoliton tau representation, two real tau functions encode the field as a continuous $4\arg (f+ig)$.

An exact-solution method that transforms a nonlinear partial differential equation into equations bilinear in auxiliary Hirota tau functions. Finite exponential expansions of the Hirota tau functions produce soliton families. The auxiliary exponential coefficients need not all be positive.

For a finite-dimensional Hamiltonian system, integrability is normally expressed by enough independent first integrals in mutual Poisson bracket involution. In classical field theories, an infinite hierarchy of compatible conserved quantities is the corresponding structure. For Sine-Gordon theory, Bäcklund transformations generate local conservation laws and explicit soliton solutions. Conservation of arbitrary quantities alone should not be confused with proof of their Hamiltonian involution.

A parameter-dependent system of differential relations maps a solution of one differential equation to a solution of another, or to a new solution of the same equation. Compatibility of the relations supplies the equations of motion. Integration constants select a particular transformed field. The Sine-Gordon Bäcklund transformation is an example of an auto-transformation and also generates conserved currents.

Image noise is unwanted variation in an observed image signal. In an additive model $g=u+\eta$, Gaussian noise motivates a squared data-fidelity penalty, while other statistics require different penalties. Image smoothing suppresses rapid fluctuations, but a filter must distinguish noise from genuine image edges.

image edge enhancement increases the apparent sharpness or contrast of an image edge. Unsharp masking gives bounded linear high-frequency amplification; backward normal diffusion in the Perona-Malik equation gives formal nonlinear sharpening with ill-posedness risks. Positive diffusion tensors chiefly preserve or connect structure rather than perform unbounded backward diffusion.

The Bohr model assumes circular electron orbits with quantized orbital angular momentum $N\hslash$. It reproduces the hydrogen energy levels, but wavefunctions in quantum mechanics describe spatial probability distributions rather than definite trajectories. In particular, spherically symmetric Coulomb bound states have zero orbital angular momentum.

The puzzle is the coexistence of reproducible spectral lines and stable chemical atoms. A classical circular orbit in a Coulomb potential can have any negative energy: Coulomb's law alone selects no special radius or emitted frequency. Moreover, its electron has centripetal acceleration. The Larmor formula then predicts positive radiated power, so conservation of energy makes the orbit shrink. Simply drawing concentric circles explains neither stability nor a discrete spectrum.

Niels Bohr postulated nonradiating stationary states and emission with $h\nu =E_i-E_f$. His circular-orbit condition is $L=n\hslash$. These assumptions override classical radiation; the emitted frequency need not equal an orbital frequency.^[ref]