Convex conjugate information

mathematical optimization, the convex conjugate of a function is a generalization of the Legendre transformation which applies to non-convex functions. It is also...

The convex conjugate of an extended real-valued function f : X → [ − ∞ , ∞ ] {\displaystyle f:X\to [-\infty ,\infty ]} (not necessarily convex) is the...

In geometry, the convex hull, convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined...

which identifies equivalent dynamical systems Convex conjugate, the ("dual") lower-semicontinuous convex function resulting from the Legendre–Fenchel transformation...

nonnegative matrix is a convex function of its diagonal elements. Concave function Convex analysis Convex conjugate Convex curve Convex optimization Geodesic...

spaces Convex function, when the line segment between any two points on the graph of the function lies above or on the graph Convex conjugate, of a function...

which means the gradient of LogSumExp is the softmax function. The convex conjugate of LogSumExp is the negative entropy. The LSE function is often encountered...

is called the convex conjugate function of f {\displaystyle f} . For historical reasons (rooted in analytic mechanics), the conjugate variable is often...

is convex if, given any two points in the subset, the subset contains the whole line segment that joins them. Equivalently, a convex set or a convex region...

{\displaystyle f^{*}} denotes the convex conjugate of f {\displaystyle f} . Since the subdifferential of a proper, convex, lower semicontinuous function...

In mathematics, the conjugate gradient method is an algorithm for the numerical solution of particular systems of linear equations, namely those whose...

negative entropy function, in physics interpreted as free entropy) is the convex conjugate of LogSumExp (in physics interpreted as the free energy). In 1953,...

Convex optimization is a subfield of mathematical optimization that studies the problem of minimizing convex functions over convex sets (or, equivalently...

Duality: If F is strictly convex, then the function F has a convex conjugate F ∗ {\displaystyle F^{*}} which is also strictly convex and continuously differentiable...

example, conjugate diameters have the property that a tangent line to the ellipse at the endpoint of one diameter is parallel to the conjugate diameter...

_{t}at-K(t)} The moment generating function is log-convex, so by a property of the convex conjugate, the Chernoff bound must be log-concave. The Chernoff...

y^{*})\leq \inf _{x\in X}F(x,0),} where F ∗ {\displaystyle F^{*}} is the convex conjugate in both variables. For any choice of perturbation function F weak duality...

order, for the convex conjugate function. Fixing an exponential family with log-normalizer A {\displaystyle A} (with convex conjugate A ∗ {\displaystyle...

_{y^{*}\in Y^{*}}[-F^{*}(0,y^{*})]} where F ∗ {\displaystyle F^{*}} is the convex conjugate in both variables. In computational optimization, another "duality...

and concave impressions. Convex body - a compact convex set in a Euclidean space whose interior is non-empty. Convex conjugate - a dual of a real functional...

the indicator function for a cone C . {\displaystyle C.} Then the convex conjugate, f ∗ ( x ∗ ) = δ ( x ∗ | C ∘ ) = δ ∗ ( x ∗ | C ) = sup x ∈ C ⟨ x ∗...

commonly used, including interior point, active set, augmented Lagrangian, conjugate gradient, gradient projection, extensions of the simplex algorithm. In...