Naum Zuselevich Shor (russian: Наум Зуселевич Шор) (1 January 1937 – 26 February 2006) was a
Soviet
The Soviet Union,. officially the Union of Soviet Socialist Republics. (USSR),. was a List of former transcontinental countries#Since 1700, transcontinental country that spanned much of Eurasia from 1922 to 1991. A flagship communist state, ...
and
Ukrainian
Ukrainian may refer to:
* Something of, from, or related to Ukraine
* Something relating to Ukrainians, an East Slavic people from Eastern Europe
* Something relating to demographics of Ukraine in terms of demography and population of Ukraine
* So ...
mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems.
Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change.
History
On ...
specializing in
optimization
Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. It is generally divided into two subfi ...
.
He made significant contributions to
nonlinear
In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many oth ...
and
stochastic programming
In the field of mathematical optimization, stochastic programming is a framework for modeling optimization problems that involve uncertainty. A stochastic program is an optimization problem in which some or all problem parameters are uncertain, ...
, numerical techniques for
non-smooth optimization,
discrete optimization
Discrete optimization is a branch of optimization in applied mathematics and computer science.
Scope
As opposed to continuous optimization, some or all of the variables used in a discrete mathematical program are restricted to be discrete varia ...
problems,
matrix optimization,
dual quadratic bounds in
multi-extremal programming problems.
Shor became a full member of the
National Academy of Science of Ukraine
The National Academy of Sciences of Ukraine (NASU; uk, Національна академія наук України, ''Natsional’na akademiya nauk Ukrayiny'', abbr: NAN Ukraine) is a self-governing state-funded organization in Ukraine t ...
in 1998.
Subgradient methods
N. Z. Shor is well known for his
method
Method ( grc, μέθοδος, methodos) literally means a pursuit of knowledge, investigation, mode of prosecuting such inquiry, or system. In recent centuries it more often means a prescribed process for completing a task. It may refer to:
*Scien ...
of
generalized gradient descent
In mathematics, gradient descent (also often called steepest descent) is a first-order iterative optimization algorithm for finding a local minimum of a differentiable function. The idea is to take repeated steps in the opposite direction of the ...
with
space dilation in the direction of the difference of two successive
subgradient
In mathematics, the subderivative, subgradient, and subdifferential generalize the derivative to convex functions which are not necessarily differentiable. Subderivatives arise in convex analysis, the study of convex functions, often in connecti ...
s (the so-called r-algorithm), that was created in collaboration with Nikolay G. Zhurbenko. The
ellipsoid method
In mathematical optimization, the ellipsoid method is an iterative method for minimizing convex functions. When specialized to solving feasible linear optimization problems with rational data, the ellipsoid method is an algorithm which finds ...
was re-invigorated by A.S. Nemirovsky and D.B. Yudin, who developed a careful
complexity analysis of its
approximation
An approximation is anything that is intentionally similar but not exactly equal to something else.
Etymology and usage
The word ''approximation'' is derived from Latin ''approximatus'', from ''proximus'' meaning ''very near'' and the prefix ' ...
properties for problems of
convex minimization
Convex optimization is a subfield of mathematical optimization that studies the problem of minimizing convex functions over convex sets (or, equivalently, maximizing concave functions over convex sets). Many classes of convex optimization probl ...
with real data. However, it was
Leonid Khachiyan who provided the rational-arithmetic complexity analysis, using an
ellipsoid
An ellipsoid is a surface that may be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation.
An ellipsoid is a quadric surface; that is, a surface that may be defined as th ...
algorithm
In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing ...
, that established that
linear programming
Linear programming (LP), also called linear optimization, is a method to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements are represented by linear relationships. Linear programming is ...
problems can be solved in polynomial time.
It has long been known that the ellipsoidal methods are special cases of these subgradient-type methods.
R-algorithm
Shor's r-algorithm is for unconstrained minimization of (possibly) non-smooth functions, which has been somewhat popular despite an unknown
convergence rate.
["The Speed of Shor's R-Algorithm", available at http://www.optimization-online.org/DB_HTML/2007/05/1656.html] It can be viewed as a
Quasi-Newton method Quasi-Newton methods are methods used to either find zeroes or local maxima and minima of functions, as an alternative to Newton's method. They can be used if the Jacobian or Hessian is unavailable or is too expensive to compute at every iteration. ...
, although it does not satisfy the secant equation. Although the method involves
subgradients, it is distinct from his so-called ''subgradient method'' described above.
References
Notes
Bibliography
* .
External links
ORB Newsletter Issue 5contains an article with a short biography
*
{{DEFAULTSORT:Shor, Naum Z.
Numerical analysts
Theoretical computer scientists
Mathematical analysts
Ukrainian mathematicians
Soviet computer scientists
Taras Shevchenko National University of Kyiv alumni
Academic staff of the Moscow Institute of Physics and Technology
Members of the National Academy of Sciences of Ukraine
Recipients of the USSR State Prize
1937 births
2006 deaths
Ukrainian Jews
Laureates of the State Prize of Ukraine in Science and Technology