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In mathematics, the direct method in the calculus of variations is a general method for constructing a proof of the existence of a minimizer for a given functional, [1] introduced by Stanisław Zaremba and David Hilbert around 1900. The method relies on methods of functional analysis and topology. As well as being used to prove the existence of ...
Micro Maniacs is set in a time where the Earth's resources are being depleted and the very planet is at risk. However, a scientist named Dr. Minimizer has an idea: using a device he calls "The Minimizer Ray", he will shrink the planet's population to 1/360 of its original size, and so create a world more suitable to our current status.
The term minimizer was coined by linguist Dwight Bolinger in his 1972 book Degree Words, where he described them as "partially stereotyped equivalents of any". [3] [4] The phenomenon had previously been remarked upon by other scholars as far back as August Friedrich Pott in 1859.
Minimizers converge to minimizers: If -converge to , and is a minimizer for , then every cluster point of the sequence is a minimizer of . Γ {\displaystyle \Gamma } -limits are always lower semicontinuous .
For any function in this class, the minimizer of the right-hand side above is unique, hence making the proximal operator well-defined. The proximal operator is used in proximal gradient methods, which is frequently used in optimization algorithms associated with non-differentiable optimization problems such as total variation denoising.
One can ask whether a minimizer point of the original, constrained optimization problem (assuming one exists) has to satisfy the above KKT conditions. This is similar to asking under what conditions the minimizer x ∗ {\displaystyle x^{*}} of a function f ( x ) {\displaystyle f(x)} in an unconstrained problem has to satisfy the condition ∇ f ...
The Moreau envelope has important applications in mathematical optimization: minimizing over and minimizing over are equivalent problems in the sense that the sets of minimizers of and are the same. However, first-order optimization algorithms can be directly applied to M f {\displaystyle M_{f}} , since f {\displaystyle f} may be non ...
For computer science, in statistical learning theory, a representer theorem is any of several related results stating that a minimizer of a regularized empirical risk functional defined over a reproducing kernel Hilbert space can be represented as a finite linear combination of kernel products evaluated on the input points in the training set data.