HOME

TheInfoList



OR:

In
computational engineering Computational science and engineering (CSE) is a relatively new discipline that deals with the development and application of computational models and simulations, often coupled with high-performance computing, to solve complex physical problems ...
, Luus–Jaakola (LJ) denotes a heuristic for global optimization of a real-valued function. In engineering use, LJ is not an algorithm that terminates with an optimal solution; nor is it an iterative method that generates a sequence of points that converges to an optimal solution (when one exists). However, when applied to a twice continuously differentiable function, the LJ heuristic is a proper iterative method, that generates a sequence that has a convergent subsequence; for this class of problems,
Newton's method In numerical analysis, Newton's method, also known as the Newton–Raphson method, named after Isaac Newton and Joseph Raphson, is a root-finding algorithm which produces successively better approximations to the roots (or zeroes) of a real-valu ...
is recommended and enjoys a quadratic rate of convergence, while no convergence rate analysis has been given for the LJ heuristic. In practice, the LJ heuristic has been recommended for functions that need be neither convex nor differentiable nor
locally Lipschitz In mathematical analysis, Lipschitz continuity, named after German mathematician Rudolf Lipschitz, is a strong form of uniform continuity for functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: there exis ...
: The LJ heuristic does not use a gradient or
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 connectio ...
when one be available, which allows its application to non-differentiable and non-convex problems. Proposed by Luus and Jaakola, LJ generates a sequence of iterates. The next iterate is selected from a sample from a neighborhood of the current position using a
uniform distribution Uniform distribution may refer to: * Continuous uniform distribution * Discrete uniform distribution * Uniform distribution (ecology) * Equidistributed sequence In mathematics, a sequence (''s''1, ''s''2, ''s''3, ...) of real numbers is said to be ...
. With each iteration, the neighborhood decreases, which forces a subsequence of iterates to converge to a cluster point. Luus has applied LJ in
optimal control Optimal control theory is a branch of mathematical optimization that deals with finding a control for a dynamical system over a period of time such that an objective function is optimized. It has numerous applications in science, engineering and ...
, transformer design, metallurgical processes, and chemical engineering.


Motivation

At each step, the LJ heuristic maintains a box from which it samples points randomly, using a uniform distribution on the box. For a unimodal function, the probability of reducing the objective function decreases as the box approach a minimum. The picture displays a one-dimensional example.


Heuristic

Let ''f'': ℝ''n'' → ℝ be the fitness or cost function which must be minimized. Let x ∈ ℝ''n'' designate a position or candidate solution in the search-space. The LJ heuristic iterates the following steps: * Initialize x ~ ''U''(blo,bup) with a random uniform position in the search-space, where blo and bup are the lower and upper boundaries, respectively. * Set the initial sampling range to cover the entire search-space (or a part of it): d = bup − blo * Until a termination criterion is met (e.g. number of iterations performed, or adequate fitness reached), repeat the following: ** Pick a random vector a ~ ''U''(−d, d) ** Add this to the current position x to create the new potential position y = x + a ** If (''f''(y) < ''f''(x)) then move to the new position by setting x = y, otherwise decrease the sampling-range: d = ''0.95'' d * Now x holds the best-found position.


Variations

Luus notes that ARS (Adaptive Random Search) algorithms proposed to date differ in regard to many aspects. * Procedure of generating random trial points. * Number of internal loops (NIL, the number of random search points in each cycle). * Number of cycles (NEL, number of external loops). * Contraction coefficient of the search region size. (Some example values are 0.95 to 0.60.) ** Whether the region reduction rate is the same for all variables or a different rate for each variable (called the M-LJ algorithm). ** Whether the region reduction rate is a constant or follows another distribution (e.g. Gaussian). * Whether to incorporate a line search. * Whether to consider constraints of the random points as acceptance criteria, or to incorporate a quadratic penalty.


Convergence

Nair proved a convergence analysis. For twice continuously differentiable functions, the LJ heuristic generates a sequence of iterates having a convergent subsequence. For this class of problems, Newton's method is the usual optimization method, and it has quadratic convergence (''regardless of the dimension'' of the space, which can be a
Banach space In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
, according to Kantorovich's analysis). The worst-case complexity of minimization on the class of unimodal functions grows exponentially in the dimension of the problem, according to the analysis of Yudin and Nemirovsky, however. The Yudin-Nemirovsky analysis implies that no method can be fast on high-dimensional problems that lack convexity:
"The catastrophic growth
n the number of iterations needed to reach an approximate solution of a given accuracy N, or n, is the fourteenth letter in the Latin alphabet, used in the modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is ''en'' (pronounced ), plural ''ens''. History ...
as
he number of dimensions increases to infinity He or HE may refer to: Language * He (pronoun), an English pronoun * He (kana), the romanization of the Japanese kana へ * He (letter), the fifth letter of many Semitic alphabets * He (Cyrillic), a letter of the Cyrillic script called ''He'' ...
shows that it is meaningless to pose the question of constructing universal methods of solving ... problems of any appreciable dimensionality 'generally'. It is interesting to note that the same onclusionholds for ... problems generated by uni-extremal
hat is, unimodal A hat is a head covering which is worn for various reasons, including protection against weather conditions, ceremonial reasons such as university graduation, religious reasons, safety, or as a fashion accessory. Hats which incorporate mech ...
(but not convex) functions." Page 7 summarizes the later discussion of .
When applied to twice continuously differentiable problems, the LJ heuristic's rate of convergence decreases as the number of dimensions increases.


See also

* Random optimization is a related family of optimization methods that sample from general distributions, for example the uniform distribution. * Random search is a related family of optimization methods that sample from general distributions, for example, a uniform distribution on the unit sphere. * Pattern search are used on noisy observations, especially in
response surface methodology In statistics, response surface methodology (RSM) explores the relationships between several explanatory variables and one or more response variables. The method was introduced by George E. P. Box and K. B. Wilson in 1951. The main idea of RSM ...
in chemical engineering. They do not require users to program gradients or hessians.


References

{{DEFAULTSORT:Luus-Jaakola Optimization algorithms and methods Heuristic algorithms