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A barrier function is also called an interior penalty function, as it is a penalty function that forces the solution to remain within the interior of the feasible region. The two most common types of barrier functions are inverse barrier functions and logarithmic barrier functions.
An interior point method was discovered by Soviet mathematician I. I. Dikin in 1967. [1] The method was reinvented in the U.S. in the mid-1980s. In 1984, Narendra Karmarkar developed a method for linear programming called Karmarkar's algorithm, [2] which runs in provably polynomial time (() operations on L-bit numbers, where n is the number of variables and constants), and is also very ...
In the above equations, (()) is the exterior penalty function while is the penalty coefficient. When the penalty coefficient is 0, f p = f . In each iteration of the method, we increase the penalty coefficient p {\displaystyle p} (e.g. by a factor of 10), solve the unconstrained problem and use the solution as the initial guess for the next ...
A self-concordant function is a function satisfying a certain differential inequality, which makes it particularly easy for optimization using Newton's method [1]: Sub.6.2.4.2 A self-concordant barrier is a particular self-concordant function, that is also a barrier function for a particular convex set.
Crowd control barriers (also referred to as crowd control barricades, with some versions called a French barrier or bike rack in the USA, and mills barriers in Hong Kong [1]) are commonly used at many public events. They are frequently visible at sporting events, parades, political rallies, demonstrations, and outdoor festivals.
A barrier certificate [1] or barrier function is used to prove that a given region is forward invariant for a given ordinary differential equation or hybrid dynamical system. [2] That is, a barrier function can be used to show that if a solution starts in a given set , then it cannot leave that set.
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It is often difficult to find a control-Lyapunov function for a given system, but if one is found, then the feedback stabilization problem simplifies considerably. For the control affine system ( 2 ), Sontag's formula (or Sontag's universal formula ) gives the feedback law k : R n → R m {\displaystyle k:\mathbb {R} ^{n}\to \mathbb {R} ^{m ...